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

2. The Model Evolution Calculus with Equality2 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)

3. The Model Evolution Calculus with Equality3 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

4. The Model Evolution Calculus with Equality4 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)

5. The Model Evolution Calculus with Equality5 DPLL procedure Input: Propositional clause set Output: Model or „unsatisfiable” Algorithm components: - Propositional semantic tree enumerates interpretations - Simplification - Split - Backtracking A : A B : C : B C N o, sp li t on C : f A, B, C gj = : A _ : B _ C _ D, ::: f A, B g ? j = : A _ : B _ C _ D, ::: Lifting to first-order logic?

6. The Model Evolution Calculus with Equality6 Model Evolution as First-Order DPLL Input: First-order clause set Output: 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 i sa " parame t er " no t qu i t e lik eavar i a bl e P ( v ) : P ( v ) : P ( a ) P ( a ) f P ( v ), : P ( a ) g ? j = P ( x ) _ Q( x )

7. The Model Evolution Calculus with Equality7 Interpretation Induced by a Branch A branch literal specifies the truth value of its ground instances unless a 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 ) N o, b ecause f : v, P ( v ), : P ( a ) g 6 j = P ( a ) _ Q( a ) ) S p li t w i t h Q( a ) t osa t i s f y P ( a ) _ Q( a ) Q( a ) B ranc h : f : v, P ( v ), : P ( a ) g I n d uce dI n t erpre t a t i on f or § = f a, b g t rue: P ( b ) f a l se: P ( a )Q( a )Q( b )

8. The Model Evolution Calculus with Equality8 Derivation Rules – Simplified (1) Split ¤ curren t con t ex t L ¾ L ¾ ¾ C _ L ©, C _ L curren t c l ausese t if 1. ¾ i sas i mu l t aneousmguo f C _ L aga i ns t ¤, 2. ne i t h er L ¾ nor L ¾ i scon t a i ne di n ¤, an d 3. L ¾ con t a i nsnovar i a bl es ¤`©, C _ L ¤, L ¾ `©, C _ L ¤, L ¾ `©, C _ L

9. The Model Evolution Calculus with Equality9 Derivation Rules – Simplified (2) Close if 1. ©6 = ; or C 6 = ? 2. t h ere i sas i mu l t aneousmgu ¾ o f C aga i ns t ¤ suc h t h a t ¤ con t a i ns t h ecomp l emen t o f eac hli t era l o f C ¾ ¤`©, C ¤`? ¤ curren t con t ex t ¾ ©, C curren t c l ausese t C ?

10. The Model Evolution Calculus with Equality10 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 `©

11. The Model Evolution Calculus with Equality11 Derivation Rules – Simplification Rules (2) 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

12. The Model Evolution Calculus with Equality12 Derivation Rules – Simplification Rules (3) Compact ¤, K, L `© ¤, K `© if 1. K con t a i nsnoparame t ers ( var i a bl es OK) 2. K ¾ = L, f orsomesu b s t i t u t i on ¾ Calculus - Derivations are trees over sequents - initial sequent - Fairness - Soundness and completeness : v ` I npu t c l ausese t

13. The Model Evolution Calculus with Equality13 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)

14. The Model Evolution Calculus with Equality14 Equality Reasoning in Instance Based Methods Inst-Gen [Ganzinger&Korovin CSL 2004]: ¾ ¾ proper i ns t an t i a t or ¾ Ã E - unsa t i s fi a bili t y

15. The Model Evolution Calculus with Equality15 Equality Reasoning in Instance Based Methods DCTP [Letz&Stenz Tableaux 02, Stenz 03]: ( ? s ¼ t L [ s 0 ] ( s ¼ t ) ¾ s 6 ¼ t L [ t ] ) ¾ Our approach: related ¾ = mg u ( s, s 0 ) s ¾ 6 ¹ t ¾

16. The Model Evolution Calculus with Equality16 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)

17. The Model Evolution Calculus with Equality17 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 L 1 _ ¢¢¢ _ L m ¢ l 1 ! r 1, :::, l n ! r n s 6 ¼ t _ C ¢ ¡ (C ¢ ¡ ) ¾ if ¾ = mg u ( s, t ) I n i t i a ll y C ¢ ; f oran i npu t c l ause C A ssumesre d uc t i onor d er i ng  l ¼ r L [ t ] _ C ¢ ¡ ( L [ r ] _ C ¢ ¡, l ! r ) ¾ if 8 < : ¾ = mg u ( l, t ) t i sno t avar i a bl e l ¾ 6 ¹ r ¾ Embed these rules in calculus

18. The Model Evolution Calculus with Equality18 Derivation Rules (1) - Reflection Ref ¤`©, s 6 ¼ t _ C ¢ ¡ ¤`©, s 6 ¼ t _ C ¢ ¡, (C ¢ ¡ ) ¾ if 1. ¾ i samguo f s an d t, 2. t h enewc l ause i sno t con t a i ne di n © [ f s 6 ¼ t _ C ¢ ¡ g

19. The Model Evolution Calculus with Equality19 Derivation Rules (2) - Paramodulation Para ¤, l ¼ r `©, L [ t ] _ C ¢ ¡ ¤, l ¼ r `©, L [ t ] _ C ¢ ¡, ( L [ r ] _ C ¢ ¡, l ! r ) ¾ 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)

