Abstract We address generating theorems from a given set of axioms, without proof goal, aiming at value from a mathematical point of view or as lemmas for automated proving. As benchmark, we convert a fragment of the Metamath database set.mm. Our techniques are centered on proof terms and condensed detachment. This ties in with automated first-order proving by proof structure enumeration, and links to Metamath and formulas-as-types. Our methods for generating theorems are based on partitioning the set of proof terms into inductively characterized levels. We study two ideas for improvement: Lemma synthesis by DAG compression of proof term sets, and incorporating combinators into proof terms. Our lemmas significantly improve solution rates of provers, e.g., of Vampire from 74% to 94%, and of leanCoP from 7% to 44%.

Abstract We show a projective Beth definability theorem for logic programs under the stable model semantics: For given programs P and Q and vocabulary V (set of predicates) the existence of a program R in V such that $$P \cup R$$ P ∪ R and $$P \cup Q$$ P ∪ Q are strongly equivalent can be expressed as a first-order entailment. Moreover, our result is effective: A program R can be constructed from a Craig interpolant for this entailment, using a known first-order encoding for testing strong equivalence, which we apply in reverse to extract programs from formulas. As a further perspective, this allows transforming logic programs via transforming their first-order encodings. In a prototypical implementation, the Craig interpolation is performed by first-order provers based on clausal tableaux or resolution calculi. Our work shows how definability and interpolation, which underlie modern logic-based approaches to advanced tasks in knowledge representation, transfer to answer set programming.

Abstract We introduce and elaborate a novel formalism for the manipulation and analysis of proofs as objects in a global manner. In this first approach the formalism is restricted to first-order problems characterized by condensed detachment. It is applied in an exemplary manner to a coherent and comprehensive formal reconstruction and analysis of historical proofs of a widely-studied problem due to Łukasiewicz. The underlying approach opens the door towards new systematic ways of generating lemmas in the course of proof search to the effects of reducing the search effort and finding shorter proofs. Among the numerous reported experiments along this line, a proof of Łukasiewicz ’s problem was automatically discovered that is much shorter than any proof found before by man or machine.

Derived datasets can be defined implicitly or explicitly. An implicit definition (of dataset O in terms of datasets I) is a logical specification involving two distinguished sets of relational symbols. One set of relations is for the "source data" I, and the other is for the "interface data" O. Such a specification is a valid definition of O in terms of I, if any two models of the specification agreeing on I agree on O. In contrast, an explicit definition is a transformation (or "query" below) that produces O from I. Variants of Beth's theorem state that one can convert implicit definitions to explicit ones. Further, this conversion can be done effectively given a proof witnessing implicit definability in a suitable proof system. We prove the analogous implicit-to-explicit result for nested relations: implicit definitions, given in the natural logic for nested relations, can be converted to explicit definitions in the nested relational calculus (NRC). We first provide a model-theoretic argument for this result, which makes some additional connections that may be of independent interest, between NRC queries, interpretations, a standard mechanism for defining structure-to-structure translation in logic, and between interpretations and implicit to definability "up to unique isomorphism". The latter connection uses a variation of a result of Gaifman concerning "relatively categorical" theories. We also provide a proof-theoretic result that provides an effective argument: from a proof witnessing implicit definability, we can efficiently produce an NRC definition. This will involve introducing the appropriate proof system for reasoning with nested sets, along with some auxiliary Beth-type results for this system. As a consequence, we can effectively extract rewritings of NRC queries in terms of NRC views, given a proof witnessing that the query is determined by the views.

Derived datasets can be defined implicitly or explicitly. An implicit definition (of dataset O in terms of datasets I) is a logical specification involving the source data I and the interface data O. It is a valid definition of O in terms of I, if any two models of the specification agreeing on I agree on O. In contrast, an explicit definition is a query that produces O from I. Variants of Beth's theorem state that one can convert implicit definitions to explicit ones. Further, this conversion can be done effectively given a proof witnessing implicit definability in a suitable proof system. We prove the analogous effective implicit-to-explicit result for nested relations: implicit definitions, given in the natural logic for nested relations, can be effectively converted to explicit definitions in the nested relational calculus (NRC). As a consequence, we can effectively extract rewritings of NRC queries in terms of NRC views, given a proof witnessing that the query is determined by the views.

Abstract Noting that lemmas are a key feature of mathematics, we engage in an investigation of the role of lemmas in automated theorem proving. The paper describes experiments with a combined system involving learning technology that generates useful lemmas for automated theorem provers, demonstrating improvement for several representative systems and solving a hard problem not solved by any system for twenty years. By focusing on condensed detachment problems we simplify the setting considerably, allowing us to get at the essence of lemmas and their role in proof search.

Abstract We show how variations of range-restriction and also the Horn property can be passed from inputs to outputs of Craig interpolation in first-order logic. The proof system is clausal tableaux, which stems from first-order ATP. Our results are induced by a restriction of the clausal tableau structure, which can be achieved in general by a proof transformation, also if the source proof is by resolution/paramodulation. Primarily addressed applications are query synthesis and reformulation with interpolation. Our methodical approach combines operations on proof structures with the immediate perspective of feasible implementation through incorporating highly optimized first-order provers.

Abstract The material presented in this paper contributes to establishing a basis deemed essential for substantial progress in Automated Deduction. It identifies and studies global features in selected problems and their proofs which offer the potential of guiding proof search in a more direct way. The studied problems are of the wide-spread form of “axiom(s) and rule(s) imply goal(s)”. The features include the well-known concept of lemmas. For their elaboration both human and automated proofs of selected theorems are taken into a close comparative consideration. The study at the same time accounts for a coherent and comprehensive formal reconstruction of historical work by Łukasiewicz, Meredith and others. First experiments resulting from the study indicate novel ways of lemma generation to supplement automated first-order provers of various families, strengthening in particular their ability to find short proofs.