1 / 55
Herbrand's Theorem via Hyper anoni al Extensions
Kazushige Terui
RIMS, Kyoto University
24/09/13, Gudauri
Outline
Introdu tion to
Substru tural Logi s
Algebrai Proof
Theory for
Substru tural Logi s
Herbrand's theorem
via hyper anoni al
extensions
Further topi s
2 / 55
1. Introdu tion to substru tural logi s
2. Algebrai proof theory for substru tural logi s
3. Herbrand's theorem via hyper anoni al extensions
Introdu tion to Substru tural Logi s
.
Introdu tion to
Substru tural
Logi s
What are SLs?
Full Lambek al ulus
Residuated Latti es
Complexity
Algebrai Proof
Theory for
Substru tural Logi s
Herbrand's theorem
via hyper anoni al
extensions
Further topi s
3 / 55
What are substru tural logi s?
Introdu tion to
Substru tural Logi s
. What are SLs?
Full Lambek al ulus
Residuated Latti es
Complexity
Algebrai Proof
Theory for
Substru tural Logi s
Herbrand's theorem
via hyper anoni al
extensions
Further topi s
4 / 55
� Logi s that may la k some of stru tural rules
(ex hange/weakening/ ontra tion)
� Axiomati extensions of Full Lambek Cal ulus FL
(= non ommutative intuitionisti linear logi without !)
� Study of universe of logi s
Why is the subje t interesting?
� Common basis for various non lassi al logi s
linear, BI, relevant, fuzzy, superintuitionisti logi s
� Common basis for various ordered algebras
latti e-ordered groups, relation algebras, ideal latti es of
rings, MV algebras, Heyting algebras
� Abundan e of weird logi s/algebras
- allows us to speak of riteria for various properties
What are substru tural logi s?
Introdu tion to
Substru tural Logi s
. What are SLs?
Full Lambek al ulus
Residuated Latti es
Complexity
Algebrai Proof
Theory for
Substru tural Logi s
Herbrand's theorem
via hyper anoni al
extensions
Further topi s
4 / 55
� Logi s that may la k some of stru tural rules
(ex hange/weakening/ ontra tion)
� Axiomati extensions of Full Lambek Cal ulus FL
(= non ommutative intuitionisti linear logi without !)
� Study of universe of logi s
Why is the subje t interesting?
� Common basis for various non lassi al logi s
linear, BI, relevant, fuzzy, superintuitionisti logi s
� Common basis for various ordered algebras
latti e-ordered groups, relation algebras, ideal latti es of
rings, MV algebras, Heyting algebras
� Abundan e of weird logi s/algebras
- allows us to speak of riteria for various properties
What are substru tural logi s?
Introdu tion to
Substru tural Logi s
. What are SLs?
Full Lambek al ulus
Residuated Latti es
Complexity
Algebrai Proof
Theory for
Substru tural Logi s
Herbrand's theorem
via hyper anoni al
extensions
Further topi s
4 / 55
� Logi s that may la k some of stru tural rules
(ex hange/weakening/ ontra tion)
� Axiomati extensions of Full Lambek Cal ulus FL
(= non ommutative intuitionisti linear logi without !)
� Study of universe of logi s
Why is the subje t interesting?
� Common basis for various non lassi al logi s
linear, BI, relevant, fuzzy, superintuitionisti logi s
� Common basis for various ordered algebras
latti e-ordered groups, relation algebras, ideal latti es of
rings, MV algebras, Heyting algebras
� Abundan e of weird logi s/algebras
- allows us to speak of riteria for various properties
What are substru tural logi s?
Introdu tion to
Substru tural Logi s
. What are SLs?
Full Lambek al ulus
Residuated Latti es
Complexity
Algebrai Proof
Theory for
Substru tural Logi s
Herbrand's theorem
via hyper anoni al
extensions
Further topi s
4 / 55
� Logi s that may la k some of stru tural rules
(ex hange/weakening/ ontra tion)
� Axiomati extensions of Full Lambek Cal ulus FL
(= non ommutative intuitionisti linear logi without !)
� Study of universe of logi s
Why is the subje t interesting?
� Common basis for various non lassi al logi s
linear, BI, relevant, fuzzy, superintuitionisti logi s
� Common basis for various ordered algebras
latti e-ordered groups, relation algebras, ideal latti es of
rings, MV algebras, Heyting algebras
� Abundan e of weird logi s/algebras
- allows us to speak of riteria for various properties
What are substru tural logi s?
Introdu tion to
Substru tural Logi s
. What are SLs?
Full Lambek al ulus
Residuated Latti es
Complexity
Algebrai Proof
Theory for
Substru tural Logi s
Herbrand's theorem
via hyper anoni al
extensions
Further topi s
4 / 55
� Logi s that may la k some of stru tural rules
(ex hange/weakening/ ontra tion)
� Axiomati extensions of Full Lambek Cal ulus FL
(= non ommutative intuitionisti linear logi without !)
� Study of universe of logi s
Why is the subje t interesting?
� Common basis for various non lassi al logi s
linear, BI, relevant, fuzzy, superintuitionisti logi s
� Common basis for various ordered algebras
latti e-ordered groups, relation algebras, ideal latti es of
rings, MV algebras, Heyting algebras
� Abundan e of weird logi s/algebras
- allows us to speak of riteria for various properties
What are substru tural logi s?
Introdu tion to
Substru tural Logi s
. What are SLs?
Full Lambek al ulus
Residuated Latti es
Complexity
Algebrai Proof
Theory for
Substru tural Logi s
Herbrand's theorem
via hyper anoni al
extensions
Further topi s
4 / 55
� Logi s that may la k some of stru tural rules
(ex hange/weakening/ ontra tion)
� Axiomati extensions of Full Lambek Cal ulus FL
(= non ommutative intuitionisti linear logi without !)
� Study of universe of logi s
Why is the subje t interesting?
� Common basis for various non lassi al logi s
linear, BI, relevant, fuzzy, superintuitionisti logi s
� Common basis for various ordered algebras
latti e-ordered groups, relation algebras, ideal latti es of
rings, MV algebras, Heyting algebras
� Abundan e of weird logi s/algebras
- allows us to speak of riteria for various properties
Full Lambek al ulus FL
Introdu tion to
Substru tural Logi s
What are SLs?
.
Full Lambek
al ulus
Residuated Latti es
Complexity
Algebrai Proof
Theory for
Substru tural Logi s
Herbrand's theorem
via hyper anoni al
extensions
Further topi s
5 / 55
� The base system for substru tural logi s (Ono 90)
� Intuitionisti logi without stru tural rules.
� Formulas:
'; ::= p j ' ^ j ' _ j ' � j 'n j '= j 1 j 0
� 0 is used to de�ne negations:
�a = an0; � a = 0=a:
� Sequents: �) �
(�: sequen e of formulas, �: at most one formula)
Inferen e Rules of FL
Introdu tion to
Substru tural Logi s
What are SLs?
.
Full Lambek
al ulus
Residuated Latti es
Complexity
Algebrai Proof
Theory for
Substru tural Logi s
Herbrand's theorem
via hyper anoni al
extensions
Further topi s
6 / 55
�) A �
1
; A;�
2
) �
�
1
;�;�
2
) �
(Cut)
A) A
(Id)
�
1
; A;�
2
) � �
1
; B;�
2
) �
�
1
; A _B;�
2
) �
(_l)
�) A
i
�) A
1
_ A
2
(_r)
�
1
; A
i
;�
2
) �
�
1
; A
1
^ A
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;�
2
) �
(^l)
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�) A ^B
(^r)
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2
) �
�
1
; A � B;�
2
) �
(�l)
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�;�) A �B
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1
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1
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(nr)
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1
; B;�
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) �
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1
; B=A;�;�
2
) �
(=l)
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�) B=A
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�
1
; 1;�
2
) �
(1l)
) 1
(1r)
0)
(0l)
�)
�) 0
(0r)
Substru tural Logi s
Introdu tion to
Substru tural Logi s
What are SLs?
.
Full Lambek
al ulus
Residuated Latti es
Complexity
Algebrai Proof
Theory for
Substru tural Logi s
Herbrand's theorem
via hyper anoni al
extensions
Further topi s
7 / 55
� Given a set � [ fsg of sequents, � `
FL
s if s is derivable
from �.
� We often identify formula ' with sequent ) '.
� We often write '! for 'n and ='.
� A substru tural logi is an axiomati extension of FL.
Some axioms:
(e) ' � ! � ' (ex hange)
(w) '! 1; 0! ' (weakening)
( ) '! ' � ' ( ontra tion)
(in) ::'! ' (involutivity)
(dist) ' ^ (
1
_
2
)! (' ^
1
) _ (' _
2
) (distributivity)
(pl) ('! ) _ ( ! ') (prelinearity)
(div) ' ^ ! ' � ('! ) (divisibility)
Substru tural Logi s
Introdu tion to
Substru tural Logi s
What are SLs?
.
Full Lambek
al ulus
Residuated Latti es
Complexity
Algebrai Proof
Theory for
Substru tural Logi s
Herbrand's theorem
via hyper anoni al
extensions
Further topi s
7 / 55
� Given a set � [ fsg of sequents, � `
FL
s if s is derivable
from �.
� We often identify formula ' with sequent ) '.
� We often write '! for 'n and ='.
� A substru tural logi is an axiomati extension of FL.
Some axioms:
(e) ' � ! � ' (ex hange)
(w) '! 1; 0! ' (weakening)
( ) '! ' � ' ( ontra tion)
(in) ::'! ' (involutivity)
(dist) ' ^ (
1
_
2
)! (' ^
1
) _ (' _
2
) (distributivity)
(pl) ('! ) _ ( ! ') (prelinearity)
(div) ' ^ ! ' � ('! ) (divisibility)
Substru tural Logi s
Introdu tion to
Substru tural Logi s
What are SLs?
.
Full Lambek
al ulus
Residuated Latti es
Complexity
Algebrai Proof
Theory for
Substru tural Logi s
Herbrand's theorem
via hyper anoni al
extensions
Further topi s
7 / 55
� Given a set � [ fsg of sequents, � `
FL
s if s is derivable
from �.
� We often identify formula ' with sequent ) '.
� We often write '! for 'n and ='.
� A substru tural logi is an axiomati extension of FL.
Some axioms:
(e) ' � ! � ' (ex hange)
(w) '! 1; 0! ' (weakening)
( ) '! ' � ' ( ontra tion)
(in) ::'! ' (involutivity)
(dist) ' ^ (
1
_
2
)! (' ^
1
) _ (' _
2
) (distributivity)
(pl) ('! ) _ ( ! ') (prelinearity)
(div) ' ^ ! ' � ('! ) (divisibility)
Substru tural Logi s
Introdu tion to
Substru tural Logi s
What are SLs?
.
Full Lambek
al ulus
Residuated Latti es
Complexity
Algebrai Proof
Theory for
Substru tural Logi s
Herbrand's theorem
via hyper anoni al
extensions
Further topi s
7 / 55
� Given a set � [ fsg of sequents, � `
FL
s if s is derivable
from �.
� We often identify formula ' with sequent ) '.
� We often write '! for 'n and ='.
� A substru tural logi is an axiomati extension of FL.
Some axioms:
(e) ' � ! � ' (ex hange)
(w) '! 1; 0! ' (weakening)
( ) '! ' � ' ( ontra tion)
(in) ::'! ' (involutivity)
(dist) ' ^ (
1
_
2
)! (' ^
1
) _ (' _
2
) (distributivity)
(pl) ('! ) _ ( ! ') (prelinearity)
(div) ' ^ ! ' � ('! ) (divisibility)
Residuated Latti es
Introdu tion to
Substru tural Logi s
What are SLs?
Full Lambek al ulus
.
Residuated
Latti es
Complexity
Algebrai Proof
Theory for
Substru tural Logi s
Herbrand's theorem
via hyper anoni al
extensions
Further topi s
9 / 55
Residuated latti e: A = hA;^;_; �; n; =; 1i su h that
� hA;^;_i is a latti e;
� hA; �; 1i is a monoid;
� a � b � , b � an , a � =b.
An FL algebra is a residuated latti e with onstant 0 2 A.
Some identities:
� a � (anb) � b
� (a _ b) � = (a � ) _ (b � )
� a! (b ^ ) = (a! b) ^ (a! )
� (a _ b)! = (a! ) ^ (b! )
� anb = b=a (with (e))
� a � b � a ^ b (with (w))
� a � b � a ^ b (with ( ))
Residuated Latti es
Introdu tion to
Substru tural Logi s
What are SLs?
Full Lambek al ulus
.
Residuated
Latti es
Complexity
Algebrai Proof
Theory for
Substru tural Logi s
Herbrand's theorem
via hyper anoni al
extensions
Further topi s
9 / 55
Residuated latti e: A = hA;^;_; �; n; =; 1i su h that
� hA;^;_i is a latti e;
� hA; �; 1i is a monoid;
� a � b � , b � an , a � =b.
An FL algebra is a residuated latti e with onstant 0 2 A.
Some identities:
� a � (anb) � b
� (a _ b) � = (a � ) _ (b � )
� a! (b ^ ) = (a! b) ^ (a! )
� (a _ b)! = (a! ) ^ (b! )
� anb = b=a (with (e))
� a � b � a ^ b (with (w))
� a � b � a ^ b (with ( ))
Residuated Latti es
Introdu tion to
Substru tural Logi s
What are SLs?
Full Lambek al ulus
.
Residuated
Latti es
Complexity
Algebrai Proof
Theory for
Substru tural Logi s
Herbrand's theorem
via hyper anoni al
extensions
Further topi s
9 / 55
Residuated latti e: A = hA;^;_; �; n; =; 1i su h that
� hA;^;_i is a latti e;
� hA; �; 1i is a monoid;
� a � b � , b � an , a � =b.
An FL algebra is a residuated latti e with onstant 0 2 A.
Some identities:
� a � (anb) � b
� (a _ b) � = (a � ) _ (b � )
� a! (b ^ ) = (a! b) ^ (a! )
� (a _ b)! = (a! ) ^ (b! )
� anb = b=a (with (e))
� a � b � a ^ b (with (w))
� a � b � a ^ b (with ( ))
Algebraization
Introdu tion to
Substru tural Logi s
What are SLs?
Full Lambek al ulus
.
Residuated
Latti es
Complexity
Algebrai Proof
Theory for
Substru tural Logi s
Herbrand's theorem
via hyper anoni al
extensions
Further topi s
10 / 55
A lass V of algebras (of the same type) is a variety if
V = HSP (V):
� H: homomorphi images
� S: subalgebras
� P : dire t produ ts
Birkho�'s Theorem
V is a variety i� V is equationally de�nable.
FL := the variety of FL algebras.
Algebraization Theorem
The substru tural logi s are in 1-1 orresponden e with the sub-
varieties of FL. If L orresponds to V,
� `
L
i� 1 � � j=
V
1 � :
Algebraization
Introdu tion to
Substru tural Logi s
What are SLs?