20. The Model Evolution Calculus with Equality20 Derivation Rules (3) - Split Split ¤`©, C _ L ¤, L ¾ `©, C _ L ¤, L ¾ `©, C _ L

21. The Model Evolution Calculus with Equality21 Derivation Example Para Simp Split (left) I n i t i a l c l auseenco d es : P ( x, y ) _ Q( x ) _ R ( y ) : : v, P ( u, u ) ¼ t ` P ( x, y ) 6 ¼ t _ Q( x ) ¼ t _ R ( y ) ¼ t ¢ ; : v, P ( u, u ) ¼ t ` P ( x, y ) 6 ¼ t _ Q( x ) ¼ t _ R ( y ) ¼ t ¢ ;, t 6 ¼ t _ Q( x ) ¼ t _ R ( x ) ¼ t ¢ P ( x, x ) ! t : v, P ( u, u ) ¼ t ` P ( x, y ) 6 ¼ 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 ) 6 ¼ t _ Q( x ) ¼ t _ R ( x ) ¼ t ¢ ;, Q( x ) ¼ t _ R ( x ) ¼ t ¢ P ( x, x ) ! t

22. The Model Evolution Calculus with Equality22 Derivation Rules (4) Close ¤`©, C ¤`? Cl oseapp li es t o C ¢ l 1 ! r 1, :::, l n ! r n U seconvers i on t oor di naryc l ause C _ l 1 6 ¼ r 1 _ ¢¢¢ _ l n 6 ¼ r n an d or di nary Cl ose:

23. The Model Evolution Calculus with Equality23 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 ¢ ;

24. The Model Evolution Calculus with Equality24 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 ¢ ¡ ¤`©, C 0 ¢ ¡ 0 if 8 > < > : C ¢ ¡ i sre d un d an t wr t. © [ f C 0 ¢ ¡ 0 g an d ¤, an d ``S oun d nesscon di t i on '' f( x ) 6 ¼ x ` a ¼ b ¢ f( a ) ! a ¡ ! f( x ) 6 ¼ x ` t ¼ t ¢ ; f( x ) ¼ x ` a ¼ b ¢ f( a ) ! a ¡ ! f( x ) ¼ x ` a ¼ b ¢ ;

25. The Model Evolution Calculus with Equality25 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)

26. The Model Evolution Calculus with Equality26 Model Evolution Calculus with Equality – How it Works Without Equality With Equality - C urren t con t ex t ¤ - C an did a t e M o d e l I ¤ - C urren t c l ause C - If I ¤ 6 j = C t h enrepa i r I ¤ (S p li t ) org i veup I ¤ (Cl ose ) R ¤ ?

27. The Model Evolution Calculus with Equality27 E-Interpretation Induced by a Branch : v b ¼ c b 6 ¼ c a 6 ¼ u a ¼ u a ¼ c a 6 ¼ c (1) R ¤ : = ; (2) can did a t e b ¼ c : R ¤ : = f b ! c g (3) can did a t e a ¼ b : R ¤ = f b ! c g I n i t i a ll y R ¤ : = ; F ora ll s ¼ t 2 I ¤, sma ll erequa t i ons fi rs t : if s  t an d s an d t are i rre d uc ibl e b ysma ll erru l es i n R ¤ t h en R ¤ : = R ¤ [ f s ! t g C an did a t eequa t i ons I ¤ = f a : ¼ b, b : ¼ c g O r d er i ng a  b  c Example I mpor t an t : R ¤ i sconvergen t

28. The Model Evolution Calculus with Equality28 Repairing the Candidate Model R ¤ 6 j = E C ° s 6 = t C ° # R ¤ = q 6 ¼ q _ ¢¢¢ _ s ¼ t _ ¢¢¢ Calculus: U se d rewr i t eru l es f rom R ¤ : l ! r, ::: lifted versions R ¤ 6 j = E ©, C R ¤ 6 j = E C ° # R ¤ (G roun d) cons t ra i ne d c l ausesseman t i cs: R ¤ j = E C ¢ ¡ iff ¡6 µ R ¤ or R ¤ j = E C S p li t w i t h s 0 ¼ t 0 t oa dd s ¼ t t o R ¤, or S p li t w i t h l 0 6 ¼ r 0 t oremove l ! r f rom R ¤ C ¢ ; ) ? P ara, R e f s 0 ¼ t 0 _ ¢¢¢¢ l 0 ! r 0, ::: Af t er S p li t C ° # R ¤ w illb e E - sa t i s fi e d, an d sow illb e C °

29. The Model Evolution Calculus with Equality29 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 l a t erpar t i a ll yre t rac t e dd ue t o f( a ) 6 ¼ a

30. The Model Evolution Calculus with Equality30 Limit Derivations : v ` I npu t c l ausese t ¤ 1 `© 1 ¤ 2 `© 2 ¤ n `© n ¤ 1 : = S i ¸ 0 T j ¸ i ¤ j © 1 : = S i ¸ 0 T j ¸ i © j c l ose d C omp l e t eness S upposea f a i r d er i va t i on t h a t i sno t ac l ose d t ree Sh ow t h a t R ¤ 1 j = © 1 Li m i t rewr i t es y s t em R ¤ 1 - Thi s i s t h e i n t en d e d mo d e l - A pprox i ma t i onsuse di n re d un d ancy t es t s Fairness?

31. The Model Evolution Calculus with Equality31 Fairness

32. The Model Evolution Calculus with Equality32 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 candidate at 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?)