The Model Evolution Calculus with Equality Peter Baumgartner Max-Planck-Institute for Informatics Saarbrücken, Germany Cesare Tinelli The University of

The Model Evolution Calculus with Equality

Peter Baumgartner

Max-Planck-Institute for Informatics

Saarbrücken, Germany

Cesare Tinelli

The University of Iowa

Iowa, USA

The Model Evolution Calculus with Equality 2

Background – Instance Based Methods

• Model Evolution is related to Instance Based Methods

– Ordered Semantic Hyper Linking [Plaisted et al]

– Primal Partial Instantiation [Hooker et al]

– Disconnection Method [Billon], DCTP [Letz&Stenz]

– Inst-Gen [Ganzinger&Korovin]

– Successor of First-Order DPLL [B.]

• Principle: Reduce proof search in first-order (clausal) logic to propositional logic in an „intelligent“ way

• Different to Resolution, Model Elimination,… (Pro‘s and Con‘s)


Background – Model Evolution

• The best modern SAT solvers (satz, MiniSat, zChaff, Berkmin,…) are based on the Davis-Putnam-Logemann-Loveland procedure [DPLL 1960-1963]

• Can DPLL be lifted to the first-order level?How to combine

– successful SAT techniques (unit propagation, backjumping, lemma learning,…)

– successful first-order techniques?(unification, redundancy concepts, ...)?

• Realization in Model Evolution calculus / Darwin implementation

– Basic approach developed (CADE-19)

– Lemma learning on the way

– This work: built-in equality reasoning


Contents

• DPLL as a starting point for the Model Evolution calculus

• Model Evolution calculus without equality

– Model construction

– Inference rules

• Equality reasoning

– Equality reasoning in instance based methods

– Inference rules

– How it works (semantical considerations)


DPLL procedure

Input: Propositional clause setOutput: Model or „unsatisfiable”

Algorithm components: - Propositional semantic tree enumerates interpretations - Simplification - Split - Backtracking

A : A

B

: C

: B

C

No, split on C: f A, B, Cg j= : A _ : B _ C _ D, : : :

f A, Bg?j= : A _ : B _ C _ D, : : :

Lifting to first-order logic?


Model Evolution as First-Order DPLL

Input: First-order clause setOutput: Model or „unsatisfiable”

if termination

Interpretation induced by a branch?

Lifing of semantic tree data structure and derivation rules to first-order

Algorithm components: - First-order semantic tree enumerates interpretations - Simplification - Split - Backtracking

v isa "parameter"not quite like a variable

P(v) : P(v)

: P(a) P(a)

f P(v), : P(a)g?j= P(x) _ Q(x)


Interpretation Induced by a BranchA branch literal specifies the truth value of its ground instances unlessa more specific branch literal specifies the opposite truth value

Calculus?

: v

P(v) : P(v)

: P(a) P(a)

Q(a) : Q(a)

f : v, P(v), : P(a)g?j= P(x) _ Q(x)

No, because f : v, P(v), : P(a)g 6j= P(a) _ Q(a)

) Split with Q(a) to satisfy P(a) _ Q(a)

Q(a)

Branch:f : v, P(v), : P(a)g

Induced Interpretation for § = f a, bgtrue: P(b)

false: P(a) Q(a) Q(b)


Derivation Rules – Simplified (1)

Split

¤ current contextL¾ L¾

¾ C _ L

©, C _ L current clause set

if

1. ¾isa simultaneousmgu of C _ L against ¤,2. neither L¾nor L¾iscontained in ¤, and3. L¾containsno variables

¤ ` ©, C _ L

¤, L¾ ` ©, C _ L ¤, L¾ ` ©, C _ L


Derivation Rules – Simplified (2)

Close

if

1. ©6= ; or C 6= ?2. there isa simultaneousmgu ¾of C against ¤ such that¤ contains the complement of each literal of C¾

¤ ` ©, C

¤ ` ?

¤ current context

¾

©, C current clause set

C

?


Derivation Rules – Simplification Rules (1)

Subsume

Propositional level:

First-order level ¼ unit subsumption:- L contains no parameters (variables OK)- Matching instead of syntactic equality

¤, L ` ©, L _ C

¤, L ` ©



Resolve

Propositional level:

First-order level ¼ restricted unit resolution- L contains no parameters (variables OK)- Unification instead of syntactic equality- The unifier must not modify C

¤, L ` ©, L _ C

¤, L ` ©, C



Compact¤, K, L ` ©

¤, K ` ©

if

1. K containsno parameters (variablesOK)2. K¾= L, for some substitution ¾

Calculus- Derivations are trees over sequents- initial sequent - Fairness- Soundness and completeness

