Dragisa Zunic - Classical computing with explicit structural rules - the *X calculus

Sequent calculus, proofs and programsThe ∗X calculus : explicit erasure and duplication

Explicit vs implicit : ∗X vs XStrong normalisation

Classical computing with explicit structural rules–the ∗X calculus–

Dragisa Zunic– together with: Silvia Ghilezan and Pierre Lescanne

Mathematical Institute SANU, BelgradeMatematicki Institut Srpske Akademije Nauka i Umetnosti, Beograd

November 19, 2012.

Dragisa Zunic Classical computing with explicit structural rules 1 / 50



Outline

1 Sequent calculus, proofs and programs

2 The ∗X calculus : explicit erasure and duplication

3 Explicit vs implicit : ∗X vs X

4 Strong normalisation




Implicit structural rules – the X calculusExplicit structural rules – the ∗X calculus

Outline

1 Sequent calculus, proofs and programsImplicit structural rules – the X calculusExplicit structural rules – the ∗X calculus

2 The ∗X calculus : explicit erasure and duplicationLogical settingFrom sequent proofs to termsThe syntax and reduction rulesDiagrammatic view







The framework

Classical logic

Sequent calculus of G.Gentzen (two important formalisms :sequent calculus and natural deduction calculus)

The Curry-Howard correspondence (the relation between prooftheory and the programming language theory)

The goal is to study classical computation, by assigningcomputational interpretation(s) to classical logic represented in thesequent calculus.





The Curry-Howard correspondence

Reveals a strong connection between logic and computation

The intuitionistic logic and simply typed λ-calculus

Proofs ⇔ TermsPropositions ⇔ Types

Normalization ⇔ Reduction

Extending the “Curry-Howard paradigm”

Classical logic : T. Griffin (1990)

M. Parigot (1992) : λµ-calculus (natural deduction)H. Herbelin (1995-2005) : λµµ-calculus (sequent calculus)





Related workThe two predecessors

Classical logic : the X calculusC. Urban (2000)S. Lengrand (2003)S. van Bakel, S.Lengrand, P. Lescanne (2005)

Intuitionistic logic : the λlxr-calculusD. Kesner, S. Lengrand (2005)

X λlxr

∗X





G3 sequent system for classical logic X -calculus

(axiom)Γ,A ` A,∆

Γ ` A,∆ Γ,B ` ∆(→ left)

Γ,A→ B ` ∆

Γ,A ` B,∆(→ right)

Γ ` A→ B,∆

Γ ` A,∆ Γ,A ` ∆(cut)

Γ ` ∆

Contexts Γ,∆ are sets

Context-sharing style

No structural rules





G1 sequent system for classical logic ∗X calculus

(axiom)A ` A

Γ ` A,∆ Γ′,B ` ∆′

(→ left)Γ, Γ′,A→ B ` ∆,∆′


Γ ` A→ B,∆

Γ ` A,∆ Γ′,A ` ∆′

(cut)Γ, Γ′ ` ∆,∆′

Γ ` ∆(left weakening)

Γ,A ` ∆

Γ ` ∆(right weakening)

Γ ` A,∆

Γ,A,A ` ∆(left contraction)

Γ,A ` ∆

Γ ` A,A,∆(right contraction)

Γ ` A,∆




Logical settingFrom sequent proofs to termsThe syntax and reduction rulesDiagrammatic view

Outline









Computational interpretation of classical proofs

Sequent calculus with explicit structural rules(weakening and contraction)

1) Terms in ∗X -calculus are in fact annotations for proofs,2) Computation in ∗X -calculus corresponds to cut-elimination

Weakening as an eraser / Contraction as a duplicator





Names

The terms are built from names.

Two categories of names : x , y , z ... in-namesα, β, γ... out-names

Binders wear “hats” : x , y , z ... α, β, γ...