Full Lambek al ulus
.
Residuated
Latti es
Complexity
Algebrai Proof
Theory for
Substru tural Logi s
Herbrand's theorem
via hyper anoni al
extensions
Further topi s
10 / 55
A lass V of algebras (of the same type) is a variety if
V = HSP (V):
� H: homomorphi images
� S: subalgebras
� P : dire t produ ts
Birkho�'s Theorem
V is a variety i� V is equationally de�nable.
FL := the variety of FL algebras.
Algebraization Theorem
The substru tural logi s are in 1-1 orresponden e with the sub-
varieties of FL. If L orresponds to V,
� `
L
i� 1 � � j=
V
1 � :
Algebraization
Introdu tion to
Substru tural Logi s
What are SLs?
Full Lambek al ulus
.
Residuated
Latti es
Complexity
Algebrai Proof
Theory for
Substru tural Logi s
Herbrand's theorem
via hyper anoni al
extensions
Further topi s
10 / 55
A lass V of algebras (of the same type) is a variety if
V = HSP (V):
� H: homomorphi images
� S: subalgebras
� P : dire t produ ts
Birkho�'s Theorem
V is a variety i� V is equationally de�nable.
FL := the variety of FL algebras.
Algebraization Theorem
The substru tural logi s are in 1-1 orresponden e with the sub-
varieties of FL. If L orresponds to V,
� `
L
i� 1 � � j=
V
1 � :
Example of resear h topi s: omputational omplexity
Introdu tion to
Substru tural Logi s
What are SLs?
Full Lambek al ulus
Residuated Latti es
. Complexity
Algebrai Proof
Theory for
Substru tural Logi s
Herbrand's theorem
via hyper anoni al
extensions
Further topi s
12 / 55
Let L be a onsistent substru tural logi and onsider the
de ision problem:
Given '; `
L
'?
Theorem (Hor ik-T. 11)
1. Any L is oNP-hard.
2. If L is �nite-valued, then L is oNP- omplete.
3. If L enjoys the disjun tion property, then L is PSPACE-hard.
(Neither 2. nor 3. is a ne essary ondition.)
oNP and PSPACE seem a natural way to lassify logi s into
\semanti ally easy" and \ omputationally expressive" ones.
Di hotomy Problem
Is there a substru tural logi whi h is neither oNP- omplete nor
PSPACE-hard?
Example of resear h topi s: omputational omplexity
Introdu tion to
Substru tural Logi s
What are SLs?
Full Lambek al ulus
Residuated Latti es
. Complexity
Algebrai Proof
Theory for
Substru tural Logi s
Herbrand's theorem
via hyper anoni al
extensions
Further topi s
12 / 55
Let L be a onsistent substru tural logi and onsider the
de ision problem:
Given '; `
L
'?
Theorem (Hor ik-T. 11)
1. Any L is oNP-hard.
2. If L is �nite-valued, then L is oNP- omplete.
3. If L enjoys the disjun tion property, then L is PSPACE-hard.
(Neither 2. nor 3. is a ne essary ondition.)
oNP and PSPACE seem a natural way to lassify logi s into
\semanti ally easy" and \ omputationally expressive" ones.
Di hotomy Problem
Is there a substru tural logi whi h is neither oNP- omplete nor
PSPACE-hard?
Example of resear h topi s: omputational omplexity
Introdu tion to
Substru tural Logi s
What are SLs?
Full Lambek al ulus
Residuated Latti es
. Complexity
Algebrai Proof
Theory for
Substru tural Logi s
Herbrand's theorem
via hyper anoni al
extensions
Further topi s
12 / 55
Let L be a onsistent substru tural logi and onsider the
de ision problem:
Given '; `
L
'?
Theorem (Hor ik-T. 11)
1. Any L is oNP-hard.
2. If L is �nite-valued, then L is oNP- omplete.
3. If L enjoys the disjun tion property, then L is PSPACE-hard.
(Neither 2. nor 3. is a ne essary ondition.)
oNP and PSPACE seem a natural way to lassify logi s into
\semanti ally easy" and \ omputationally expressive" ones.
Di hotomy Problem
Is there a substru tural logi whi h is neither oNP- omplete nor
PSPACE-hard?
Example of resear h topi s: omputational omplexity
Introdu tion to
Substru tural Logi s
What are SLs?
Full Lambek al ulus
Residuated Latti es
. Complexity
Algebrai Proof
Theory for
Substru tural Logi s
Herbrand's theorem
via hyper anoni al
extensions
Further topi s
12 / 55
Let L be a onsistent substru tural logi and onsider the
de ision problem:
Given '; `
L
'?
Theorem (Hor ik-T. 11)
1. Any L is oNP-hard.
2. If L is �nite-valued, then L is oNP- omplete.
3. If L enjoys the disjun tion property, then L is PSPACE-hard.
(Neither 2. nor 3. is a ne essary ondition.)
oNP and PSPACE seem a natural way to lassify logi s into
\semanti ally easy" and \ omputationally expressive" ones.
Di hotomy Problem
Is there a substru tural logi whi h is neither oNP- omplete nor
PSPACE-hard?
Example of resear h topi s: omputational omplexity
Introdu tion to
Substru tural Logi s
What are SLs?
Full Lambek al ulus
Residuated Latti es
. Complexity
Algebrai Proof
Theory for
Substru tural Logi s
Herbrand's theorem
via hyper anoni al
extensions
Further topi s
12 / 55
Let L be a onsistent substru tural logi and onsider the
de ision problem:
Given '; `
L
'?
Theorem (Hor ik-T. 11)
1. Any L is oNP-hard.
2. If L is �nite-valued, then L is oNP- omplete.
3. If L enjoys the disjun tion property, then L is PSPACE-hard.
(Neither 2. nor 3. is a ne essary ondition.)
oNP and PSPACE seem a natural way to lassify logi s into
\semanti ally easy" and \ omputationally expressive" ones.
Di hotomy Problem
Is there a substru tural logi whi h is neither oNP- omplete nor
PSPACE-hard?
Example of resear h topi s: omputational omplexity
Introdu tion to
Substru tural Logi s
What are SLs?
Full Lambek al ulus
Residuated Latti es
. Complexity
Algebrai Proof
Theory for
Substru tural Logi s
Herbrand's theorem
via hyper anoni al
extensions
Further topi s
12 / 55
Let L be a onsistent substru tural logi and onsider the
de ision problem:
Given '; `
L
'?
Theorem (Hor ik-T. 11)
1. Any L is oNP-hard.
2. If L is �nite-valued, then L is oNP- omplete.
3. If L enjoys the disjun tion property, then L is PSPACE-hard.
(Neither 2. nor 3. is a ne essary ondition.)
oNP and PSPACE seem a natural way to lassify logi s into
\semanti ally easy" and \ omputationally expressive" ones.
Di hotomy Problem
Is there a substru tural logi whi h is neither oNP- omplete nor
PSPACE-hard?
Algebrai Proof Theory for Substru tural
Logi s
Introdu tion to
Substru tural Logi s
.
Algebrai Proof
Theory for
Substru tural
Logi s
APT for SL
Completions
Ma Neille omp.
Residuated frames
From axioms to rules
Subst. hierar hy
Limitation
Hypersequent al .
Hyper-Ma Neille
Class N
3
Summary
Herbrand's theorem
via hyper anoni al
extensions
Further topi s
13 / 55
Substru tural proof theory is dying ...
Introdu tion to
Substru tural Logi s
Algebrai Proof
Theory for
Substru tural Logi s
APT for SL
Completions
Ma Neille omp.
Residuated frames
From axioms to rules
Subst. hierar hy
Limitation
Hypersequent al .
Hyper-Ma Neille
Class N
3
Summary
Herbrand's theorem
via hyper anoni al
extensions
Further topi s
14 / 55
� Full of bureau ra y and ad ho studies,
� Few of appli ations,
� Nevertheless there are some brilliant ideas.
Our aim: to salvage those brilliant ideas from the sea of
bureau ra y.
Substru tural proof theory is dying ...
Introdu tion to
Substru tural Logi s
Algebrai Proof
Theory for
Substru tural Logi s
APT for SL
Completions
Ma Neille omp.
Residuated frames
From axioms to rules
Subst. hierar hy
Limitation
Hypersequent al .
Hyper-Ma Neille
Class N
3
Summary
Herbrand's theorem
via hyper anoni al
extensions
Further topi s
14 / 55
� Full of bureau ra y and ad ho studies,
� Few of appli ations,
� Nevertheless there are some brilliant ideas.
Our aim: to salvage those brilliant ideas from the sea of
bureau ra y.
Substru tural proof theory is dying ...
Introdu tion to
Substru tural Logi s
Algebrai Proof
Theory for
Substru tural Logi s
APT for SL
Completions
Ma Neille omp.
Residuated frames
From axioms to rules
Subst. hierar hy
Limitation
Hypersequent al .
Hyper-Ma Neille
Class N
3
Summary
Herbrand's theorem
via hyper anoni al
extensions
Further topi s
14 / 55
� Full of bureau ra y and ad ho studies,
� Few of appli ations,
� Nevertheless there are some brilliant ideas.
Our aim: to salvage those brilliant ideas from the sea of
bureau ra y.
Substru tural proof theory is dying ...
Introdu tion to
Substru tural Logi s
Algebrai Proof
Theory for
Substru tural Logi s
APT for SL
Completions
Ma Neille omp.
Residuated frames
From axioms to rules
Subst. hierar hy
Limitation
Hypersequent al .
Hyper-Ma Neille
Class N
3
Summary
Herbrand's theorem
via hyper anoni al
extensions
Further topi s
14 / 55
� Full of bureau ra y and ad ho studies,
� Few of appli ations,
� Nevertheless there are some brilliant ideas.
Our aim: to salvage those brilliant ideas from the sea of
bureau ra y.
Algebrai proof theory for SL
Introdu tion to
Substru tural Logi s
Algebrai Proof
Theory for
Substru tural Logi s
. APT for SL
Completions
Ma Neille omp.
Residuated frames
From axioms to rules
Subst. hierar hy
Limitation
Hypersequent al .
Hyper-Ma Neille
Class N
3
Summary
Herbrand's theorem
via hyper anoni al
extensions
Further topi s
15 / 55
� Collaborators: Paolo Baldi, Agata Ciabattoni,
Nikolaos Galatos, Rostislav Hor� ��k, Lutz Stra�burger, . . .
� Explore the onne tion between proof theory and ordered
algebra.
{ Uniform proof theory
{ Appli ations to ordered algebra
� Core: ut elimination � algebrai ompletion
� Origin:
{ omputability/redu ibility argument (Tait/Girard)
{ phase semanti ut elimination (Okada 96)
{ algebrai meaning of ut elimination
(Belardinelli-Jipsen-Ono 04)
{ residuated frames (Jipsen-Galatos 13)
� Today I will fo us on ut elimination � algebrai ompletion.
Algebrai proof theory for SL
Introdu tion to
Substru tural Logi s
Algebrai Proof
Theory for
Substru tural Logi s
. APT for SL
Completions
Ma Neille omp.
Residuated frames
From axioms to rules
Subst. hierar hy
Limitation
Hypersequent al .
Hyper-Ma Neille
Class N
3
Summary
Herbrand's theorem
via hyper anoni al
extensions
Further topi s
15 / 55
� Collaborators: Paolo Baldi, Agata Ciabattoni,
Nikolaos Galatos, Rostislav Hor� ��k, Lutz Stra�burger, . . .
� Explore the onne tion between proof theory and ordered
algebra.
{ Uniform proof theory
{ Appli ations to ordered algebra
� Core: ut elimination � algebrai ompletion
� Origin:
{ omputability/redu ibility argument (Tait/Girard)
{ phase semanti ut elimination (Okada 96)
{ algebrai meaning of ut elimination
(Belardinelli-Jipsen-Ono 04)
{ residuated frames (Jipsen-Galatos 13)
� Today I will fo us on ut elimination � algebrai ompletion.
Algebrai proof theory for SL
Introdu tion to
Substru tural Logi s
Algebrai Proof
Theory for
Substru tural Logi s
. APT for SL
Completions
Ma Neille omp.
Residuated frames
From axioms to rules
Subst. hierar hy
Limitation
Hypersequent al .
Hyper-Ma Neille
Class N
3
Summary
Herbrand's theorem
via hyper anoni al
extensions
Further topi s
15 / 55
� Collaborators: Paolo Baldi, Agata Ciabattoni,
Nikolaos Galatos, Rostislav Hor� ��k, Lutz Stra�burger, . . .
� Explore the onne tion between proof theory and ordered
algebra.
{ Uniform proof theory
{ Appli ations to ordered algebra
� Core: ut elimination � algebrai ompletion
� Origin:
{ omputability/redu ibility argument (Tait/Girard)
{ phase semanti ut elimination (Okada 96)
{ algebrai meaning of ut elimination
(Belardinelli-Jipsen-Ono 04)
{ residuated frames (Jipsen-Galatos 13)
� Today I will fo us on ut elimination � algebrai ompletion.
Algebrai proof theory for SL
Introdu tion to
Substru tural Logi s
Algebrai Proof
Theory for
Substru tural Logi s
. APT for SL
Completions
Ma Neille omp.
Residuated frames
From axioms to rules
Subst. hierar hy
Limitation
Hypersequent al .
Hyper-Ma Neille
Class N
3
Summary
Herbrand's theorem
via hyper anoni al
extensions
Further topi s
15 / 55
� Collaborators: Paolo Baldi, Agata Ciabattoni,
Nikolaos Galatos, Rostislav Hor� ��k, Lutz Stra�burger, . . .
� Explore the onne tion between proof theory and ordered
algebra.
{ Uniform proof theory
{ Appli ations to ordered algebra
� Core: ut elimination � algebrai ompletion
� Origin:
{ omputability/redu ibility argument (Tait/Girard)
{ phase semanti ut elimination (Okada 96)
{ algebrai meaning of ut elimination
(Belardinelli-Jipsen-Ono 04)
{ residuated frames (Jipsen-Galatos 13)
� Today I will fo us on ut elimination � algebrai ompletion.
Algebrai proof theory for SL
Introdu tion to
Substru tural Logi s
Algebrai Proof
Theory for
Substru tural Logi s
. APT for SL
Completions
Ma Neille omp.
Residuated frames
From axioms to rules
Subst. hierar hy
Limitation
Hypersequent al .
Hyper-Ma Neille
Class N
3
Summary
Herbrand's theorem
via hyper anoni al
extensions
Further topi s
15 / 55
� Collaborators: Paolo Baldi, Agata Ciabattoni,
Nikolaos Galatos, Rostislav Hor� ��k, Lutz Stra�burger, . . .
� Explore the onne tion between proof theory and ordered
algebra.
