Unification with Abstraction and Theory Instantiation in Saturation-based Reasoning

Abstract

This paper explores two new inference rules for reasoning with quantifiers and theories in a saturation-based first-order theorem prover. The focus here is on non-ground clauses, complementing our recent work on AVATAR modulo theories for ground theory reasoning. The current implementation focuses on complete theories, e.g. various versions of arithmetic, but we also sketch how to work with incomplete theories. The first new rule utilises theory constraint solving (an SMT solver) to perform reasoning within a clause to find an instance where we can remove one or more theory literals. This utilises the power of SMT solvers for theory reasoning with non-ground clauses, reasoning which is currently achieved by the addition of prolific theory axioms. The second new rule is unification with abstraction where the notion of unification is extended to introduce constraints where theory terms may not otherwise unify e.g. p(2) may unify with ¬p(x + 1) ∨ q(x) to produce 2 x + 1 ∨ q(x). This abstraction is performed lazily, as needed, to allow the superposition theorem prover to make as much progress as possible without the search space growing too quickly. Additionally, the first rule can be used to discharge the constraints introduced by the second. These rules were implemented within the Vampire theorem prover and experimental results show that they are useful for solving a considerable number of previously unsolved problems.

Making Theory Reasoning Simpler

“…AVATAR modulo theories [16] uses an SMT solver within the context of clause splitting to ensure that the ground part of any chosen clause splits are theory-consistent. The previously mentioned unification with abstraction and theory instantiation [20] rules support lazy unification modulo theories and pragmatic instantiation. Theory axiom usage can be controlled by the set of support strategy [17] or layered clause selection [10].…”

“…rules that make reasoning in arithmetic simpler. This work was motivated by our previous attempt [20] to find useful instances of first-order clauses that would be otherwise difficult to find via reasoning with theory axioms. For example, when considering the two clauses r(7x) ¬r(6 + y) ∨ p(y) our previous work would apply resolution on r(7x) and ¬r(6+y) using unification with abstraction to produce the clause 7x = 6 + y ∨ p(y) and then applied theory instantiation, utilising an SMT solver to find the substitution {x → 1, y → 1}, producing the instance p(1).…”

Loop Analysis by Quantification over Iterations

“…For first-order theorem provers based on saturation, reasoning about linear arithmetic is traditionally accomplished by including a (partial) axiomatization in the set of clauses to saturate. Recently, these provers have taken advantage of SMT solvers to perform theory reasoning on ground clauses [31] as well as non-ground clauses [32]. Regarding term algebras, the SMT solvers Z3 and CVC4 both include theory solvers for the ground theory.…”

Experimenting with Theory Instantiation in Vampire

“…Theory instantiation [13] introduces a new inference rule that uses the theory part of a clause to instantiate it. The idea is that an instantiation that conflicts with the the theory part is sufficiently precise to be useful for proof search on the uninterpreted part.…”

“…Although it is admissible the theory instantiation rule has shown to be helpful for problems that combine (nonlinear) arithmetic and reasoning with uninterpreted functions [13]. Since arrays roughly correspond to the lambda-free fragment of higher-order logic with pointwise function updates, better reasoning over arrays could be a useful alternative to full higher-order reasoning.…”