Examples :

〈x .α〉 x 〈x .β〉 β . α [P〉βγ >α





Name 6= Variable

In λ-calculus : variables

(λy .xyz)Mβ−→ xMz

An arbitrary term M substitutes the variable y

In ∗X -calculus : names

A name can never be substituted for a termA name can only be renamed





G1 sequent system for classical logic

(axiom)A ` A

Γ ` A,∆ Γ′,B ` ∆′

(→ left)Γ, Γ′,A→ B ` ∆,∆′


Γ ` A→ B,∆

Γ ` A,∆ Γ′,A ` ∆′

(cut)Γ, Γ′ ` ∆,∆′

Γ ` ∆(left weakening)

Γ,A ` ∆

Γ ` ∆(right weakening)

Γ ` A,∆

Γ,A,A ` ∆(left contraction)

Γ,A ` ∆

Γ ` A,A,∆(right contraction)

Γ ` A,∆





The terms correspond to proofs :

(caps)〈x .α〉 :. x : A ` α : A

P :. Γ ` α : A,∆ Q :. Γ′, x : B ` ∆′

(imp)Pα [y ] xQ :. Γ, Γ′, y : A→ B ` ∆,∆′

P :. Γ, x : A ` α : B,∆(exp)

x P α . β :. Γ ` β : A→ B,∆

P :. Γ ` α : A,∆ Q :. Γ′, x : A ` ∆′

(cut)Pα † xQ :. Γ, Γ′ ` ∆,∆′

P :. Γ ` ∆(left eraser)

x � P :. Γ, x : A ` ∆

P :. Γ ` ∆(right eraser)

P � α :. Γ ` α : A,∆

P :. Γ, y : A, z : A ` ∆(left dupl.)

x< yz〈P] :. Γ, x : A ` ∆

P :. Γ ` β : A, γ : A,∆(right dupl.)

[P〉βγ >α :. Γ ` α : A,∆Dragisa Zunic Classical computing with explicit structural rules 14 / 50





(caps)〈x .α〉 :. x : A ` α : A

P :. Γ ` α : A,∆ Q :. Γ′, x : B ` ∆′

(imp)Pα [y ] xQ :. Γ, Γ′, y : A→ B ` ∆,∆′

P :. Γ, x : A ` α : B,∆(exp)

x P α . β :. Γ ` β : A→ B,∆

P :. Γ ` α : A,∆ Q :. Γ′, x : A ` ∆′

(cut)Pα † xQ :. Γ, Γ′ ` ∆,∆′


x � P :. Γ, x : A ` ∆


P � α :. Γ ` α : A,∆

P :. Γ, y : A, z : A ` ∆(left dupl.)

x< yz〈P] :. Γ, x : A ` ∆







(caps)〈x .α〉 :. x : A ` α : A

P :. Γ ` α : A,∆ Q :. Γ′, x : B ` ∆′

(imp)Pα [y ] xQ :. Γ, Γ′, y : A→ B ` ∆,∆′

P :. Γ, x : A ` α : B,∆(exp)

x P α . β :. Γ ` β : A→ B,∆

P :. Γ ` α : A,∆ Q :. Γ′, x : A ` ∆′

(cut)Pα † xQ :. Γ, Γ′ ` ∆,∆′


x � P :. Γ, x : A ` ∆


P � α :. Γ ` α : A,∆

P :. Γ, y : A, z : A ` ∆(left dupl.)

x< yz〈P] :. Γ, x : A ` ∆







(caps)〈x .α〉 :. x : A ` α : A

P :. Γ ` α : A,∆ Q :. Γ′, x : B ` ∆′

(imp)Pα [y ] xQ :. Γ, Γ′, y : A→ B ` ∆,∆′

P :. Γ, x : A ` α : B,∆(exp)

x P α . β :. Γ ` β : A→ B,∆

P :. Γ ` α : A,∆ Q :. Γ′, x : A ` ∆′

(cut)Pα † xQ :. Γ, Γ′ ` ∆,∆′


x � P :. Γ, x : A ` ∆


P � α :. Γ ` α : A,∆

P :. Γ, y : A, z : A ` ∆(left dupl.)

x< yz〈P] :. Γ, x : A ` ∆







(caps)〈x .α〉 :. x : A ` α : A

P :. Γ ` α : A,∆ Q :. Γ′, x : B ` ∆′

(imp)Pα [y ] xQ :. Γ, Γ′, y : A→ B ` ∆,∆′

P :. Γ, x : A ` α : B,∆(exp)

x P α . β :. Γ ` β : A→ B,∆

P :. Γ ` α : A,∆ Q :. Γ′, x : A ` ∆′

(cut)Pα † xQ :. Γ, Γ′ ` ∆,∆′


x � P :. Γ, x : A ` ∆


P � α :. Γ ` α : A,∆

P :. Γ, y : A, z : A ` ∆(left dupl.)