{ Uniform proof theory
{ Appli ations to ordered algebra
� Core: ut elimination � algebrai ompletion
� Origin:
{ omputability/redu ibility argument (Tait/Girard)
{ phase semanti ut elimination (Okada 96)
{ algebrai meaning of ut elimination
(Belardinelli-Jipsen-Ono 04)
{ residuated frames (Jipsen-Galatos 13)
� Today I will fo us on ut elimination � algebrai ompletion.
Algebrai proof theory for SL
Introdu tion to
Substru tural Logi s
Algebrai Proof
Theory for
Substru tural Logi s
. APT for SL
Completions
Ma Neille omp.
Residuated frames
From axioms to rules
Subst. hierar hy
Limitation
Hypersequent al .
Hyper-Ma Neille
Class N
3
Summary
Herbrand's theorem
via hyper anoni al
extensions
Further topi s
15 / 55
� Collaborators: Paolo Baldi, Agata Ciabattoni,
Nikolaos Galatos, Rostislav Hor� ��k, Lutz Stra�burger, . . .
� Explore the onne tion between proof theory and ordered
algebra.
{ Uniform proof theory
{ Appli ations to ordered algebra
� Core: ut elimination � algebrai ompletion
� Origin:
{ omputability/redu ibility argument (Tait/Girard)
{ phase semanti ut elimination (Okada 96)
{ algebrai meaning of ut elimination
(Belardinelli-Jipsen-Ono 04)
{ residuated frames (Jipsen-Galatos 13)
� Today I will fo us on ut elimination � algebrai ompletion.
Algebrai proof theory for SL
Introdu tion to
Substru tural Logi s
Algebrai Proof
Theory for
Substru tural Logi s
. APT for SL
Completions
Ma Neille omp.
Residuated frames
From axioms to rules
Subst. hierar hy
Limitation
Hypersequent al .
Hyper-Ma Neille
Class N
3
Summary
Herbrand's theorem
via hyper anoni al
extensions
Further topi s
15 / 55
� Collaborators: Paolo Baldi, Agata Ciabattoni,
Nikolaos Galatos, Rostislav Hor� ��k, Lutz Stra�burger, . . .
� Explore the onne tion between proof theory and ordered
algebra.
{ Uniform proof theory
{ Appli ations to ordered algebra
� Core: ut elimination � algebrai ompletion
� Origin:
{ omputability/redu ibility argument (Tait/Girard)
{ phase semanti ut elimination (Okada 96)
{ algebrai meaning of ut elimination
(Belardinelli-Jipsen-Ono 04)
{ residuated frames (Jipsen-Galatos 13)
� Today I will fo us on ut elimination � algebrai ompletion.
Algebrai proof theory for SL
Introdu tion to
Substru tural Logi s
Algebrai Proof
Theory for
Substru tural Logi s
. APT for SL
Completions
Ma Neille omp.
Residuated frames
From axioms to rules
Subst. hierar hy
Limitation
Hypersequent al .
Hyper-Ma Neille
Class N
3
Summary
Herbrand's theorem
via hyper anoni al
extensions
Further topi s
15 / 55
� Collaborators: Paolo Baldi, Agata Ciabattoni,
Nikolaos Galatos, Rostislav Hor� ��k, Lutz Stra�burger, . . .
� Explore the onne tion between proof theory and ordered
algebra.
{ Uniform proof theory
{ Appli ations to ordered algebra
� Core: ut elimination � algebrai ompletion
� Origin:
{ omputability/redu ibility argument (Tait/Girard)
{ phase semanti ut elimination (Okada 96)
{ algebrai meaning of ut elimination
(Belardinelli-Jipsen-Ono 04)
{ residuated frames (Jipsen-Galatos 13)
� Today I will fo us on ut elimination � algebrai ompletion.
Completions
Introdu tion to
Substru tural Logi s
Algebrai Proof
Theory for
Substru tural Logi s
APT for SL
. Completions
Ma Neille omp.
Residuated frames
From axioms to rules
Subst. hierar hy
Limitation
Hypersequent al .
Hyper-Ma Neille
Class N
3
Summary
Herbrand's theorem
via hyper anoni al
extensions
Further topi s
16 / 55
Let A be an FL algebra.
A ompletion of A is a pair of a omplete FL algebra B and an
embedding e : A ,! B.
We may assume A � B.
We onsider 4 types of ompletion:
� Ma Neille ompletions
(Dedekind, Ma Neille, S hmidt, Banas hewski . . . )
� Canoni al extensions
(Tarski, J�onson, Gehrke, Harding . . . )
� Hyper-Ma Neille ompletions
� Hyper anoni al extensions
Completions
Introdu tion to
Substru tural Logi s
Algebrai Proof
Theory for
Substru tural Logi s
APT for SL
. Completions
Ma Neille omp.
Residuated frames
From axioms to rules
Subst. hierar hy
Limitation
Hypersequent al .
Hyper-Ma Neille
Class N
3
Summary
Herbrand's theorem
via hyper anoni al
extensions
Further topi s
16 / 55
Let A be an FL algebra.
A ompletion of A is a pair of a omplete FL algebra B and an
embedding e : A ,! B.
We may assume A � B.
We onsider 4 types of ompletion:
� Ma Neille ompletions
(Dedekind, Ma Neille, S hmidt, Banas hewski . . . )
� Canoni al extensions
(Tarski, J�onson, Gehrke, Harding . . . )
� Hyper-Ma Neille ompletions
� Hyper anoni al extensions
Ma Neille ompletions
Introdu tion to
Substru tural Logi s
Algebrai Proof
Theory for
Substru tural Logi s
APT for SL
Completions
. Ma Neille omp.
Residuated frames
From axioms to rules
Subst. hierar hy
Limitation
Hypersequent al .
Hyper-Ma Neille
Class N
3
Summary
Herbrand's theorem
via hyper anoni al
extensions
Further topi s
17 / 55
Dedekind ompletion: ([0; 1℄
Q
;min;max) ,! ([0; 1℄
R
;min;max).
What is the distin tive feature of this ompletion?
For every x 2 [0; 1℄
R
,
x = supfa 2 [0; 1℄
Q
: a � xg = inffa 2 [0; 1℄
Q
: a � xg:
Let A be a latti e. Its ompletion B is
� join-dense if for every x 2 B, x =
W
fa 2 A : a � xg.
� meet-dense if for every x 2 B, x =
V
fa 2 A : a � xg.
Theorem (S hmidt 56, Banas hewski 56)
Every latti e A has a join-dense and meet-dense ompletion A,
unique up to isomorphism, alled the Ma Neille ompletion.
It an be extended to FL algebras too.
Ma Neille ompletions
Introdu tion to
Substru tural Logi s
Algebrai Proof
Theory for
Substru tural Logi s
APT for SL
Completions
. Ma Neille omp.
Residuated frames
From axioms to rules
Subst. hierar hy
Limitation
Hypersequent al .
Hyper-Ma Neille
Class N
3
Summary
Herbrand's theorem
via hyper anoni al
extensions
Further topi s
17 / 55
Dedekind ompletion: ([0; 1℄
Q
;min;max) ,! ([0; 1℄
R
;min;max).
What is the distin tive feature of this ompletion?
For every x 2 [0; 1℄
R
,
x = supfa 2 [0; 1℄
Q
: a � xg = inffa 2 [0; 1℄
Q
: a � xg:
Let A be a latti e. Its ompletion B is
� join-dense if for every x 2 B, x =
W
fa 2 A : a � xg.
� meet-dense if for every x 2 B, x =
V
fa 2 A : a � xg.
Theorem (S hmidt 56, Banas hewski 56)
Every latti e A has a join-dense and meet-dense ompletion A,
unique up to isomorphism, alled the Ma Neille ompletion.
It an be extended to FL algebras too.
Ma Neille ompletions
Introdu tion to
Substru tural Logi s
Algebrai Proof
Theory for
Substru tural Logi s
APT for SL
Completions
. Ma Neille omp.
Residuated frames
From axioms to rules
Subst. hierar hy
Limitation
Hypersequent al .
Hyper-Ma Neille
Class N
3
Summary
Herbrand's theorem
via hyper anoni al
extensions
Further topi s
17 / 55
Dedekind ompletion: ([0; 1℄
Q
;min;max) ,! ([0; 1℄
R
;min;max).
What is the distin tive feature of this ompletion?
For every x 2 [0; 1℄
R
,
x = supfa 2 [0; 1℄
Q
: a � xg = inffa 2 [0; 1℄
Q
: a � xg:
Let A be a latti e. Its ompletion B is
� join-dense if for every x 2 B, x =
W
fa 2 A : a � xg.
� meet-dense if for every x 2 B, x =
V
fa 2 A : a � xg.
Theorem (S hmidt 56, Banas hewski 56)
Every latti e A has a join-dense and meet-dense ompletion A,
unique up to isomorphism, alled the Ma Neille ompletion.
It an be extended to FL algebras too.
Ma Neille ompletions
Introdu tion to
Substru tural Logi s
Algebrai Proof
Theory for
Substru tural Logi s
APT for SL
Completions
. Ma Neille omp.
Residuated frames
From axioms to rules
Subst. hierar hy
Limitation
Hypersequent al .
Hyper-Ma Neille
Class N
3
Summary
Herbrand's theorem
via hyper anoni al
extensions
Further topi s
17 / 55
Dedekind ompletion: ([0; 1℄
Q
;min;max) ,! ([0; 1℄
R
;min;max).
What is the distin tive feature of this ompletion?
For every x 2 [0; 1℄
R
,
x = supfa 2 [0; 1℄
Q
: a � xg = inffa 2 [0; 1℄
Q
: a � xg:
Let A be a latti e. Its ompletion B is
� join-dense if for every x 2 B, x =
W
fa 2 A : a � xg.
� meet-dense if for every x 2 B, x =
V
fa 2 A : a � xg.
Theorem (S hmidt 56, Banas hewski 56)
Every latti e A has a join-dense and meet-dense ompletion A,
unique up to isomorphism, alled the Ma Neille ompletion.
It an be extended to FL algebras too.
Residuated frames
Introdu tion to
Substru tural Logi s
Algebrai Proof
Theory for
Substru tural Logi s
APT for SL
Completions
Ma Neille omp.
.
Residuated
frames
From axioms to rules
Subst. hierar hy
Limitation
Hypersequent al .
Hyper-Ma Neille
Class N
3
Summary
Herbrand's theorem
via hyper anoni al
extensions
Further topi s
18 / 55
A preframe is W = (W;W
0
; N; Æ; "; �) su h that
� N �W �W
0
,
� (W; Æ; ") is a monoid, � 2W
0
.
A residuated frame is a preframe where for every x 2W; z 2W
0
there are elements x z and z�x 2W
0
su h that
x Æ y N z () x N z�y () y N x z:
Lemma
If W is a preframe, then
~
W := (W; W�W
0
�W;
~
N; Æ; "; ("; �; "))
x
~
N (u; z; v) () uÆxÆv N z
is a residuated frame.
Residuated frames
Introdu tion to
Substru tural Logi s
Algebrai Proof
Theory for
Substru tural Logi s
APT for SL
Completions
Ma Neille omp.
.
Residuated
frames
From axioms to rules
Subst. hierar hy
Limitation
Hypersequent al .
Hyper-Ma Neille
Class N
3
Summary
Herbrand's theorem
via hyper anoni al
extensions
Further topi s
18 / 55
A preframe is W = (W;W
0
; N; Æ; "; �) su h that
� N �W �W
0
,
� (W; Æ; ") is a monoid, � 2W
0
.
A residuated frame is a preframe where for every x 2W; z 2W
0
there are elements x z and z�x 2W
0
su h that
x Æ y N z () x N z�y () y N x z:
Lemma
If W is a preframe, then
~
W := (W; W�W
0
�W;
~
N; Æ; "; ("; �; "))
x
~
N (u; z; v) () uÆxÆv N z
is a residuated frame.
Residuated frames
Introdu tion to
Substru tural Logi s
Algebrai Proof
Theory for
Substru tural Logi s
APT for SL
Completions
Ma Neille omp.
.
Residuated
frames
From axioms to rules
Subst. hierar hy
Limitation
Hypersequent al .
Hyper-Ma Neille
Class N
3
Summary
Herbrand's theorem
via hyper anoni al
extensions
Further topi s
18 / 55
A preframe is W = (W;W
0
; N; Æ; "; �) su h that
� N �W �W
0
,
� (W; Æ; ") is a monoid, � 2W
0
.
A residuated frame is a preframe where for every x 2W; z 2W
0
there are elements x z and z�x 2W
0
su h that
x Æ y N z () x N z�y () y N x z:
Lemma
If W is a preframe, then
~
W := (W; W�W
0
�W;
~
N; Æ; "; ("; �; "))
x
~
N (u; z; v) () uÆxÆv N z
is a residuated frame.
Residuated frames
Introdu tion to
Substru tural Logi s
Algebrai Proof
Theory for
Substru tural Logi s
APT for SL
Completions
Ma Neille omp.
.
Residuated
frames
From axioms to rules
Subst. hierar hy
Limitation
Hypersequent al .
Hyper-Ma Neille
Class N
3
Summary
Herbrand's theorem
via hyper anoni al
extensions
Further topi s
19 / 55
Residuated frames are e�e tive tools to build omplete FL
algeblras.
Given X �W and Z �W
0
,
X
B
:= fz 2W
0
: x N z for every x 2 Xg
Z
C
:= fx 2W : x N z for every z 2 Zg
(
B
;
C
) forms a Galois onne tion:
X � Z
C
() X
B
� Z
that indu es a losure operator (X) := X
BC
on }(W ).
Residuated frames
Introdu tion to
Substru tural Logi s
Algebrai Proof
Theory for
Substru tural Logi s
APT for SL
Completions
Ma Neille omp.
.
Residuated
frames
From axioms to rules
Subst. hierar hy
Limitation
Hypersequent al .
Hyper-Ma Neille
Class N
3
Summary
Herbrand's theorem
via hyper anoni al
extensions
Further topi s
19 / 55
Residuated frames are e�e tive tools to build omplete FL
algeblras.
Given X �W and Z �W
0
,
X
B
:= fz 2W
0
: x N z for every x 2 Xg
Z
C
:= fx 2W : x N z for every z 2 Zg
(
B
;
C
) forms a Galois onne tion:
X � Z
C
() X
B
� Z
that indu es a losure operator (X) := X
BC
on }(W ).
Residuated frames
Introdu tion to
Substru tural Logi s
Algebrai Proof
Theory for
Substru tural Logi s
APT for SL
Completions
Ma Neille omp.
.
Residuated
frames
From axioms to rules
Subst. hierar hy
Limitation
Hypersequent al .
