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Register a new userLayered Clause Selection for Theory Reasoning
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Added by
Bernhard Gleiss
Publication date
2020
fields
Informatics Engineering
and research's language is
English
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Explicit theory axioms are added by a saturation-based theorem prover as one of the techniques for supporting theory reasoning. While simple and effective, adding theory axioms can also pollute the search space with many irrelevant consequences. As a result, the prover often gets lost in parts of the search space where the chance to find a proof is low. In this paper we describe a new strategy for controlling the amount of reasoning with explicit theory axioms. The strategy refines a recently proposed two-layer-queue clause selection and combines it with a heuristical measure of the amount of theory reasoning in the derivation of a clause. We implemented the new strategy in the automatic theorem prover Vampire and present an evaluation showing that our work dramatically improves the state-of-the-art clause-selection strategy in the presence of theory axioms.
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Structural proof theory is praised for being a symbolic approach to reasoning and proofs, in which one can define schemas for reasoning steps and manipulate proofs as a mathematical structure. For this to be possible, proof systems must be designed as a set of rules such that proofs using those rules are correct by construction. Therefore, one must consider all ways these rules can interact and prove that they satisfy certain properties which makes them well-behaved. This is called the meta-theory of a proof system. Meta-theory proofs typically involve many cases on structures with lots of symbols. The majority of cases are usually quite similar, and when a proof fails, it might be because of a sub-case on a very specific configuration of rules. Developing these proofs by hand is tedious and error-prone, and their combinatorial nature suggests they could be automated. There are various approaches on how to automate, either partially or completely, meta-theory proofs. In this paper, I will present some techniques that I have been involved in for facilitating meta-theory reasoning.
In this paper we use results from Computable Set Theory as a means to represent and reason about description logics and rule languages for the semantic web. Specifically, we introduce the description logic $mathcal{DL}langle 4LQS^Rrangle(D)$--admitting features such as min/max cardinality constructs on the left-hand/right-hand side of inclusion axioms, role chain axioms, and datatypes--which turns out to be quite expressive if compared with $mathcal{SROIQ}(D)$, the description logic underpinning the Web Ontology Language OWL. Then we show that the consistency problem for $mathcal{DL}langle 4LQS^Rrangle(D)$-knowledge bases is decidable by reducing it, through a suitable translation process, to the satisfiability problem of the stratified fragment $4LQS^R$ of set theory, involving variables of four sorts and a restricted form of quantification. We prove also that, under suitable not very restrictive constraints, the consistency problem for $mathcal{DL}langle 4LQS^Rrangle(D)$-knowledge bases is textbf{NP}-complete. Finally, we provide a $4LQS^R$-translation of rules belonging to the Semantic Web Rule Language (SWRL).
Graded modal types systems and coeffects are becoming a standard formalism to deal with context-dependent computations where code usage plays a central role. The theory of program equivalence for modal and coeffectful languages, however, is considerably underdeveloped if compared to the denotational and operational semantics of such languages. This raises the question of how much of the theory of ordinary program equivalence can be given in a modal scenario. In this work, we show that coinductive equivalences can be extended to a modal setting, and we do so by generalising Abramskys applicative bisimilarity to coeffectful behaviours. To achieve this goal, we develop a general theory of ternary program relations based on the novel notion of a comonadic lax extension, on top of which we define a modal extension of Abramskys applicative bisimilarity (which we dub modal applicative bisimilarity). We prove such a relation to be a congruence, this way obtaining a compositional technique for reasoning about modal and coeffectful behaviours. But this is not the end of the story: we also establish a correspondence between modal program relations and program distances. This correspondence shows that modal applicative bisimilarity and (a properly extended) applicative bisimilarity distance coincide, this way revealing that modal program equivalences and program distances are just two sides of the same coin.
We re-examine the topic of machine-learned clause selection guidance in saturation-based theorem provers. The central idea, recently popularized by the ENIGMA system, is to learn a classifier for recognizing clauses that appeared in previously discovered proofs. In subsequent runs, clauses classified positively are prioritized for selection. We propose several improvements to this approach and experimentally confirm their viability. For the demonstration, we use a recursive neural network to classify clauses based on their derivation history and the presence or absence of automatically supplied theory axioms therein. The automatic theorem prover Vampire guided by the network achieves a 41% improvement on a relevant subset of SMT-LIB in a real time evaluation.
Programming languages with countable nondeterministic choice are computationally interesting since countable nondeterminism arises when modeling fairness for concurrent systems. Because countable choice introduces non-continuous behaviour, it is well-known that developing semantic models for programming languages with countable nondeterminism is challenging. We present a step-indexed logical relations model of a higher-order functional programming language with countable nondeterminism and demonstrate how it can be used to reason about contextually defined may- and must-equivalence. In earlier step-indexed models, the indices have been drawn from {omega}. Here the step-indexed relations for must-equivalence are indexed over an ordinal greater than {omega}.
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