A tailored integration of SAT solving for detecting variants of subsumption in superposition using the Vampire prover and showing that SAT encodings improve literal matching, and thus subsumption, in first-order theorem proving is presented.
This talk overviews recent progress in automating inductive reasoning in quantified logic, with applications to code synthesis, and shows that induction and synthesis are better together in saturation, allowing not only to prove quantified properties F, but also generate a functional implementation of F during proof search 1.
Solving Boolean polynomial systems, or equivalently, SAT-solving, based on the XNF and the XLIN proof system outperforms classical methods not only theoretically, but based on the new conflict-driven XNF clause learning methods developed here, it is possible to implement an XNF solver with good practical efficiency.
Julian Danner, Martin Kreuzer· Journal of Artificial Intell...· 0 citations
This paper introduces the strategy language of Pgeon, a meta-prover with a clear separation between inference rules and proof search. We give the semantics of strategies as functions over proof states, and of the operators that are used to combine them, allowing for sequential composition, choice, repetition and interleaving of strategies. This language is designed to handle the challenge of fair proof search in semi-decidable logics, where simple depth-first exploration of the proof space is not guaranteed to achieve completeness. We showcase the expressiveness and effectiveness of the approach through case studies in first-order and modal logics.
Romain Sidhoum, Simon Robillard, David Delahaye· Electronic Proceedings in Th...· 0 citations
The satisfiability problem of propositional logic, SAT for short, is the first algorithmic problem that was shown to be NP-complete, and is the cornerstone of virtually all NP-completeness proofs. The SAT problem consists of deciding whether a given Boolean formula has a “solution”, in the sense of an assignment to the variables making the entire formula to evaluate to true. Over the last few years very powerful algorithms have been devised being able to solve SAT problems with hundreds of thousands of variables. For difficult (or randomly generated) formulas these algorithms can be compared to the proverbial search for the needle in a haystack. This book explains how such algorithms work, for example, by exploiting the structure of the SAT problem with an appropriate logical calculus, like resolution. But also algorithms based on “physical” principles are considered.
The top-down solver TD is a generic fixpoint algorithm that can be used to compute partial post-solutions of equation systems for abstract interpretation. We consider two extensions of the TD to deal with infinite strictly ascending chains. For the TD extended with warrowing, we formally prove that it always returns partial post-solutions, while for the TD extended with widening and narrowing in phases, we prove termination provided that the set of unknowns is finite. By proving the equivalence of the two extensions, we deduce the total correctness of both solvers. For the equivalence to hold, we in particular assume the equation system to have right-hand sides that are both monotonic and have monotonic dependencies. We demonstrate with counterexamples that the violation of any of the assumptions may compromise the equivalence. All proofs have been formalized using the interactive theorem prover Isabelle.
Sarah Tilscher, Alexandra Graß, Helmut Seidl et al.· Journal of automated reasoni...· 0 citations
GK is a query-directed first-order prover that extends ordinary resolution-based proof search with explicit positive and negative claims, numerical confidence values, and prioritized default rules with exceptions. It works directly with non-ground clauses, including equality and function terms. Candidate proofs are found by bounded first-order proof search; exception conditions of defaults are checked by further bounded searches, recursively when exceptions themselves depend on defaults. This avoids requiring a finite global grounding, while allowing incomplete searches to be reported as such. This paper adds structure-preserving quantitative reporting to that framework. Retained proof histories are used in two calculations. The first reconstructs the uncertain ground premises used by each proof and computes the probability that at least one retained proof is available, without counting shared premises independently. The second resolves positive and negative support at intermediate atoms before that support is propagated through later rules; the same calculation evaluates uncertain exception conditions for individual rule applications. Reports separate positive support, negative support, conflict, and ignorance and identify detected incomplete calculations or fallbacks. The implementation performs bounded reconstruction and dependency traversal after proof search and still requires no global grounding. Analytic examples and independent simulators reproduce the reference calculations on their stated fragments. Comparisons with probabilistic logic, probabilistic ASP, default logic, and goal-directed ASP identify cases of agreement, semantic difference, unsupported translation, and incomplete computation.
T. Tammet· 1 citation
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