Hyper-Ma Neille
Class N
3
Summary
Herbrand's theorem
via hyper anoni al
extensions
Further topi s
20 / 55
Given W = (W;W
0
; N; Æ; "; �) and X;Y �W ,
G(W ) := the set of Galois- losed subsets of W
(X = (X) = X
BC
)
XnY := fy : x Æ y 2 Y for every x 2 Xg
Y=X := fy : y Æ x 2 Y for every x 2 Xg
X Æ
Y := (X Æ Y )
X [
Y := (X [ Y )
Lemma
W
+
:= (G(W );\;[
; Æ
; n; =; ("); �
C
)
is a omplete FL algebra, alled the omplex algebra of W.
Appli ations of residuated frames
Introdu tion to
Substru tural Logi s
Algebrai Proof
Theory for
Substru tural Logi s
APT for SL
Completions
Ma Neille omp.
.
Residuated
frames
From axioms to rules
Subst. hierar hy
Limitation
Hypersequent al .
Hyper-Ma Neille
Class N
3
Summary
Herbrand's theorem
via hyper anoni al
extensions
Further topi s
21 / 55
Fm := the set of formulas.
Fm
�
:= the set of formula sequen es.
� Let W
f
:= (Fm
�
; Fm [ f;g; N
f
; Æ; ;; ;) where
� N
f
� i� �) � is ut-free derivable.
Then W
+
f
is an FL algebra su h that
j=
W
+
f
1 � ' implies ) ' is ut-free derivable.
) Algebrai ut elimination.
� Given an FL algebra A, let W
A
:= (A;A;�
A
; �; 1; 0).
Then W
+
A
is the Ma Neille ompletion of A.
Appli ations of residuated frames
Introdu tion to
Substru tural Logi s
Algebrai Proof
Theory for
Substru tural Logi s
APT for SL
Completions
Ma Neille omp.
.
Residuated
frames
From axioms to rules
Subst. hierar hy
Limitation
Hypersequent al .
Hyper-Ma Neille
Class N
3
Summary
Herbrand's theorem
via hyper anoni al
extensions
Further topi s
21 / 55
Fm := the set of formulas.
Fm
�
:= the set of formula sequen es.
� Let W
f
:= (Fm
�
; Fm [ f;g; N
f
; Æ; ;; ;) where
� N
f
� i� �) � is ut-free derivable.
Then W
+
f
is an FL algebra su h that
j=
W
+
f
1 � ' implies ) ' is ut-free derivable.
) Algebrai ut elimination.
� Given an FL algebra A, let W
A
:= (A;A;�
A
; �; 1; 0).
Then W
+
A
is the Ma Neille ompletion of A.
Appli ations of residuated frames
Introdu tion to
Substru tural Logi s
Algebrai Proof
Theory for
Substru tural Logi s
APT for SL
Completions
Ma Neille omp.
.
Residuated
frames
From axioms to rules
Subst. hierar hy
Limitation
Hypersequent al .
Hyper-Ma Neille
Class N
3
Summary
Herbrand's theorem
via hyper anoni al
extensions
Further topi s
21 / 55
Fm := the set of formulas.
Fm
�
:= the set of formula sequen es.
� Let W
f
:= (Fm
�
; Fm [ f;g; N
f
; Æ; ;; ;) where
� N
f
� i� �) � is ut-free derivable.
Then W
+
f
is an FL algebra su h that
j=
W
+
f
1 � ' implies ) ' is ut-free derivable.
) Algebrai ut elimination.
� Given an FL algebra A, let W
A
:= (A;A;�
A
; �; 1; 0).
Then W
+
A
is the Ma Neille ompletion of A.
From axioms to rules
Introdu tion to
Substru tural Logi s
Algebrai Proof
Theory for
Substru tural Logi s
APT for SL
Completions
Ma Neille omp.
Residuated frames
.
From axioms to
rules
Subst. hierar hy
Limitation
Hypersequent al .
Hyper-Ma Neille
Class N
3
Summary
Herbrand's theorem
via hyper anoni al
extensions
Further topi s
22 / 55
Let L be a SL (axiomati extension of FL). To obtain an analyti
al ulus for L, axioms have to be transformed into stru tural rules:
'! ' � ' ' � '! ' :(' ^ :')
�;�;�;�) �
�;�;�) �
�;�;�) � �;�;�) �
�;�;�;�) �
�;�)
�)
Let V be a subvariety of FL. To show that V is losed under
ompletions, identities have to be transformed into quasi-identities:
x � xx xx � x x ^ :x � 0
xx � z
x � z
x � z y � z
xy � z
xx � 0
x � 0
Fundamental Question
Whi h axioms/identities an be transformed into \good" stru tural
rules/quasi-identities?
From axioms to rules
Introdu tion to
Substru tural Logi s
Algebrai Proof
Theory for
Substru tural Logi s
APT for SL
Completions
Ma Neille omp.
Residuated frames
.
From axioms to
rules
Subst. hierar hy
Limitation
Hypersequent al .
Hyper-Ma Neille
Class N
3
Summary
Herbrand's theorem
via hyper anoni al
extensions
Further topi s
22 / 55
Let L be a SL (axiomati extension of FL). To obtain an analyti
al ulus for L, axioms have to be transformed into stru tural rules:
'! ' � ' ' � '! ' :(' ^ :')
�;�;�;�) �
�;�;�) �
�;�;�) � �;�;�) �
�;�;�;�) �
�;�)
�)
Let V be a subvariety of FL. To show that V is losed under
ompletions, identities have to be transformed into quasi-identities:
x � xx xx � x x ^ :x � 0
xx � z
x � z
x � z y � z
xy � z
xx � 0
x � 0
Fundamental Question
Whi h axioms/identities an be transformed into \good" stru tural
rules/quasi-identities?
From axioms to rules
Introdu tion to
Substru tural Logi s
Algebrai Proof
Theory for
Substru tural Logi s
APT for SL
Completions
Ma Neille omp.
Residuated frames
.
From axioms to
rules
Subst. hierar hy
Limitation
Hypersequent al .
Hyper-Ma Neille
Class N
3
Summary
Herbrand's theorem
via hyper anoni al
extensions
Further topi s
22 / 55
Let L be a SL (axiomati extension of FL). To obtain an analyti
al ulus for L, axioms have to be transformed into stru tural rules:
'! ' � ' ' � '! ' :(' ^ :')
�;�;�;�) �
�;�;�) �
�;�;�) � �;�;�) �
�;�;�;�) �
�;�)
�)
Let V be a subvariety of FL. To show that V is losed under
ompletions, identities have to be transformed into quasi-identities:
x � xx xx � x x ^ :x � 0
xx � z
x � z
x � z y � z
xy � z
xx � 0
x � 0
Fundamental Question
Whi h axioms/identities an be transformed into \good" stru tural
rules/quasi-identities?
Substru tural hierar hy
Introdu tion to
Substru tural Logi s
Algebrai Proof
Theory for
Substru tural Logi s
APT for SL
Completions
Ma Neille omp.
Residuated frames
From axioms to rules
. Subst. hierar hy
Limitation
Hypersequent al .
Hyper-Ma Neille
Class N
3
Summary
Herbrand's theorem
via hyper anoni al
extensions
Further topi s
23 / 55
P
3
N
3
P
2
N
2
P
1
N
1
P
0
N
0
pp
pp
pp
pp
p6
pp
pp
pp
pp
p6
6
�
�
�
�� 6
�
�
�
�I
6
�
�
�
�� 6
�
�
�
�I
6
�
�
�
�� 6
�
�
�
�I
Classi� ation of axioms
P
0
;N
0
::= the set of variables
P
n
::= N
n�1
j 1 j P
n
_ P
n
j P
n
� P
n
N
n
::= P
n�1
j 0 j N
n
^N
n
j P
n
! N
n
Some N
2
axioms:
�! 1, 0! � weakening
�! � � � ontra tion
� � �! � expansion
�
n
! �
m
knotted axioms (n;m � 0)
:(� ^ :�) no- ontradi tion
N
2
orresponds to sequent al ulus and Ma Neille ompletions
Introdu tion to
Substru tural Logi s
Algebrai Proof
Theory for
Substru tural Logi s
APT for SL
Completions
Ma Neille omp.
Residuated frames
From axioms to rules
. Subst. hierar hy
Limitation
Hypersequent al .
Hyper-Ma Neille
Class N
3
Summary
Herbrand's theorem
via hyper anoni al
extensions
Further topi s
24 / 55
P
3
N
3
P
2
N
2
P
1
N
1
P
0
N
0
pp
pp
pp
pp
p6
pp
pp
pp
pp
p6
6
�
�
�
�� 6
�
�
�
�I
6
�
�
�
�� 6
�
�
�
�I
6
�
�
�
�� 6
�
�
�
�I
Theorem (Ciabattoni-Galatos-T. 12)
1. Every N
2
axiom an be transformed into a
set of stru tural rules in sequent al ulus FL.
2. For every set E of N
2
axioms, the following
are equivalent.
� FL(E) admits a strongly analyti sequent
al ulus ( ut elimination for derivations with
assumptions + subformula property).
� FL(E) is losed under Ma Neille omple-
tions.
� E is a y li (a synta ti riterion).
3. The above three hold whenever (w) 2 E.
N
2
orresponds to sequent al ulus and Ma Neille ompletions
Introdu tion to
Substru tural Logi s
Algebrai Proof
Theory for
Substru tural Logi s
APT for SL
Completions
Ma Neille omp.
Residuated frames
From axioms to rules
. Subst. hierar hy
Limitation
Hypersequent al .
Hyper-Ma Neille
Class N
3
Summary
Herbrand's theorem
via hyper anoni al
extensions
Further topi s
24 / 55
P
3
N
3
P
2
N
2
P
1
N
1
P
0
N
0
pp
pp
pp
pp
p6
pp
pp
pp
pp
p6
6
�
�
�
�� 6
�
�
�
�I
6
�
�
�
�� 6
�
�
�
�I
6
�
�
�
�� 6
�
�
�
�I
Theorem (Ciabattoni-Galatos-T. 12)
1. Every N
2
axiom an be transformed into a
set of stru tural rules in sequent al ulus FL.
2. For every set E of N
2
axioms, the following
are equivalent.
� FL(E) admits a strongly analyti sequent
al ulus ( ut elimination for derivations with
assumptions + subformula property).
� FL(E) is losed under Ma Neille omple-
tions.
� E is a y li (a synta ti riterion).
3. The above three hold whenever (w) 2 E.
N
2
orresponds to sequent al ulus and Ma Neille ompletions
Introdu tion to
Substru tural Logi s
Algebrai Proof
Theory for
Substru tural Logi s
APT for SL
Completions
Ma Neille omp.
Residuated frames
From axioms to rules
. Subst. hierar hy
Limitation
Hypersequent al .
Hyper-Ma Neille
Class N
3
Summary
Herbrand's theorem
via hyper anoni al
extensions
Further topi s
24 / 55
P
3
N
3
P
2
N
2
P
1
N
1
P
0
N
0
pp
pp
pp
pp
p6
pp
pp
pp
pp
p6
6
�
�
�
�� 6
�
�
�
�I
6
�
�
�
�� 6
�
�
�
�I
6
�
�
�
�� 6
�
�
�
�I
Theorem (Ciabattoni-Galatos-T. 12)
1. Every N
2
axiom an be transformed into a
set of stru tural rules in sequent al ulus FL.
2. For every set E of N
2
axioms, the following
are equivalent.
� FL(E) admits a strongly analyti sequent
al ulus ( ut elimination for derivations with
assumptions + subformula property).
� FL(E) is losed under Ma Neille omple-
tions.
� E is a y li (a synta ti riterion).
3. The above three hold whenever (w) 2 E.
Let's limb up the hierar hy
Introdu tion to
Substru tural Logi s
Algebrai Proof
Theory for
Substru tural Logi s
APT for SL
Completions
Ma Neille omp.
Residuated frames
From axioms to rules
. Subst. hierar hy
Limitation
Hypersequent al .
Hyper-Ma Neille
Class N
3
Summary
Herbrand's theorem
via hyper anoni al
extensions
Further topi s
25 / 55
P
3
N
3
P
2
N
2
P
1
N
1
P
0
N
0
pp
pp
pp
pp
p6
pp
pp
pp
pp
p6
6
�
�
�
�� 6
�
�
�
�I
6
�
�
�
�� 6
�
�
�
�I
6
�
�
�
�� 6
�
�
�
�I
Classi� ation of axioms
P
0
;N
0
::= the set of variables
P
n
::= N
n�1
j 1 j P
n
_ P
n
j P
n
� P
n
N
n
::= P
n�1
j 0 j N
n
^N
n
j P
n
! N
n
Some P
3
axioms:
(�! �) _ (� ! �) prelinearity
� _ :� ex luded middle
:� _ ::� weak ex luded middle
:(� � �) _ (� ^ � ! � � �) weak nilpotent minimum
W
ki=0
(�
i
!
W
j 6=i
�
j
) bounded width � k
W
ki=0
(�
0
^ � � � ^ �
i�1
! �
i
) bounded size � k
Limitation of sequent al ulus/Ma Neille ompletions
Introdu tion to
Substru tural Logi s
Algebrai Proof
Theory for
Substru tural Logi s
APT for SL
Completions
Ma Neille omp.
Residuated frames
From axioms to rules
Subst. hierar hy
. Limitation
Hypersequent al .
Hyper-Ma Neille
Class N
3
Summary
Herbrand's theorem
via hyper anoni al
extensions
Further topi s
26 / 55
Fa t
Every stru tural rule in sequent al ulus is either derivable or on-
tradi tory in Int.
�) � �) �
�;�) �
�) �
) �
Theorem (G.Bezhanishvili-Harding 04)
There is no intermediate variety between HA and BA that is losed
under Ma Neille ompletions.
Eg. prelinearity annot be dealt with by sequent
al ulus/Ma Neille ompletions.
Limitation of sequent al ulus/Ma Neille ompletions
Introdu tion to
Substru tural Logi s
Algebrai Proof
Theory for
Substru tural Logi s
APT for SL
Completions
Ma Neille omp.
Residuated frames
From axioms to rules
Subst. hierar hy
. Limitation
Hypersequent al .
Hyper-Ma Neille
Class N
3
Summary
Herbrand's theorem
via hyper anoni al
extensions
Further topi s
26 / 55
Fa t
Every stru tural rule in sequent al ulus is either derivable or on-
tradi tory in Int.
�) � �) �
�;�) �
�) �
) �
Theorem (G.Bezhanishvili-Harding 04)
There is no intermediate variety between HA and BA that is losed
under Ma Neille ompletions.
Eg. prelinearity annot be dealt with by sequent
al ulus/Ma Neille ompletions.
Hypersequent al ulus (Avron 91)
Introdu tion to
Substru tural Logi s
Algebrai Proof
Theory for
Substru tural Logi s
APT for SL
Completions
Ma Neille omp.
Residuated frames
From axioms to rules
Subst. hierar hy
Limitation
.
Hypersequent
al .