: v ` Input clause set


Contents




– Inference rules



– Inference rules



Equality Reasoning in Instance Based Methods

Inst-Gen [Ganzinger&Korovin CSL 2004]:

¾

¾proper instantiator

¾ Ã E-unsatisfiability


Equality Reasoning in Instance Based Methods

DCTP [Letz&Stenz Tableaux 02, Stenz 03]:

(?

s ¼ t

L[s0]

( s ¼ t )¾

s 6¼ t L[t] )¾

Our approach: related

¾= mgu(s, s0)s¾6¹ t¾


Contents




– Inference rules



– Inference rules



Model Evolution Calculus with Equality - Overview

- Split and Close: same- Simplification rules: general rule based on redundancy concept- Clauses: with constraints now: -Equality reasoning: Reflection and (Ordered) Paramodulation rules

Paramodulation

Reflection

L1 _ ¢¢¢_ Lm ¢l1 ! r1, :::, ln ! rn

s 6¼ t _ C ¢¡(C ¢¡ )¾

if ¾= mgu(s, t)

Initially C ¢; for an input clause C

Assumesreduction ordering Â

l ¼ r L[t] _ C ¢¡(L[r] _ C ¢¡ , l ! r)¾

if

8<

:

¾= mgu(l, t)t isnot a variablel¾6¹ r¾

Embed these rules in calculus


Derivation Rules (1) - Reflection

Ref¤ ` ©, s 6¼ t _ C ¢¡

¤ ` ©, s 6¼ t _ C ¢¡ , (C ¢¡ )¾

if

1. ¾isa mgu of s and t,2. the new clause isnot contained in ©[ f s 6¼ t _ C ¢¡g


Derivation Rules (2) - Paramodulation

Para¤, l ¼ r ` ©, L[t] _ C ¢¡

¤, l ¼ r ` ©, L[t] _ C ¢¡ , (L[r] _ C ¢¡ , l ! r)¾

if

1. ¾isa mgu of l and t,2. t isnot a variable,3. l¾6¹ r¾,4. the new clause containsno parameters, and5. the new clause isnot contained in ©[ f L _ C ¢¡g

NB – This is not a resolution calculus:- Paramodulation only from unit equations- Clause part does not grow in length, and no paramodulation into constrained part- (No paramodulation from context equations into context literals)


Derivation Rules (3) - Split

Split¤ ` ©, C _ L

¤, L¾ ` ©, C _ L ¤, L¾ ` ©, C _ L

Split applies to

C ¢l1 ! r1, : : : , ln ! rn

only if C isa positive clause

Use conversion to ordinaryclause

C _ l1 6¼ r1 _ ¢¢¢_ ln 6¼ rn

and ordinary Split:


Derivation Example

Para

Simp

Split (left)

Initial clause encodes : P(x, y) _ Q(x) _ R(y):

: v, P(u, u) ¼ t ` P(x, y) 6¼ t _ Q(x) ¼ t _ R(y) ¼ t ¢;

: v, P(u, u) ¼ t `P(x, y) 6¼ t _ Q(x) ¼ t _ R(y) ¼ t ¢; ,

t 6¼ t _ Q(x) ¼ t _ R(x) ¼ t ¢P(x, x) ! t

: v, P(u, u) ¼ t `P(x, y) 6¼ t _ Q(x) ¼ t _ R(x) ¼ t ¢; ,

Q(x) ¼ t _ R(x) ¼ t ¢P(x, x) ! t

: v, P(u, u) ¼ t,Q(u) ¼ t

`P(x, y) 6¼ t _ Q(x) ¼ t _ R(x) ¼ t ¢; ,

Q(x) ¼ t _ R(x) ¼ t ¢P(x, x) ! t


Derivation Rules (4)

Close¤ ` ©, C

¤ ` ?

Close applies to

C ¢l1 ! r1, : : : , ln ! rn

Use conversion to ordinaryclause

C _ l1 6¼ r1 _ ¢¢¢_ ln 6¼ rn

and ordinary Close:


Optional Derivation Rules (1)

Assert if L is not subsumed by a context literal and "soundness condition" holds

Examples

No Split for unit clauses:

With equality reasoning:

¤ ` ©

¤, L ` ©

¤ ` ©, P(x) ¢;¡ ! ¤, P(x) ` ©, P(x) ¢;

P(u, b), b ¼ c ` : P(x, y) _ f(x) ¼ y ¢;¡ ! P(u, b), b ¼ c, f(u) ¼ c ` : P(x, y) _ f(x) ¼ y ¢;


Optional Derivation Rules (2)

Simp

Examples

Delete a clause whose constraint will never be satisfied:

Simplify constraint:

