SUPERPOSITION FOR LAMBDA-FREE HIGHER-ORDER LOGIC Motivation: - PowerPoint PPT Presentation

� 1 ALEXANDER BENTKAMP JASMIN BLANCHETTE SIMON CRUANES UWE WALDMANN SUPERPOSITION FOR 
 LAMBDA-FREE HIGHER-ORDER LOGIC

Motivation: Sledgehammer � 2 Proof goal 
 from Isabelle Fact selection Translation to FOL A complete HO superposition LEO-II/III Satallax prover Superposition provers SMT provers Proof reconstruction Proof text 
 in Isabelle

DESIGN PRINCIPLE: BE GRACEFUL � 3 HO superposition on first-order problems should coincide with FO superposition

Our way to higher-order superposition � 4 HOL predicate-free HOL boolean formulas 
 nested in terms λ -free HOL / λ -expressions / 
 applicative FOL comprehension 
 axioms partial application 
 FOL & applied variables

Translation to FOL: applicative encoding � 5 f ( H f) app(f, app( H , f)) is translated to λ -free HOL FOL NOT GRACEFUL!

Term orders for λ -free HOL � 6 Compatibility with arguments? 
 t > s ⇒ t u > s u KBO without argument 
 KBO with argument 
 coefficients coefficients LPO Yes: 
 No: 
 Petar’s talk This talk

The superposition rule � 7 D ∨ t = t’ C ∨ (¬) s[u] = s’ σ = mgu(t,u) (D ∨ C ∨ (¬) s[t’] = s’) σ + order conditions

Superposition only at argument subterms � 8 f a (h b c) Argument subterms: f a (h b c) Prefix subterms: g = f g a ≠ b S UP f a ≠ b

Argument congruence rule � 9 C ∨ t = s A RG C ONG C ∨ t X = s X Example: g = f A RG C ONG g X = f X g a ≠ b S UP f a ≠ b

Argument congruence rule � 10 C ∨ t = s A RG C ONG C ∨ t X = s X BUT ISN’T THIS RULE ALWAYS REDUNDANT?

Floor encoding � 11 Encode ground λ -free HOL terms into FOL: ⎣ f ⎦ = f 0 ⎣ f a ⎦ = f 1 (a 0 ) Redundancy is defined with respect to this encoding.

Floor encoding � 12 Example: g 0 = f 0 g = f A RG C ONG g 1 a 0 = f 1 a 0 g X = f X Not redundant!

What changes in the proof? � 13 Refutational completeness: 
 Let N be saturated up to redundancy, ⊥∉ N. 
 Then N has a model. Proof sketch for FOL: N model of N G(N) model of G(N) model construction

What changes in the proof? � 14 Refutational completeness: 
 Let N be saturated up to redundancy, ⊥∉ N. 
 Then N has a model. Proof sketch for λ -free HOL: N model of N G(N) model of G(N) ⎣ G(N) ⎦ model of ⎣ G(N) ⎦ model construction

Issue: superposition at variables � 15 Example: C = … X … X a … Given g > f, it is unclear whether X := g or X := f 
 will yield the smaller clause Solution #1: 
 Solution #2: 
 purifying calculus nonpurifying calculus … X u ̅ … X v ̅ … Perform superpositions at variables 
 if the order situation is unclear is purified to … X u ̅ … Y v ̅ … ∨ X ≠ Y if u ̅ ≠ v ̅

Evaluation of our prototype � 16 using the Zipperposition theorem prover Judgment Day 
 TPTP benchmarks λ -free HOL benchmarks # unsat FO HO 32 facts 512 facts 181 - - - first-order mode 151 677 873 843 applicative encoding 180 647 851 908 purifying calculus 179 669 866 889 nonpurifying calculus

In summary � 17 ‣ We developed refutationally complete calculi 
 for λ -free HOL 
 ‣ They reduce the gap between HO proof assistants 
 and superposition provers 
 ‣ They are promising as a stepping stone towards a 
 HO superposition calculus