Hyper-Ma Neille
Class N
3
Summary
Herbrand's theorem
via hyper anoni al
extensions
Further topi s
28 / 55
� Hypersequents: �
1
) �
1
j � � � j �
m
) �
m
(meaning (�
1
! �
1
) _ � � � _ (�
m
! �
m
))
� Hypersequent al ulus for FL onsists of
Rules of FL Ext-Weakening Ext-Contra tion
� j �;�) �
� j �) �! �
�
� j �) �
� j �) � j �) �
� j �) �
� (pl) (�! �) _ (� ! �) is equivalent (in FLew) to
� j �
1
;�
1
) � � j �
2
;�
2
) �
� j �
1
;�
2
) � j �
1
;�
2
) �
( om)
�) � � ) �
�) � j � ) �
( om)
) �! � j ) � ! �
(! r)
) (�! �) _ (� ! �) j ) (�! �) _ (� ! �)
(_r)
) (�! �) _ (� ! �)
(EC)
Hypersequent al ulus (Avron 91)
Introdu tion to
Substru tural Logi s
Algebrai Proof
Theory for
Substru tural Logi s
APT for SL
Completions
Ma Neille omp.
Residuated frames
From axioms to rules
Subst. hierar hy
Limitation
.
Hypersequent
al .
Hyper-Ma Neille
Class N
3
Summary
Herbrand's theorem
via hyper anoni al
extensions
Further topi s
28 / 55
� Hypersequents: �
1
) �
1
j � � � j �
m
) �
m
(meaning (�
1
! �
1
) _ � � � _ (�
m
! �
m
))
� Hypersequent al ulus for FL onsists of
Rules of FL Ext-Weakening Ext-Contra tion
� j �;�) �
� j �) �! �
�
� j �) �
� j �) � j �) �
� j �) �
� (pl) (�! �) _ (� ! �) is equivalent (in FLew) to
� j �
1
;�
1
) � � j �
2
;�
2
) �
� j �
1
;�
2
) � j �
1
;�
2
) �
( om)
�) � � ) �
�) � j � ) �
( om)
) �! � j ) � ! �
(! r)
) (�! �) _ (� ! �) j ) (�! �) _ (� ! �)
(_r)
) (�! �) _ (� ! �)
(EC)
Hypersequent al ulus (Avron 91)
Introdu tion to
Substru tural Logi s
Algebrai Proof
Theory for
Substru tural Logi s
APT for SL
Completions
Ma Neille omp.
Residuated frames
From axioms to rules
Subst. hierar hy
Limitation
.
Hypersequent
al .
Hyper-Ma Neille
Class N
3
Summary
Herbrand's theorem
via hyper anoni al
extensions
Further topi s
28 / 55
� Hypersequents: �
1
) �
1
j � � � j �
m
) �
m
(meaning (�
1
! �
1
) _ � � � _ (�
m
! �
m
))
� Hypersequent al ulus for FL onsists of
Rules of FL Ext-Weakening Ext-Contra tion
� j �;�) �
� j �) �! �
�
� j �) �
� j �) � j �) �
� j �) �
� (pl) (�! �) _ (� ! �) is equivalent (in FLew) to
� j �
1
;�
1
) � � j �
2
;�
2
) �
� j �
1
;�
2
) � j �
1
;�
2
) �
( om)
�) � � ) �
�) � j � ) �
( om)
) �! � j ) � ! �
(! r)
) (�! �) _ (� ! �) j ) (�! �) _ (� ! �)
(_r)
) (�! �) _ (� ! �)
(EC)
Hypersequent al ulus (Avron 91)
Introdu tion to
Substru tural Logi s
Algebrai Proof
Theory for
Substru tural Logi s
APT for SL
Completions
Ma Neille omp.
Residuated frames
From axioms to rules
Subst. hierar hy
Limitation
.
Hypersequent
al .
Hyper-Ma Neille
Class N
3
Summary
Herbrand's theorem
via hyper anoni al
extensions
Further topi s
28 / 55
� Hypersequents: �
1
) �
1
j � � � j �
m
) �
m
(meaning (�
1
! �
1
) _ � � � _ (�
m
! �
m
))
� Hypersequent al ulus for FL onsists of
Rules of FL Ext-Weakening Ext-Contra tion
� j �;�) �
� j �) �! �
�
� j �) �
� j �) � j �) �
� j �) �
� (pl) (�! �) _ (� ! �) is equivalent (in FLew) to
� j �
1
;�
1
) � � j �
2
;�
2
) �
� j �
1
;�
2
) � j �
1
;�
2
) �
( om)
�) � � ) �
�) � j � ) �
( om)
) �! � j ) � ! �
(! r)
) (�! �) _ (� ! �) j ) (�! �) _ (� ! �)
(_r)
) (�! �) _ (� ! �)
(EC)
Hypersequent al ulus
Introdu tion to
Substru tural Logi s
Algebrai Proof
Theory for
Substru tural Logi s
APT for SL
Completions
Ma Neille omp.
Residuated frames
From axioms to rules
Subst. hierar hy
Limitation
.
Hypersequent
al .
Hyper-Ma Neille
Class N
3
Summary
Herbrand's theorem
via hyper anoni al
extensions
Further topi s
29 / 55
Theorem (CGT 08)
Every P
3
axiom is equivalent (in FLew) to a set of stru tural
rules in hypersequent al ulus.
:(� � �) _ (� ^ � ! � � �)
is equivalent to
Hypersequent al ulus
Introdu tion to
Substru tural Logi s
Algebrai Proof
Theory for
Substru tural Logi s
APT for SL
Completions
Ma Neille omp.
Residuated frames
From axioms to rules
Subst. hierar hy
Limitation
.
Hypersequent
al .
Hyper-Ma Neille
Class N
3
Summary
Herbrand's theorem
via hyper anoni al
extensions
Further topi s
29 / 55
Theorem (CGT 08)
Every P
3
axiom is equivalent (in FLew) to a set of stru tural
rules in hypersequent al ulus.
:(� � �) _ (� ^ � ! � � �)
is equivalent to
� j ) (� � �)! 0 j ) � ^ � ! � � �
Hypersequent al ulus
Introdu tion to
Substru tural Logi s
Algebrai Proof
Theory for
Substru tural Logi s
APT for SL
Completions
Ma Neille omp.
Residuated frames
From axioms to rules
Subst. hierar hy
Limitation
.
Hypersequent
al .
Hyper-Ma Neille
Class N
3
Summary
Herbrand's theorem
via hyper anoni al
extensions
Further topi s
29 / 55
Theorem (CGT 08)
Every P
3
axiom is equivalent (in FLew) to a set of stru tural
rules in hypersequent al ulus.
:(� � �) _ (� ^ � ! � � �)
is equivalent to
� j � � � ) 0 j ) � ^ � ! � � �
Hypersequent al ulus
Introdu tion to
Substru tural Logi s
Algebrai Proof
Theory for
Substru tural Logi s
APT for SL
Completions
Ma Neille omp.
Residuated frames
From axioms to rules
Subst. hierar hy
Limitation
.
Hypersequent
al .
Hyper-Ma Neille
Class N
3
Summary
Herbrand's theorem
via hyper anoni al
extensions
Further topi s
29 / 55
Theorem (CGT 08)
Every P
3
axiom is equivalent (in FLew) to a set of stru tural
rules in hypersequent al ulus.
:(� � �) _ (� ^ � ! � � �)
is equivalent to
� j �; � ) 0 j ) � ^ � ! � � �
Hypersequent al ulus
Introdu tion to
Substru tural Logi s
Algebrai Proof
Theory for
Substru tural Logi s
APT for SL
Completions
Ma Neille omp.
Residuated frames
From axioms to rules
Subst. hierar hy
Limitation
.
Hypersequent
al .
Hyper-Ma Neille
Class N
3
Summary
Herbrand's theorem
via hyper anoni al
extensions
Further topi s
29 / 55
Theorem (CGT 08)
Every P
3
axiom is equivalent (in FLew) to a set of stru tural
rules in hypersequent al ulus.
:(� � �) _ (� ^ � ! � � �)
is equivalent to
� j �; � ) j ) � ^ � ! � � �
Hypersequent al ulus
Introdu tion to
Substru tural Logi s
Algebrai Proof
Theory for
Substru tural Logi s
APT for SL
Completions
Ma Neille omp.
Residuated frames
From axioms to rules
Subst. hierar hy
Limitation
.
Hypersequent
al .
Hyper-Ma Neille
Class N
3
Summary
Herbrand's theorem
via hyper anoni al
extensions
Further topi s
29 / 55
Theorem (CGT 08)
Every P
3
axiom is equivalent (in FLew) to a set of stru tural
rules in hypersequent al ulus.
:(� � �) _ (� ^ � ! � � �)
is equivalent to
� j �; � ) j � ^ � ) � � �
Hypersequent al ulus
Introdu tion to
Substru tural Logi s
Algebrai Proof
Theory for
Substru tural Logi s
APT for SL
Completions
Ma Neille omp.
Residuated frames
From axioms to rules
Subst. hierar hy
Limitation
.
Hypersequent
al .
Hyper-Ma Neille
Class N
3
Summary
Herbrand's theorem
via hyper anoni al
extensions
Further topi s
29 / 55
Theorem (CGT 08)
Every P
3
axiom is equivalent (in FLew) to a set of stru tural
rules in hypersequent al ulus.
:(� � �) _ (� ^ � ! � � �)
is equivalent to
� j ) � ^ �
� j �; � ) j ) � � �
Hypersequent al ulus
Introdu tion to
Substru tural Logi s
Algebrai Proof
Theory for
Substru tural Logi s
APT for SL
Completions
Ma Neille omp.
Residuated frames
From axioms to rules
Subst. hierar hy
Limitation
.
Hypersequent
al .
Hyper-Ma Neille
Class N
3
Summary
Herbrand's theorem
via hyper anoni al
extensions
Further topi s
29 / 55
Theorem (CGT 08)
Every P
3
axiom is equivalent (in FLew) to a set of stru tural
rules in hypersequent al ulus.
:(� � �) _ (� ^ � ! � � �)
is equivalent to
� j ) � � j ) �
� j �; � ) j ) � � �
Hypersequent al ulus
Introdu tion to
Substru tural Logi s
Algebrai Proof
Theory for
Substru tural Logi s
APT for SL
Completions
Ma Neille omp.
Residuated frames
From axioms to rules
Subst. hierar hy
Limitation
.
Hypersequent
al .
Hyper-Ma Neille
Class N
3
Summary
Herbrand's theorem
via hyper anoni al
extensions
Further topi s
29 / 55
Theorem (CGT 08)
Every P
3
axiom is equivalent (in FLew) to a set of stru tural
rules in hypersequent al ulus.
:(� � �) _ (� ^ � ! � � �)
is equivalent to
� j ) � � j ) � � j � � � ) Æ
� j �; � ) j ) Æ
Hypersequent al ulus
Introdu tion to
Substru tural Logi s
Algebrai Proof
Theory for
Substru tural Logi s
APT for SL
Completions
Ma Neille omp.
Residuated frames
From axioms to rules
Subst. hierar hy
Limitation
.
Hypersequent
al .
Hyper-Ma Neille
Class N
3
Summary
Herbrand's theorem
via hyper anoni al
extensions
Further topi s
29 / 55
Theorem (CGT 08)
Every P
3
axiom is equivalent (in FLew) to a set of stru tural
rules in hypersequent al ulus.
:(� � �) _ (� ^ � ! � � �)
is equivalent to
� j ) � � j ) � � j �; � ) Æ
� j �; � ) j ) Æ
Hypersequent al ulus
Introdu tion to
Substru tural Logi s
Algebrai Proof
Theory for
Substru tural Logi s
APT for SL
Completions
Ma Neille omp.
Residuated frames
From axioms to rules
Subst. hierar hy
Limitation
.
Hypersequent
al .
Hyper-Ma Neille
Class N
3
Summary
Herbrand's theorem
via hyper anoni al
extensions
Further topi s
29 / 55
Theorem (CGT 08)
Every P
3
axiom is equivalent (in FLew) to a set of stru tural
rules in hypersequent al ulus.
:(� � �) _ (� ^ � ! � � �)
is equivalent to
� j ) � � j ) � � j �; � ) Æ
� j �; � ) j ) Æ
Hypersequent al ulus
Introdu tion to
Substru tural Logi s
Algebrai Proof
Theory for
Substru tural Logi s
APT for SL
Completions
Ma Neille omp.
Residuated frames
From axioms to rules
Subst. hierar hy
Limitation
.
Hypersequent
al .
Hyper-Ma Neille
Class N
3
Summary
Herbrand's theorem
via hyper anoni al
extensions
Further topi s
29 / 55
Theorem (CGT 08)
Every P
3
axiom is equivalent (in FLew) to a set of stru tural
rules in hypersequent al ulus.
:(� � �) _ (� ^ � ! � � �)
is equivalent to
� j ) � � j ) � � j �; � ) Æ
� j �) �
� j �; � ) j ) Æ
Hypersequent al ulus
Introdu tion to
Substru tural Logi s
Algebrai Proof
Theory for
Substru tural Logi s
APT for SL
Completions
Ma Neille omp.
Residuated frames
From axioms to rules
Subst. hierar hy
Limitation
.
Hypersequent
al .
Hyper-Ma Neille
Class N
3
Summary
Herbrand's theorem
via hyper anoni al
extensions
Further topi s
29 / 55
Theorem (CGT 08)
Every P
3
axiom is equivalent (in FLew) to a set of stru tural
rules in hypersequent al ulus.
:(� � �) _ (� ^ � ! � � �)
is equivalent to
� j ) � � j ) � � j �; � ) Æ
� j �) � � j �) �
� j �;�) j ) Æ
Hypersequent al ulus
Introdu tion to
Substru tural Logi s
Algebrai Proof
Theory for
Substru tural Logi s
APT for SL
Completions
Ma Neille omp.
Residuated frames
From axioms to rules
Subst. hierar hy
Limitation
.
Hypersequent
al .
Hyper-Ma Neille
Class N
3
Summary
Herbrand's theorem
via hyper anoni al
extensions
Further topi s
29 / 55
Theorem (CGT 08)
Every P
3
axiom is equivalent (in FLew) to a set of stru tural
rules in hypersequent al ulus.
:(� � �) _ (� ^ � ! � � �)
is equivalent to
� j ) � � j ) � � j �; � ) Æ
� j �) � � j �) � � j �)
� j �;�) j �) Æ
Hypersequent al ulus
Introdu tion to
Substru tural Logi s
Algebrai Proof
Theory for
Substru tural Logi s
APT for SL
Completions
Ma Neille omp.
Residuated frames
From axioms to rules
Subst. hierar hy
Limitation
.
Hypersequent
al .
Hyper-Ma Neille
Class N
3
Summary
Herbrand's theorem
via hyper anoni al
extensions
Further topi s
29 / 55
Theorem (CGT 08)
Every P
3
axiom is equivalent (in FLew) to a set of stru tural
rules in hypersequent al ulus.