Generic Simp rule covers most simplification rules so far as special cases

¤ ` ©, C ¢¡

¤ ` ©, C0 ¢¡ 0if

8><

>:

C ¢¡ is redundant wrt.©[ f C0 ¢¡ 0g and ¤, and

`̀Soundnesscondition''

f(x) 6¼ x ` a ¼ b ¢f(a) ! a¡ ! f(x) 6¼ x ` t ¼ t ¢;

f(x) ¼ x ` a ¼ b ¢f(a) ! a¡ ! f(x) ¼ x ` a ¼ b ¢;


Contents




– Inference rules



– Inference rules



Model Evolution Calculus with Equality – How it Works

Without Equality

With Equality

- Current context ¤

- Candidate Model I¤

- Current clause C

- If I¤ 6j= C then repair I¤ (Split) or give up I¤ (Close)

R¤ ?

- Current context ¤

- Candidate E-model R¤ -- a ground rewrite system

- Current clause C

- If R¤ 6j=E C then repair R¤ (Split) or give up R¤ (Close)


E-Interpretation Induced by a Branch

: v

b ¼ c b 6¼ c

a 6¼ ua ¼ u

a ¼ ca 6¼ c

(1) R¤ := ;(2) candidate b ¼ c: R¤ := f b ! cg(3) candidate a ¼ b: R¤ = f b ! cg

Initially R¤ := ;

For all s ¼ t 2 I¤, smaller equations first:if s Â t and

s and t are irreducible by smaller rules in R¤then R¤ := R¤ [ f s ! tg

Candidate equations I¤ = f a:¼ b, b

:¼ cg

Ordering a Â b Â c

Example

Important: R¤ is convergent


Repairing the Candidate Model

R¤ 6j=E C°

s 6= t

C° #R¤ = q 6¼ q _ ¢¢¢_ s ¼ t _ ¢¢¢

Calculus:

Used rewrite rules from R¤:l ! r, : : :

lifted versions

R¤ 6j=E ©, C R¤ 6j=E C° #R¤

(Ground) constrained clausessemantics:R¤ j=E C ¢¡ iff ¡ 6µ R¤ or R¤ j=E C

Split with s0 ¼ t0 to add s ¼ t to R¤, orSplit with l0 6¼ r0 to remove l ! r from R¤

C ¢; ) ?Para,Ref s0 ¼ t0_ ¢¢¢¢l0 ! r0, : : :

After Split C° #R¤ will be E-satisfied, and so will be C°


Model Construction Considerations

• The model construction technique has been developed for the Superposition calculus [Bachmair&Ganzinger]

• Differences due to parametric literals

– Nonmonotonicity:

– Have to work with two orderings: term ordering and instantiation ordering

– Model construction:smaller sides of equations must be irreducible, too, in order to be turned into rewrite rules

– In consequence, paramodulation into smaller sides is necessary (really?)

e.g. f(u) ¼ u later partially retracted due to f(a) 6¼ a


Limit Derivations

: v ` Input clause set

¤1 ` ©1

¤2 ` ©2

¤n ` ©n

¤1 :=S

i¸ 0

Tj¸ i ¤j

©1 :=S

i¸ 0

Tj¸ i ©j

closedCompletenessSuppose a fair derivationthat isnot a closed tree

Show that R¤1 j= ©1

Limit rewrite system R¤1- This is the intended model- Approximationsused in

redundancy tests

Fairness?


FairnessDef. (Fairness)

Para Suppose timepoint i in derivation such that

Para¤i, l ¼ r ` ©i, C ¢¡

¤i, l ¼ r ` ©i, C ¢¡ , C0 ¢¡ 0

where C ¢¡ ) Para(l¼r,¾) C0 ¢¡ 0

If1. l ¼ r 2 ¤B,2. ¤B produces (l ¼ r)¾, and3. (C ¢¡ )¾isnot redundant wrt.©i [ f C ¢¡g and R¤B

thenthere isa j such that the inferenceC ¢¡ ) Para(l¼r,¾) C0 ¢¡ 0 is redundant wrt. ©j and R¤B

Split, Ref, Close...


Conclusions

• Main result: soundness and refutational completeness

• Nice features (perhaps):

– Paramodulation only from unit equations

– No paramodulation inferences between context equations or

into constraint part

– Clause part of constrained clauses does not grow in length(decide Bernays-Schoenfinkel clauses with equality)

– Works with explicitly represented model candidateat the calculus level (the context)

• Not so nice features (perhaps):

– Semantic redundancy criterion based on model candidate difficult to exploit

– Need paramodulation into smaller sides of equations (really?)

Documents

The Model Evolution Calculus with Equality Peter Baumgartner Max-Planck-Institute for Informatics Saarbrücken, Germany Cesare Tinelli The University of