Beagle – A Hierarchic Superposition Theorem Prover

Baumgartner, Peter; Bax, Joshua; Waldmann, Uwe

doi:10.1007/978-3-319-21401-6_25

Cited by 16 publications

(22 citation statements)

References 15 publications

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“…We took all such problems from the TPTP library [31] and SMT-LIB [5] and Table 2 shows the results. In the table, new solutions are meant with respect to what VAMPIRE could previously solve and uniquely solved stands for the number of new problems with respect to what can be solved by other entrants into SMT-COMP 1 and CASC 2 where the main competitors are SMT solvers such as Z3 [11] and CVC4 [4] and ATPs such as Beagle [6] and Princess [27,28]. Figure 2 lists the problems uniquely solved by VAMPIRE (the last two are from TPTP, the rest from SMT-LIB).…”

Section: Resultsmentioning

confidence: 99%

Unification with Abstraction and Theory Instantiation in Saturation-based Reasoning

Reger¹,

Suda²,

Воронков³

2017

EasyChair Preprints

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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.

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Section: Resultsmentioning

confidence: 99%

Unification with Abstraction and Theory Instantiation in Saturation-based Reasoning

Reger¹,

Suda²,

Воронков³

2017

EasyChair Preprints

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Section: Methodsmentioning

confidence: 99%

“…Beagle's proof procedure and background reasoning, in particular for linear integer arithmetic, and experimental results have been described in [7]. Here we only provide an update on the experiments and report on complementary aspects not discussed in [7]. More specifically, our new experiments are based on a more recent version of the TPTP problem library [26] (by four years), and we discuss in more detail the impact of the various calculus variants introduced in this paper.…”

Section: Methodsmentioning

confidence: 99%

“…These conversions exploit the associativity and commutativity laws for + and ·. We refer the reader to [7] for additional aggressive simplification rules, but we note here that aggressive simplification does not always preserve sufficient completeness. For example, in the clause set N = {P(1 + (2 + f(X))), ¬P(1 + (X + f(X)))} the first clause is aggressively simplified, giving N ′ = {P(3 + f(X)), ¬P(1 + (X + f(X)))}.…”

Section: Ordinary Variables Abstraction Variablesmentioning

confidence: 99%

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Hierarchic Superposition Revisited

Baumgartner

Waldmann

2019

Lecture Notes in Computer Science

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Many applications of automated deduction require reasoning in first-order logic modulo background theories, in particular some form of integer arithmetic. A major unsolved research challenge is to design theorem provers that are "reasonably complete" even in the presence of free function symbols ranging into a background theory sort. The hierarchic superposition calculus of Bachmair, Ganzinger, and Waldmann already supports such symbols, but, as we demonstrate, not optimally. This paper aims to rectify the situation by introducing a novel form of clause abstraction, a core component in the hierarchic superposition calculus for transforming clauses into a form needed for internal operation. We argue for the benefits of the resulting calculus and provide two new completeness results: one for the fragment where all background-sorted terms are ground and another one for a special case of linear (integer or rational) arithmetic as a background theory. ⋆ The final authenticated version is available online at https://doi.org/TBD.

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“…A number of automated theorem provers such as Vampire [30], E [44], and SPASS [48] were developed to target quantified formulas in single-sorted first-order logic. More recently, these tools have been extended with dedicated support for background theories [1,7]. For the most part, tools from this community are superposition-based, although some are instantiation-based such as the Inst-Gen calculus used by iProver [22].…”

Section: Introductionmentioning

confidence: 99%

Conflicts, Models and Heuristics for Quantifier Instantiation in SMT

Reynolds¹

EPiC Series in Computing

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Satisfiability Modulo Theories (SMT) solvers have emerged as prominent tools in formal methods applications. While originally targeted towards quantifier-free inputs, SMT solvers are now often used for handling quantified formulas in automated theorem proving and software verification applications. The most common technique for handling quantified formulas in modern SMT solvers in quantifier instantiation. This paper gives an overview of recent advances in quantifier instantiation in SMT. In addition to the well-known technique known as E-matching, we discuss the use of conflicts and models for accelerating the search for (un)satisfiably. We further mention new instantiation-based techniques that are specialized to background theories such as linear real and integer arithmetic, and future work in this direction.

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Beagle – A Hierarchic Superposition Theorem Prover

Cited by 16 publications

References 15 publications

Unification with Abstraction and Theory Instantiation in Saturation-based Reasoning

Unification with Abstraction and Theory Instantiation in Saturation-based Reasoning

Hierarchic Superposition Revisited

Conflicts, Models and Heuristics for Quantifier Instantiation in SMT

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