:(� � �) _ (� ^ � ! � � �)
is equivalent to
� j ) � � j ) � � j �; � ) Æ
� j �) � � j �) � � j �) � j Æ;�) �
� j �;�) j �;�) �
Hypersequent al ulus
Introdu tion to
Substru tural Logi s
Algebrai Proof
Theory for
Substru tural Logi s
APT for SL
Completions
Ma Neille omp.
Residuated frames
From axioms to rules
Subst. hierar hy
Limitation
.
Hypersequent
al .
Hyper-Ma Neille
Class N
3
Summary
Herbrand's theorem
via hyper anoni al
extensions
Further topi s
29 / 55
Theorem (CGT 08)
Every P
3
axiom is equivalent (in FLew) to a set of stru tural
rules in hypersequent al ulus.
:(� � �) _ (� ^ � ! � � �)
is equivalent to
� j ) � � j ) � � j �; �;�) �
� j �) � � j �) � � j �)
� j �;�) j �;�) �
Hypersequent al ulus
Introdu tion to
Substru tural Logi s
Algebrai Proof
Theory for
Substru tural Logi s
APT for SL
Completions
Ma Neille omp.
Residuated frames
From axioms to rules
Subst. hierar hy
Limitation
.
Hypersequent
al .
Hyper-Ma Neille
Class N
3
Summary
Herbrand's theorem
via hyper anoni al
extensions
Further topi s
29 / 55
Theorem (CGT 08)
Every P
3
axiom is equivalent (in FLew) to a set of stru tural
rules in hypersequent al ulus.
:(� � �) _ (� ^ � ! � � �)
is equivalent to
� j �) � � j �) � � j �; �;�) �
� j �) � � j �) �
� j �;�) j �;�) �
Hypersequent al ulus
Introdu tion to
Substru tural Logi s
Algebrai Proof
Theory for
Substru tural Logi s
APT for SL
Completions
Ma Neille omp.
Residuated frames
From axioms to rules
Subst. hierar hy
Limitation
.
Hypersequent
al .
Hyper-Ma Neille
Class N
3
Summary
Herbrand's theorem
via hyper anoni al
extensions
Further topi s
29 / 55
Theorem (CGT 08)
Every P
3
axiom is equivalent (in FLew) to a set of stru tural
rules in hypersequent al ulus.
:(� � �) _ (� ^ � ! � � �)
is equivalent to
� j �; �;�) � � j �; �;�) �
� j �) � � j �) �
� j �;�) j �;�) �
Hypersequent al ulus
Introdu tion to
Substru tural Logi s
Algebrai Proof
Theory for
Substru tural Logi s
APT for SL
Completions
Ma Neille omp.
Residuated frames
From axioms to rules
Subst. hierar hy
Limitation
.
Hypersequent
al .
Hyper-Ma Neille
Class N
3
Summary
Herbrand's theorem
via hyper anoni al
extensions
Further topi s
29 / 55
Theorem (CGT 08)
Every P
3
axiom is equivalent (in FLew) to a set of stru tural
rules in hypersequent al ulus.
:(� � �) _ (� ^ � ! � � �)
is equivalent to
� j �;�;�) � � j �;�;�) �
� j �;�;�) � � j �;�;�) �
� j �;�) j �;�) �
Hypersequent al ulus
Introdu tion to
Substru tural Logi s
Algebrai Proof
Theory for
Substru tural Logi s
APT for SL
Completions
Ma Neille omp.
Residuated frames
From axioms to rules
Subst. hierar hy
Limitation
.
Hypersequent
al .
Hyper-Ma Neille
Class N
3
Summary
Herbrand's theorem
via hyper anoni al
extensions
Further topi s
29 / 55
Theorem (CGT 08)
Every P
3
axiom is equivalent (in FLew) to a set of stru tural
rules in hypersequent al ulus.
:(� � �) _ (� ^ � ! � � �)
is equivalent to
� j �;�;�) � � j �;�;�) �
� j �;�;�) � � j �;�;�) �
� j �;�) j �;�) �
� Cf. A kermann Lemma-based Algorithm
(Conradie-Palmigiano)
� Implemented by (Ciabattoni-Spendier)
Hyper-Ma Neille ompletions
Introdu tion to
Substru tural Logi s
Algebrai Proof
Theory for
Substru tural Logi s
APT for SL
Completions
Ma Neille omp.
Residuated frames
From axioms to rules
Subst. hierar hy
Limitation
Hypersequent al .
. Hyper-Ma Neille
Class N
3
Summary
Herbrand's theorem
via hyper anoni al
extensions
Further topi s
30 / 55
Let A be an FLew algebra. De�ne
W
h
A
:= (A�A;A�A;N; (�;_); (1; 0); (0; 0))
(a; h) N (b; k) () 1 = (a! b)_h _ k
Theorem
W
h+
A
is a ompletion of A, alled the hyper-Ma Neill ompletion.
Theorem (CGT)
For every set E of P
3
axioms,
� FLew(E) admits a strongly analyti hypersequent al ulus.
� FLew(E) is losed under hyper-Ma Neille ompletions.
Hyper-Ma Neille ompletions
Introdu tion to
Substru tural Logi s
Algebrai Proof
Theory for
Substru tural Logi s
APT for SL
Completions
Ma Neille omp.
Residuated frames
From axioms to rules
Subst. hierar hy
Limitation
Hypersequent al .
. Hyper-Ma Neille
Class N
3
Summary
Herbrand's theorem
via hyper anoni al
extensions
Further topi s
30 / 55
Let A be an FLew algebra. De�ne
W
h
A
:= (A�A;A�A;N; (�;_); (1; 0); (0; 0))
(a; h) N (b; k) () 1 = (a! b)_h _ k
Theorem
W
h+
A
is a ompletion of A, alled the hyper-Ma Neill ompletion.
Theorem (CGT)
For every set E of P
3
axioms,
� FLew(E) admits a strongly analyti hypersequent al ulus.
� FLew(E) is losed under hyper-Ma Neille ompletions.
Hyper-Ma Neille ompletions
Introdu tion to
Substru tural Logi s
Algebrai Proof
Theory for
Substru tural Logi s
APT for SL
Completions
Ma Neille omp.
Residuated frames
From axioms to rules
Subst. hierar hy
Limitation
Hypersequent al .
. Hyper-Ma Neille
Class N
3
Summary
Herbrand's theorem
via hyper anoni al
extensions
Further topi s
30 / 55
Let A be an FLew algebra. De�ne
W
h
A
:= (A�A;A�A;N; (�;_); (1; 0); (0; 0))
(a; h) N (b; k) () 1 = (a! b)_h _ k
Theorem
W
h+
A
is a ompletion of A, alled the hyper-Ma Neill ompletion.
Theorem (CGT)
For every set E of P
3
axioms,
� FLew(E) admits a strongly analyti hypersequent al ulus.
� FLew(E) is losed under hyper-Ma Neille ompletions.
Class N
3
Introdu tion to
Substru tural Logi s
Algebrai Proof
Theory for
Substru tural Logi s
APT for SL
Completions
Ma Neille omp.
Residuated frames
From axioms to rules
Subst. hierar hy
Limitation
Hypersequent al .
Hyper-Ma Neille
. Class N
3
Summary
Herbrand's theorem
via hyper anoni al
extensions
Further topi s
31 / 55
P
3
N
3
P
2
N
2
P
1
N
1
P
0
N
0
pp
pp
pp
pp
p6
pp
pp
pp
pp
p6
6
�
�
�
�� 6
�
�
�
�I
6
�
�
�
�� 6
�
�
�
�I
6
�
�
�
�� 6
�
�
�
�I
Some N
3
axioms:
� ^ (� _ )! (� ^ �) _ (� ^ ) distributivity
(�! � � �)! � an ellativity
� ^ � ! � � (�! �) divisibility
BL := FLew + (pl) + (div)
L := BL+ (in)
Theorem ( f. Kowalski-Litak 08)
The varieties BL, MV(= V( L)) are not losed under
any ompletions. Hen e the logi s BL and L do not
admit any strongly analyti al ulus.
Limitation of uniform proof theory!
Class N
3
Introdu tion to
Substru tural Logi s
Algebrai Proof
Theory for
Substru tural Logi s
APT for SL
Completions
Ma Neille omp.
Residuated frames
From axioms to rules
Subst. hierar hy
Limitation
Hypersequent al .
Hyper-Ma Neille
. Class N
3
Summary
Herbrand's theorem
via hyper anoni al
extensions
Further topi s
31 / 55
P
3
N
3
P
2
N
2
P
1
N
1
P
0
N
0
pp
pp
pp
pp
p6
pp
pp
pp
pp
p6
6
�
�
�
�� 6
�
�
�
�I
6
�
�
�
�� 6
�
�
�
�I
6
�
�
�
�� 6
�
�
�
�I
Some N
3
axioms:
� ^ (� _ )! (� ^ �) _ (� ^ ) distributivity
(�! � � �)! � an ellativity
� ^ � ! � � (�! �) divisibility
BL := FLew + (pl) + (div)
L := BL+ (in)
Theorem ( f. Kowalski-Litak 08)
The varieties BL, MV(= V( L)) are not losed under
any ompletions. Hen e the logi s BL and L do not
admit any strongly analyti al ulus.
Limitation of uniform proof theory!
Class N
3
Introdu tion to
Substru tural Logi s
Algebrai Proof
Theory for
Substru tural Logi s
APT for SL
Completions
Ma Neille omp.
Residuated frames
From axioms to rules
Subst. hierar hy
Limitation
Hypersequent al .
Hyper-Ma Neille
. Class N
3
Summary
Herbrand's theorem
via hyper anoni al
extensions
Further topi s
31 / 55
P
3
N
3
P
2
N
2
P
1
N
1
P
0
N
0
pp
pp
pp
pp
p6
pp
pp
pp
pp
p6
6
�
�
�
�� 6
�
�
�
�I
6
�
�
�
�� 6
�
�
�
�I
6
�
�
�
�� 6
�
�
�
�I
Some N
3
axioms:
� ^ (� _ )! (� ^ �) _ (� ^ ) distributivity
(�! � � �)! � an ellativity
� ^ � ! � � (�! �) divisibility
BL := FLew + (pl) + (div)
L := BL+ (in)
Theorem ( f. Kowalski-Litak 08)
The varieties BL, MV(= V( L)) are not losed under
any ompletions. Hen e the logi s BL and L do not
admit any strongly analyti al ulus.
Limitation of uniform proof theory!
Summary
Introdu tion to
Substru tural Logi s
Algebrai Proof
Theory for
Substru tural Logi s
APT for SL
Completions
Ma Neille omp.
Residuated frames
From axioms to rules
Subst. hierar hy
Limitation
Hypersequent al .
Hyper-Ma Neille
Class N
3
. Summary
Herbrand's theorem
via hyper anoni al
extensions
Further topi s
32 / 55
P
3
N
3
P
2
N
2
P
1
N
1
P
0
N
0
pp
pp
pp
pp
p6
pp
pp
pp
pp
p6
6
�
�
�
�� 6
�
�
�
�I
6
�
�
�
�� 6
�
�
�
�I
6
�
�
�
�� 6
�
�
�
�I
N
2
: sequent al ulus
Ma Neille ompletions
P
3
: hypersequent al ulus
hyper-Ma Neille ompletions
N
3
: limitation of uniform proof theory
� Axioms ) rules is important in both proof
theory and algebra.
� The hyper- onstru tion (proof theory) is
useful for ompletions (algebra) too.
� Limitation of proof theory is imposed by al-
gebrai fa ts.
Summary
Introdu tion to
Substru tural Logi s
Algebrai Proof
Theory for
Substru tural Logi s
APT for SL
Completions
Ma Neille omp.
Residuated frames
From axioms to rules
Subst. hierar hy
Limitation
Hypersequent al .
Hyper-Ma Neille
Class N
3
. Summary
Herbrand's theorem
via hyper anoni al
extensions
Further topi s
32 / 55
P
3
N
3
P
2
N
2
P
1
N
1
P
0
N
0
pp
pp
pp
pp
p6
pp
pp
pp
pp
p6
6
�
�
�
�� 6
�
�
�
�I
6
�
�
�
�� 6
�
�
�
�I
6
�
�
�
�� 6
�
�
�
�I
N
2
: sequent al ulus
Ma Neille ompletions
P
3
: hypersequent al ulus
hyper-Ma Neille ompletions
N
3
: limitation of uniform proof theory
� Axioms ) rules is important in both proof
theory and algebra.
� The hyper- onstru tion (proof theory) is
useful for ompletions (algebra) too.
� Limitation of proof theory is imposed by al-
gebrai fa ts.
Herbrand's theorem via hyper anoni al
extensions
Introdu tion to
Substru tural Logi s
Algebrai Proof
Theory for
Substru tural Logi s
.
Herbrand's
theorem via
hyper anoni al
extensions
Predi ate SL
Herbrand property
Canoni al extensions
HP ompa t ompl.
HP for �nite
HP for N2
HP for P3
Class N
3
Herbrand's theorem
for 98
Further topi s
33 / 55
Predi ate substru tural logi s
Introdu tion to
Substru tural Logi s
Algebrai Proof
Theory for
Substru tural Logi s
Herbrand's theorem
via hyper anoni al
extensions
. Predi ate SL
Herbrand property
Canoni al extensions
HP ompa t ompl.
HP for �nite
HP for N2
HP for P3
Class N
3
Herbrand's theorem
for 98
Further topi s
34 / 55
Let L be a propositional substru tural logi .
QL := the predi ate extension of L obtained by adding
8x:�(x)! �(t) �(t)! 9x:�(x)
� ! �(x)
� ! 8x:�(x)
�(x)! �
9x:�(x)! � (x not free in �)
Herbrand property
Introdu tion to
Substru tural Logi s
Algebrai Proof
Theory for
Substru tural Logi s
Herbrand's theorem
via hyper anoni al
extensions
Predi ate SL
.
Herbrand
property
Canoni al extensions
HP ompa t ompl.
HP for �nite
HP for N2
HP for P3
Class N
3
Herbrand's theorem
for 98
Further topi s
35 / 55
De�nition
L satis�es the Herbrand property if for every set of universal
formulas and every quanti�er-free formula '(x),
`
QL
9x:'(x) ()
Æ
`
L
'(t
1
) _ � � � _ '(t
n
)
for some t
1
; : : : ; t
n
;
where
Æ
:= f (t) : 8x: (x) 2 g.
Herbrand property is related to ompa tness phenomena
(previous talk).
What is the algebrai form of ompa tness?
Herbrand property
Introdu tion to
Substru tural Logi s
Algebrai Proof
Theory for
Substru tural Logi s
Herbrand's theorem
via hyper anoni al
extensions
Predi ate SL
.
Herbrand
property
Canoni al extensions
HP ompa t ompl.