x< yz〈P] :. Γ, x : A ` ∆






The ∗X calculusThe syntax

P, Q ::= 〈x .α〉 capsule

| x P β . α exporter

| Pα [y ] xQ importer

| Pα † xQ cut

| x � P left-eraser

| P � α right-eraser

| x<yz〈P] left-duplicator

| [P〉βγ >α right-duplicator





Linearity

In ∗X -calculus only linear terms are considered :

� every free name occurs only once� every binder does bind an occurrence of a free name

(and therefore only one)

Examples of non-linear terms :

x 〈y .β〉 β . α and 〈x .α〉 � α

Every non-linear term has a linear representation :

x (x�〈y .β〉) β . α and [〈x .α1〉 � α2〉α1α2>α





Linearity





x 〈y .β〉 β . α and 〈x .α〉 � α







Linearity





x 〈y .β〉 β . α and 〈x .α〉 � α







Linearity





x 〈y .β〉 β . α and 〈x .α〉 � α







Linearity





x 〈y .β〉 β . α and 〈x .α〉 � α







The ∗X calculusGrouping the rules :

Logical rules (L-principal names involved)

Structural rules (S-principal names involved)

(Activation rules)

(Deactivation rules)

Propagation rules









| Pα † xQ cut





The notion of principal name of a term.1 L-principal name2 S–principal name









| Pα † xQ cut





The notion of principal name of a term.1 L–principal name2 S-principal name





∗X : logical rules

Renaming and inserting

Renaming :

(ren-L) : 〈y .α〉α † xQ → Q{y/x}

(ren-R) : Pα † x〈x .β〉 → P{β/α}

Diagrammatically :

(ren-L) :y

Qy α x Q

(ren-R) : Pxα β

Pβ





∗X : logical rules

Inserting (ei-insert) :

(y P β . α)α † x(Qγ [x ] zR) → either

{(Qγ † yP)β † zR

Qγ † y(Pβ † zR)

Diagrammatically :

z

IQ R

γ

Ry β

PQ γ z

y

E

α x

βP

Notice : logical rules define reducing when a cut bindsL-principal names





∗X : structural rules

Left erasure :

( † -eras) : (P � α)α † xQ → IQ � P �OQ ,

where IQ = I (Q) \x , OQ = O(Q)

Diagrammatically :

IQ

O} Q

IQ{

P{O} Q

P x Qα





∗X : structural rules

Left duplication ( † -dupl) :

([P〉α1

α2>α)α † xQ → IQ<

IQ1

IQ2

⟨(Pα1

† x1Q1)α2† x2Q2

⟩ OQ1

OQ2

>OQ ,

Diagrammatically :

P

IQ{

O} QQα x

β

γ

PQ

1

Q2

IQ{

O} Q

β

γ x

x





Symmetry...The † -erasure

Q

QPα

O

I

I

O

P

P

P

P

{ }

}

{ x

An illustration of the symmetry...





Symmetry...The † -duplication

PI {

OP}

P Qα

y

z

x

PI {

OP}

QP

P1

2

α

α

y

z

An illustration of the symmetry...





Duplication, informally

Classical Logic Intuitionistic Logic

Component duplicated, interface preserved.

Similarly for erasure





∗X : propagation rules (left subgroup)

(exp † − prop) : (x P γ . α)β † yR → x (Pβ † yR) γ . α

(imp † − prop1) : (Pα [x] zQ)β † yR → (Pβ † yR)α [x] zQ, β ∈ O(P)

(imp † − prop2) : (Pα [x] zQ)β † yR → Pα [x] z(Qβ † yR), β ∈ O(Q)

(cut(c)† − prop) : (Pα † x〈x.β〉)β † yR → Pα † yR

(cut † − prop1) : (Pα † xQ)β † yR → (Pβ † yR)α † xQ, β ∈ O(P), Q 6= 〈x.β〉

(cut † − prop2) : (Pα † xQ)β † yR → Pα † x(Qβ † yR), β ∈ O(Q), Q 6= 〈x.β〉

(L–eras † − prop) : (x � M)β † yR → x � (Mβ † yR)

(R–eras † − prop) : (M � α)β † yR → (Mβ † yR)� α, α 6= β

(L–dupl † − prop) : (x< x1x2〈M])β † yR → x< x1

x2〈Mβ † yR]