HP for �nite
HP for N2
HP for P3
Class N
3
Herbrand's theorem
for 98
Further topi s
35 / 55
De�nition
L satis�es the Herbrand property if for every set of universal
formulas and every quanti�er-free formula '(x),
`
QL
9x:'(x) ()
Æ
`
L
'(t
1
) _ � � � _ '(t
n
)
for some t
1
; : : : ; t
n
;
where
Æ
:= f (t) : 8x: (x) 2 g.
Herbrand property is related to ompa tness phenomena
(previous talk).
What is the algebrai form of ompa tness?
Canoni al extensions
Introdu tion to
Substru tural Logi s
Algebrai Proof
Theory for
Substru tural Logi s
Herbrand's theorem
via hyper anoni al
extensions
Predi ate SL
Herbrand property
.
Canoni al
extensions
HP ompa t ompl.
HP for �nite
HP for N2
HP for P3
Class N
3
Herbrand's theorem
for 98
Further topi s
36 / 55
Let C be a Boolean algebra and X
C
be its Stone spa e. Then
C
�
:= (P(X
C
);\;[;
C
)
is a ompletion of C.
Let D be a bounded distributive latti e and Y
D
be its Priestly
spa e. Then
D
�
:= (P
#
(Y
D
);\;[)
is a ompletion of D.
Re all that
Ma Neille ompletions = join-dense, meet-dense ompletions
Do we have a similar abstra t hara terization for C
�
;D
�
?
Canoni al extensions
Introdu tion to
Substru tural Logi s
Algebrai Proof
Theory for
Substru tural Logi s
Herbrand's theorem
via hyper anoni al
extensions
Predi ate SL
Herbrand property
.
Canoni al
extensions
HP ompa t ompl.
HP for �nite
HP for N2
HP for P3
Class N
3
Herbrand's theorem
for 98
Further topi s
36 / 55
Let C be a Boolean algebra and X
C
be its Stone spa e. Then
C
�
:= (P(X
C
);\;[;
C
)
is a ompletion of C.
Let D be a bounded distributive latti e and Y
D
be its Priestly
spa e. Then
D
�
:= (P
#
(Y
D
);\;[)
is a ompletion of D.
Re all that
Ma Neille ompletions = join-dense, meet-dense ompletions
Do we have a similar abstra t hara terization for C
�
;D
�
?
Canoni al extensions
Introdu tion to
Substru tural Logi s
Algebrai Proof
Theory for
Substru tural Logi s
Herbrand's theorem
via hyper anoni al
extensions
Predi ate SL
Herbrand property
.
Canoni al
extensions
HP ompa t ompl.
HP for �nite
HP for N2
HP for P3
Class N
3
Herbrand's theorem
for 98
Further topi s
36 / 55
Let C be a Boolean algebra and X
C
be its Stone spa e. Then
C
�
:= (P(X
C
);\;[;
C
)
is a ompletion of C.
Let D be a bounded distributive latti e and Y
D
be its Priestly
spa e. Then
D
�
:= (P
#
(Y
D
);\;[)
is a ompletion of D.
Re all that
Ma Neille ompletions = join-dense, meet-dense ompletions
Do we have a similar abstra t hara terization for C
�
;D
�
?
Canoni al extensions
Introdu tion to
Substru tural Logi s
Algebrai Proof
Theory for
Substru tural Logi s
Herbrand's theorem
via hyper anoni al
extensions
Predi ate SL
Herbrand property
.
Canoni al
extensions
HP ompa t ompl.
HP for �nite
HP for N2
HP for P3
Class N
3
Herbrand's theorem
for 98
Further topi s
37 / 55
Let A be a latti e. Its ompletion B is
� dense if for every x 2 B, there exist C
i
;D
j
� A
(i 2 I; j 2 J) su h that
x =
_
i2I
^
C
i
=
^
j2J
_
D
j
:
� ompa t if for every C;D � A,
^
C �
_
D =)
^
C
0
�
_
D
0
for some �nite C
0
� C and D
0
� D.
Theorem (Gehrke-Harding 01)
Every latti e A has a dense and ompa t ompletion A
�
, unique
up to isomorphism, alled the anoni al extension.
Canoni al extensions
Introdu tion to
Substru tural Logi s
Algebrai Proof
Theory for
Substru tural Logi s
Herbrand's theorem
via hyper anoni al
extensions
Predi ate SL
Herbrand property
.
Canoni al
extensions
HP ompa t ompl.
HP for �nite
HP for N2
HP for P3
Class N
3
Herbrand's theorem
for 98
Further topi s
37 / 55
Let A be a latti e. Its ompletion B is
� dense if for every x 2 B, there exist C
i
;D
j
� A
(i 2 I; j 2 J) su h that
x =
_
i2I
^
C
i
=
^
j2J
_
D
j
:
� ompa t if for every C;D � A,
^
C �
_
D =)
^
C
0
�
_
D
0
for some �nite C
0
� C and D
0
� D.
Theorem (Gehrke-Harding 01)
Every latti e A has a dense and ompa t ompletion A
�
, unique
up to isomorphism, alled the anoni al extension.
Canoni al extensions
Introdu tion to
Substru tural Logi s
Algebrai Proof
Theory for
Substru tural Logi s
Herbrand's theorem
via hyper anoni al
extensions
Predi ate SL
Herbrand property
.
Canoni al
extensions
HP ompa t ompl.
HP for �nite
HP for N2
HP for P3
Class N
3
Herbrand's theorem
for 98
Further topi s
37 / 55
Let A be a latti e. Its ompletion B is
� dense if for every x 2 B, there exist C
i
;D
j
� A
(i 2 I; j 2 J) su h that
x =
_
i2I
^
C
i
=
^
j2J
_
D
j
:
� ompa t if for every C;D � A,
^
C �
_
D =)
^
C
0
�
_
D
0
for some �nite C
0
� C and D
0
� D.
Theorem (Gehrke-Harding 01)
Every latti e A has a dense and ompa t ompletion A
�
, unique
up to isomorphism, alled the anoni al extension.
Herbrand property via ompa t ompletions
Introdu tion to
Substru tural Logi s
Algebrai Proof
Theory for
Substru tural Logi s
Herbrand's theorem
via hyper anoni al
extensions
Predi ate SL
Herbrand property
Canoni al extensions
.
HP ompa t
ompl.
HP for �nite
HP for N2
HP for P3
Class N
3
Herbrand's theorem
for 98
Further topi s
38 / 55
A ompletion of A is ompa t if for every C;D � A,
^
C �
_
D =)
^
C
0
�
_
D
0
for some �nite C
0
� C and D
0
� D.
Theorem
If V(L) is losed under ompa t ompletions, then L satis�es the
Herbrand property.
Herbrand property for �nite-valued logi s
Introdu tion to
Substru tural Logi s
Algebrai Proof
Theory for
Substru tural Logi s
Herbrand's theorem
via hyper anoni al
extensions
Predi ate SL
Herbrand property
Canoni al extensions
HP ompa t ompl.
. HP for �nite
HP for N2
HP for P3
Class N
3
Herbrand's theorem
for 98
Further topi s
39 / 55
Theorem (Gehrke-Harding 01)
Let V be a variety of monotone latti e expansions.
If V is generated by a �nite algebra, then V is losed under anon-
i al extensions.
Corollary
Every �nite-valued substru tural logi satis�es the Herbrand prop-
erty.
It a tually applies to a mu h wider range of �nite-valued logi s.
The GH theorem is an algebrai ounterpart of the uniform
midsequent theorem for �nite-valued logi s
(Baaz-Ferm�uller-Za h 94).
Herbrand property for �nite-valued logi s
Introdu tion to
Substru tural Logi s
Algebrai Proof
Theory for
Substru tural Logi s
Herbrand's theorem
via hyper anoni al
extensions
Predi ate SL
Herbrand property
Canoni al extensions
HP ompa t ompl.
. HP for �nite
HP for N2
HP for P3
Class N
3
Herbrand's theorem
for 98
Further topi s
39 / 55
Theorem (Gehrke-Harding 01)
Let V be a variety of monotone latti e expansions.
If V is generated by a �nite algebra, then V is losed under anon-
i al extensions.
Corollary
Every �nite-valued substru tural logi satis�es the Herbrand prop-
erty.
It a tually applies to a mu h wider range of �nite-valued logi s.
The GH theorem is an algebrai ounterpart of the uniform
midsequent theorem for �nite-valued logi s
(Baaz-Ferm�uller-Za h 94).
Herbrand property for �nite-valued logi s
Introdu tion to
Substru tural Logi s
Algebrai Proof
Theory for
Substru tural Logi s
Herbrand's theorem
via hyper anoni al
extensions
Predi ate SL
Herbrand property
Canoni al extensions
HP ompa t ompl.
. HP for �nite
HP for N2
HP for P3
Class N
3
Herbrand's theorem
for 98
Further topi s
39 / 55
Theorem (Gehrke-Harding 01)
Let V be a variety of monotone latti e expansions.
If V is generated by a �nite algebra, then V is losed under anon-
i al extensions.
Corollary
Every �nite-valued substru tural logi satis�es the Herbrand prop-
erty.
It a tually applies to a mu h wider range of �nite-valued logi s.
The GH theorem is an algebrai ounterpart of the uniform
midsequent theorem for �nite-valued logi s
(Baaz-Ferm�uller-Za h 94).
Herbrand property for N
2
logi s
Introdu tion to
Substru tural Logi s
Algebrai Proof
Theory for
Substru tural Logi s
Herbrand's theorem
via hyper anoni al
extensions
Predi ate SL
Herbrand property
Canoni al extensions
HP ompa t ompl.
HP for �nite
. HP for N2
HP for P3
Class N
3
Herbrand's theorem
for 98
Further topi s
40 / 55
Theorem (Gehrke-Harding-Venema 05)
Let V be a variety of bounded monotone latti e expansions. If
V is losed under Ma Neille ompletions, it is also losed under
anoni al extensions.
Ma Neille ompletions preserve (in) ::�! �.
Canoni al extensions preserve (dist)
(� _ �) ^ $ (� ^ ) _ (� ^ ).
Corollary
Every substru tural logi axiomatized by
� a y li N
2
axioms
� and/or (in), (dist)
satis�es the Herbrand property.
Herbrand property for N
2
logi s
Introdu tion to
Substru tural Logi s
Algebrai Proof
Theory for
Substru tural Logi s
Herbrand's theorem
via hyper anoni al
extensions
Predi ate SL
Herbrand property
Canoni al extensions
HP ompa t ompl.
HP for �nite
. HP for N2
HP for P3
Class N
3
Herbrand's theorem
for 98
Further topi s
40 / 55
Theorem (Gehrke-Harding-Venema 05)
Let V be a variety of bounded monotone latti e expansions. If
V is losed under Ma Neille ompletions, it is also losed under
anoni al extensions.
Ma Neille ompletions preserve (in) ::�! �.
Canoni al extensions preserve (dist)
(� _ �) ^ $ (� ^ ) _ (� ^ ).
Corollary
Every substru tural logi axiomatized by
� a y li N
2
axioms
� and/or (in), (dist)
satis�es the Herbrand property.
Herbrand property for N
2
logi s
Introdu tion to
Substru tural Logi s
Algebrai Proof
Theory for
Substru tural Logi s
Herbrand's theorem
via hyper anoni al
extensions
Predi ate SL
Herbrand property
Canoni al extensions
HP ompa t ompl.
HP for �nite
. HP for N2
HP for P3
Class N
3
Herbrand's theorem
for 98
Further topi s
40 / 55
Theorem (Gehrke-Harding-Venema 05)
Let V be a variety of bounded monotone latti e expansions. If
V is losed under Ma Neille ompletions, it is also losed under
anoni al extensions.
Ma Neille ompletions preserve (in) ::�! �.
Canoni al extensions preserve (dist)
(� _ �) ^ $ (� ^ ) _ (� ^ ).
Corollary
Every substru tural logi axiomatized by
� a y li N
2
axioms
� and/or (in), (dist)
satis�es the Herbrand property.
Herbrand property for N
2
logi s
Introdu tion to
Substru tural Logi s
Algebrai Proof
Theory for
Substru tural Logi s
Herbrand's theorem
via hyper anoni al
extensions
Predi ate SL
Herbrand property
Canoni al extensions
HP ompa t ompl.
HP for �nite
. HP for N2
HP for P3
Class N
3
Herbrand's theorem
for 98
Further topi s
41 / 55
The GHV theorem states:
Ma Neille ompletions =) Canoni al extensions:
It onforms to the proof theoreti intuition:
Cut elimination =) Herbrand's theorem:
What about P
3
logi s?
Introdu tion to
Substru tural Logi s
Algebrai Proof
Theory for
Substru tural Logi s
Herbrand's theorem
via hyper anoni al
extensions
Predi ate SL
Herbrand property
Canoni al extensions
HP ompa t ompl.
HP for �nite
. HP for N2
HP for P3
Class N
3
Herbrand's theorem
for 98
Further topi s
42 / 55
P
3
N
3
P
2
N
2
P
1
N
1
P
0
N
0
pp
pp
pp
pp
p6
pp
pp
pp
pp
p6
6
�
�
�
�� 6
�
�
�
�I
6
�
�
�
�� 6
�
�
�
�I
6
�
�
�
�� 6
�
�
�
�I
Some P
3
axioms:
(�! �) _ (� ! �) prelinearity
� _ :� ex luded middle
:� _ ::� weak ex luded middle
:(� � �) _ (� ^ � ! � � �) weak nilpotent minimum
W
ki=0
(�
i
!
W
j 6=i
�
j
) bounded width � k
W
ki=0
(�
0
^ � � � ^ �
i�1
! �
i
) bounded size � k
We want ompa t ompletions that preserve P
3
axioms.
) Hyper anoni al extensions.
What about P
3
logi s?
Introdu tion to
Substru tural Logi s
Algebrai Proof
Theory for
Substru tural Logi s
Herbrand's theorem
via hyper anoni al
extensions
Predi ate SL
Herbrand property
Canoni al extensions
HP ompa t ompl.
HP for �nite
. HP for N2
HP for P3
Class N
3
Herbrand's theorem
for 98
Further topi s
42 / 55
P
3
N
3
P
2
N
2
P
1
N
1
P
0
N
0
pp
pp
pp
pp
p6
pp
pp
pp
pp
p6
6
�
�
�
�� 6
�
�
�
�I
6
�
�
�
�� 6
�
�
�
�I
6
�
�
�
�� 6
�
�
�
�I
Some P
3
axioms:
(�! �) _ (� ! �) prelinearity
� _ :� ex luded middle
:� _ ::� weak ex luded middle
:(� � �) _ (� ^ � ! � � �) weak nilpotent minimum
W
ki=0
(�
i
!
W
j 6=i
�
j
) bounded width � k
W
ki=0
(�
0
^ � � � ^ �
i�1
! �
i
) bounded size � k
We want ompa t ompletions that preserve P
3
axioms.
) Hyper anoni al extensions.
Reminder: Ma Neille and hyper-Ma Neille ompletions
Introdu tion to
Substru tural Logi s
Algebrai Proof
Theory for
Substru tural Logi s
Herbrand's theorem
via hyper anoni al
extensions
Predi ate SL
Herbrand property
Canoni al extensions
HP ompa t ompl.
HP for �nite
. HP for N2
HP for P3
Class N
3
Herbrand's theorem
for 98
Further topi s
43 / 55
Let A be an FL algebra.
Ma Neille ompletion of A is W
+
A
where
W
A
:= (A;A;�; �; 1; 0):
Assuming A is FLew, hyper-Ma Neille ompletion is W
h+
A
where
W
h
A
:= (A�A;A�A;N; � � � )
(a; h) N (b; k) () 1 = (a! b)_h _ k:
Reminder: Ma Neille and hyper-Ma Neille ompletions
Introdu tion to
Substru tural Logi s
Algebrai Proof
Theory for
Substru tural Logi s
Herbrand's theorem
via hyper anoni al
extensions
Predi ate SL
Herbrand property
Canoni al extensions
HP ompa t ompl.
HP for �nite
. HP for N2
HP for P3
Class N
3
Herbrand's theorem
for 98
Further topi s
43 / 55
Let A be an FL algebra.
Ma Neille ompletion of A is W
+
A
where
W
A
:= (A;A;�; �; 1; 0):
Assuming A is FLew, hyper-Ma Neille ompletion is W
h+
A
where
W
h
A
:= (A�A;A�A;N; � � � )
(a; h) N (b; k) () 1 = (a! b)_h _ k:
Canoni al and hyper anoni al extensions
Introdu tion to
Substru tural Logi s
Algebrai Proof
Theory for
Substru tural Logi s
Herbrand's theorem
via hyper anoni al
extensions
Predi ate SL
Herbrand property
Canoni al extensions
HP ompa t ompl.
HP for �nite
. HP for N2
HP for P3
Class N
3
Herbrand's theorem
for 98
Further topi s
44 / 55
Canoni al extension of A is W
�+
A
where
W
�
A
:= (F
A
; I
A
; N; Æ; "1; #0);
F
A
:= the �lters of A
I
A
:= the ideals of A
f N i () f \ i 6= ;
Assuming A is FLew, hyper anoni al extension of A is W
H+
A
where
W
H
A
:= (F
A
�I
A
; I
A
�I
A
; N; � � � )
(f; j) N (i; k) () 1 2 (f ! i)_j _ k
Canoni al and hyper anoni al extensions
Introdu tion to
Substru tural Logi s
Algebrai Proof
Theory for
Substru tural Logi s
Herbrand's theorem
via hyper anoni al
extensions
Predi ate SL
Herbrand property
Canoni al extensions
HP ompa t ompl.
HP for �nite
. HP for N2
HP for P3
Class N
3
Herbrand's theorem
for 98
Further topi s
44 / 55
Canoni al extension of A is W
�+
A
where
W
�
A
:= (F
A
; I
A
; N; Æ; "1; #0);
F
A
:= the �lters of A
I
A
:= the ideals of A
f N i () f \ i 6= ;
Assuming A is FLew, hyper anoni al extension of A is W
H+
A
where
W
H
A
:= (F
A
�I
A
; I
A
�I
A
; N; � � � )
(f; j) N (i; k) () 1 2 (f ! i)_j _ k
Herbrand property for P
3
logi s
Introdu tion to
Substru tural Logi s
Algebrai Proof
Theory for
Substru tural Logi s
Herbrand's theorem
via hyper anoni al
extensions
Predi ate SL
Herbrand property
Canoni al extensions
HP ompa t ompl.
HP for �nite
HP for N2
. HP for P3
Class N
3
Herbrand's theorem
for 98
Further topi s
45 / 55
Theorem
Hyper anoni al extensions are ompa t ompletions. They pre-
serve all P
3
identities.
Corollary
Every substru tural logi over FLew axiomatized by P
3
axioms
satis�es the Herbrand property.
It applies to MTL, G, LQ and many more uniformly.
Herbrand property for P
3
logi s
Introdu tion to
Substru tural Logi s
Algebrai Proof
Theory for
Substru tural Logi s
Herbrand's theorem
via hyper anoni al
extensions
Predi ate SL
Herbrand property
Canoni al extensions
HP ompa t ompl.
HP for �nite
HP for N2
. HP for P3
Class N
3
Herbrand's theorem
for 98
Further topi s
45 / 55
Theorem
Hyper anoni al extensions are ompa t ompletions. They pre-
serve all P
3
identities.
Corollary
Every substru tural logi over FLew axiomatized by P
3
axioms
satis�es the Herbrand property.
It applies to MTL, G, LQ and many more uniformly.
Class N
3
Introdu tion to
Substru tural Logi s
Algebrai Proof
Theory for
Substru tural Logi s
Herbrand's theorem
via hyper anoni al
extensions
Predi ate SL
Herbrand property
Canoni al extensions
HP ompa t ompl.
HP for �nite
HP for N2
HP for P3
. Class N
3
Herbrand's theorem
for 98
Further topi s
46 / 55
P
3
N
3
P
2
N
2
P
1
N
1
P
0
N
0
pp
pp
pp
pp
p6
pp
pp
pp
pp
p6
6
�
�
�
�� 6
�
�
�
�I
6
�
�
�
�� 6
�
�
�
�I
6
�
�
�
�� 6
�
�
�
�I
Re all that MV(= V( L)) is not losed under any
ompletions.
Theorem (Baaz-Met alfe 08)
L does not satisfy the Herbrand property, al-
though it does satisfy an \approximate" Her-
brand theorem.
Herbrand's theorem for 98-formulas
Introdu tion to
Substru tural Logi s
Algebrai Proof
Theory for
Substru tural Logi s
Herbrand's theorem
via hyper anoni al
extensions
Predi ate SL
Herbrand property
Canoni al extensions
HP ompa t ompl.
HP for �nite
HP for N2
HP for P3
Class N
3
.
Herbrand's
theorem for 98
Further topi s
47 / 55
Herbrand's theorem for 98-formulas:
� ` 9x8y:'(x; y) () � ` '(t
1
; y
1
) _ � � � _ '(t
n
; y
n
)
where t
i
does not ontain y
i
; : : : ; y
n
.
The general form requires the onstant domain axiom ( d):
8x:(�(x) _ �)$ (8x:�(x)) _ �:
Its algebrai ounterpart is meet in�nite distributivity:
(mid)
^
i2I
(x
i
_ y) = (
^
i2I
x
i
) _ y:
Herbrand's theorem for 98-formulas
Introdu tion to
Substru tural Logi s
Algebrai Proof
Theory for
Substru tural Logi s
Herbrand's theorem
via hyper anoni al
extensions
Predi ate SL
Herbrand property
Canoni al extensions
HP ompa t ompl.
HP for �nite
HP for N2
HP for P3
Class N
3
.
Herbrand's
theorem for 98
Further topi s
48 / 55
Lemma
Let A be an FL algebra.
� If A is distributive, then A
�
satis�es (mid).
� If A is an MTL algebra, then A
h
satis�es (mid).
Theorem
Let L be a substru tural logi . Herbrand's theorem for 98-
formulas holds for QL( d) if
� either L is axiomatized by distributivity and some N
2
axioms,
� or L is axiomatized by (e), (w), (pl) and and some P
3
axioms.
Herbrand's theorem for 98-formulas
Introdu tion to
Substru tural Logi s
Algebrai Proof
Theory for
Substru tural Logi s
Herbrand's theorem
via hyper anoni al
extensions
Predi ate SL
Herbrand property
Canoni al extensions
HP ompa t ompl.
HP for �nite
HP for N2
HP for P3
Class N
3
.
Herbrand's
theorem for 98
Further topi s
48 / 55
Lemma
Let A be an FL algebra.
� If A is distributive, then A
�
satis�es (mid).
� If A is an MTL algebra, then A
h
satis�es (mid).
Theorem
Let L be a substru tural logi . Herbrand's theorem for 98-
formulas holds for QL( d) if
� either L is axiomatized by distributivity and some N
2
axioms,
� or L is axiomatized by (e), (w), (pl) and and some P
3
axioms.
Further topi s
Introdu tion to
Substru tural Logi s
Algebrai Proof
Theory for
Substru tural Logi s
Herbrand's theorem
via hyper anoni al
extensions
. Further topi s
Extra ting info
Interpolation
Density rule elim.
Con lusion
49 / 55
Extra ting algebrai information from proof theory
Introdu tion to
Substru tural Logi s
Algebrai Proof
Theory for
Substru tural Logi s
Herbrand's theorem
via hyper anoni al
extensions
Further topi s
. Extra ting info
Interpolation
Density rule elim.
Con lusion
50 / 55
� Apparently there is no ni e duality between FL algebras and
residuated frames.
� Residuated frames are lose to syntax so that one an
en ode synta ti information into frames.
� By en oding proof theoreti arguments into frames and
taking the omplex algebra, one an obtain an algebrai
onstru tion.
Case study: Interpolation ) Amalgamation
Introdu tion to
Substru tural Logi s
Algebrai Proof
Theory for
Substru tural Logi s
Herbrand's theorem
via hyper anoni al
extensions
Further topi s
Extra ting info
. Interpolation
Density rule elim.
Con lusion
51 / 55
Let X;Y be sets of propositional variables.
Maehara's lemma
If `
FLe
�;�) � with � � Fm(X) and �;� � Fm(Y ),
there is � 2 Fm(X \ Y ) su h that
`
FLe
�)� and `
FLe
�;�) �:
Case study: Interpolation ) Amalgamation
Introdu tion to
Substru tural Logi s
Algebrai Proof
Theory for
Substru tural Logi s
Herbrand's theorem
via hyper anoni al
extensions
Further topi s
Extra ting info
. Interpolation
Density rule elim.
Con lusion
52 / 55
Let A;B;C be FLe algebras.
Suppose that A is a subalgebra of both B and C.
B
A
C
�
��
inj
�
�R
inj
De�ne
W
I
:= (B � C;B [ C;N; � � � )
(b; ) N
0
() 9i 2 A: b �
B
i and i �
C
0
(b; ) N b
0
() 9i 2 A: �
C
i and ib �
B
b
0
:
Case study: Interpolation ) Amalgamation
Introdu tion to
Substru tural Logi s
Algebrai Proof
Theory for
Substru tural Logi s
Herbrand's theorem
via hyper anoni al
extensions
Further topi s
Extra ting info
. Interpolation
Density rule elim.
Con lusion
53 / 55
Then the omplex algebra gives rise to an amalgam.
B
A
W
+
I
C
p
p
p
p
p
R
inj
�
�
��
inj
�
�
�R
inj
p
p
p
p
p
�
inj
(interpolation)
+
= amalgamation
Case study: Interpolation ) Amalgamation
Introdu tion to
Substru tural Logi s
Algebrai Proof
Theory for
Substru tural Logi s
Herbrand's theorem
via hyper anoni al
extensions
Further topi s
Extra ting info
. Interpolation
Density rule elim.
Con lusion
53 / 55
Then the omplex algebra gives rise to an amalgam.
B
A
W
+
I
C
p
p
p
p
p
R
inj
�
�
��
inj
�
�
�R
inj
p
p
p
p
p
�
inj
(interpolation)
+
= amalgamation
Another su ess: density rule elimination ) densi� ation
Introdu tion to
Substru tural Logi s
Algebrai Proof
Theory for
Substru tural Logi s
Herbrand's theorem
via hyper anoni al
extensions
Further topi s
Extra ting info
Interpolation
. Density rule elim.
Con lusion
54 / 55
Likewise, the next talk by Hor� ��k is an out ome of:
(density rule elimination)
+
= densi� ation
The slogan is:
(proof theoreti argument)
+
= algebrai onstru tion
This way we an salvage ni e proof theoreti ideas and bring
them to ordered algebras.
Another su ess: density rule elimination ) densi� ation
Introdu tion to
Substru tural Logi s
Algebrai Proof
Theory for
Substru tural Logi s
Herbrand's theorem
via hyper anoni al
extensions
Further topi s
Extra ting info
Interpolation
. Density rule elim.
Con lusion
54 / 55
Likewise, the next talk by Hor� ��k is an out ome of:
(density rule elimination)
+
= densi� ation
The slogan is:
(proof theoreti argument)
+
= algebrai onstru tion
This way we an salvage ni e proof theoreti ideas and bring
them to ordered algebras.
Con lusion
Introdu tion to
Substru tural Logi s
Algebrai Proof
Theory for
Substru tural Logi s
Herbrand's theorem
via hyper anoni al
extensions
Further topi s
Extra ting info
Interpolation
Density rule elim.
. Con lusion
55 / 55
We have explored the onne tion between proof theoreti
arugments and algebrai ompletions based on the substru tural
hierar hy:
sequent al . (N
2
) hypersequent al . (P
3
)
ut elimination Ma Neille ompl. hyper-Ma Neille ompl.
Herbran's theorem anoni al ext. hyper anoni al ext.
It is residuated frames that onne t the two:
(proof theoreti argument)
+
= algebrai onstru tion.
Substru tural proof theory is full of burea ra y, but hopefully
there are still something good to be salvaged.
Con lusion
Introdu tion to
Substru tural Logi s
Algebrai Proof
Theory for
Substru tural Logi s
Herbrand's theorem
via hyper anoni al
extensions
Further topi s
Extra ting info
Interpolation
Density rule elim.
. Con lusion
55 / 55
We have explored the onne tion between proof theoreti
arugments and algebrai ompletions based on the substru tural
hierar hy:
sequent al . (N
2
) hypersequent al . (P
3
)
ut elimination Ma Neille ompl. hyper-Ma Neille ompl.
Herbran's theorem anoni al ext. hyper anoni al ext.
It is residuated frames that onne t the two:
(proof theoreti argument)
+
= algebrai onstru tion.
Substru tural proof theory is full of burea ra y, but hopefully
there are still something good to be salvaged.
Con lusion
Introdu tion to
Substru tural Logi s
Algebrai Proof
Theory for
Substru tural Logi s
Herbrand's theorem
via hyper anoni al
extensions
Further topi s
Extra ting info
Interpolation
Density rule elim.
. Con lusion
55 / 55
We have explored the onne tion between proof theoreti
arugments and algebrai ompletions based on the substru tural
hierar hy:
sequent al . (N
2
) hypersequent al . (P
3
)
ut elimination Ma Neille ompl. hyper-Ma Neille ompl.
Herbran's theorem anoni al ext. hyper anoni al ext.
It is residuated frames that onne t the two:
(proof theoreti argument)
+
= algebrai onstru tion.
Substru tural proof theory is full of burea ra y, but hopefully
there are still something good to be salvaged.