(R–dupl † − prop) : ([M〉α1α2>α)β † yR → [Mβ † yR〉α1

α2>α, α 6= β

Propagation rules do not have a diagrammatic representation





The ∗X calculusPropagation rules

^α x^ Qα






^α x^ Qα






^α x^ Qα






^α x^ Qα






^α x^ Qα






^α x^ Qα

We may think of α † xQ as an explicit substitution





∗X : the basic properties

1 Linearity preservation

If P is linear and P → Q then Q is linear

2 Free names (“interface”) preservation

If P → Q then N(P) = N(Q)

3 Type preservation – computation ——–can be (has to be) seen asproof-transformation

If P :. Γ ` ∆ and P → P ′, then P ′ :. Γ ` ∆

4 Strong normalisation (termination of reduction)





Peirce’s law in ∗X(capsule)

〈x.α1〉 :. x : A ` α1 : A

(R − eraser)

〈x.α1〉 � β :. x : A ` α1 : A, β : B

(exporter)

x (〈x.α1〉 � β) β . γ :. ` α1 : A, γ : A→ B

(capsule)

〈y.α2〉 :. y : A ` α2 : A

(importer)

(x (〈x.α1〉 � β) β . γ) γ [z] y 〈y.α2〉 :. z : (A→ B)→ A ` α1 : A, α2 : A

(R − duplicator)

[(x (〈x.α1〉 � β) β . γ) γ [z] y 〈y.α2〉〉α1α2>α :. z : (A→ B)→ A ` α : A

(exporter)

z ([(x (〈x.α1〉 � β) β . γ) γ [z] y 〈y.α2〉〉α1α2>α) α . δ :. ` δ : ((A→ B)→ A)→ A

(axiom)

A ` A

(R − weakening)

A ` A, B

(→ R)

` A, A→ B

(axiom)

A ` A

(→ L)

(A→ B)→ A ` A, A

(R − contraction)

(A→ B)→ A ` A

(→ R)

` ((A→ B)→ A)→ A





*X :(axiom)

A ` A

(R − weakening)

A ` A, B

(→ R)

` A, A→ B

(axiom)

A ` A

(→ L)

(A→ B)→ A ` A, A

(R − contraction)

(A→ B)→ A ` A

(→ R)

` ((A→ B)→ A)→ A

X :

(axiom)

A ` A, B

(→ R)

` A, A→ B

(axiom)

A ` A

(→ L)

(A→ B)→ A ` A

(→ R)

` ((A→ B)→ A)→ A





The terms of ∗X are convenient for two-dimensional representation,due to

the presence of erasers and duplicators

but also the linearity constraints





The diagrammatic calculusThe syntax

P Q αx

P

E

x β

α

PI

Qα y

x

α

β

γP

z

x

y P

P Qα x

αP x

P

::=,





The main source of non-confluence

P QI O

P

Q Q

and

P Q QI O

P P

non-confluence is a feature of sequent calculus

an intrinsic property of classical computation

the “choice” represented by activation rules

in literature this is known as “Lafont’s example”





An exampleThe Peirce’s law

E

E

I





An exampleThe Peirce’s law - with types assigned

E

E

((A B) A)

(A

Aα1 :

I

B) A

Aδ:

z:

α: A

γ: A B

y: A

Ax:

Bβ: Aα :2




Outline








Encoding ∗X in X calculus

Definition (Encoding)




Encoding preserves types

Lemma (Type preservation)

For an arbitrary ∗X -term




Simulating ∗X reduction

Theorem (Reduction simulation)




Outline








Strong normalisation of ∗X -calculus

Theorem (SN)




Future work

define equivalent terms, the c©X calculus

formalize the diagrammatic calculus (capturing the essence ofcomputation)

a 3d computational model ?

possible applications of non-confluent systems ?




Some references

� S. Ghilezan, P. Lescanne, D. Zunic

Comp. interpretation of classical logic with expl. struct. rules, 2012.

� S. Ghilezan, J. Ivetic, P. Lescanne, D. Zunic

Intuitionistic sequent-style calculus with explicit struct. rules, 2010.

� D. Kesner and S. Lengrand

Explicit operators for λ-calculus, 2006.

� C. Urban and G. M. Bierman

Strong normalisation of cut-elimination in classical logic, 1999.

� S. van Bakel, S. Lengrand, P. Lescanne

The language X . Circuits, computations and classical logic, 2005.

� E. Robinson

Proof nets for classical logic, 2005.




Thank you. Hvala na paznji !


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Dragisa Zunic - Classical computing with explicit structural rules - the *X calculus