Beyond formulas-as-cographs: an extension of Boolean logic to arbitrary graphs

We propose a graph-based extension of Boolean logic called Boolean Graph Logic (BGL). Construing formula trees as the cotrees of cographs, we may state semantic notions such as evaluation and entailment in purely graph-theoretic terms, whence we recover the definition of BGL. Naturally, it is conservative over usual Boolean logic. Our contributions are the following: (1) We give a natural semantics of BGL based on Boolean relations, i.e. it is a multivalued semantics, and show adequacy of this semantics for the corresponding notions of entailment. (2) We show that the complexity of evaluation is NP-complete for arbitrary graphs (as opposed to ALOGTIME-complete for formulas), while entailment is $\Pi^p_2$-complete (as opposed to coNP-complete for formulas). (3) We give a 'recursive' algorithm for evaluation by induction on the modular decomposition of graphs. (Though this is not polynomial-time, cf. point (2) above). (4) We characterise evaluation in a game-theoretic setting, in terms of both static and sequentical strategies, extending the classical notion of positional game forms beyond cographs. (5) We give an axiomatisation of BGL, inspired by deep-inference proof theory, and show soundness and completeness for the corresponding notions of entailment. One particular feature of the graph-theoretic setting is that it escapes certain no-go theorems such as a recent result of Das and Strassburger, that there is no linear axiomatisation of the linear fragment of Boolean logic (equivalently the multiplicative fragment of Japaridze's Computability Logic or Blass' game semantics for Mutliplicative Linear Logic).Comment: 47 pages, 2 figures, 2 table

The Bernays-Schönfinkel-Ramsey Class of Separation Logic with Uninterpreted Predicates

International audienceThis paper investigates the satisfiability problem for Separation Logic with k record fields, with unrestricted nesting of separating conjunctions and implications. It focuses on prenex formulae with a quantifier prefix in the language ∃ * ∀ * , that contain uninterpreted (heap-independent) predicate symbols. In analogy with first-order logic, we call this fragment Bernays-Schönfinkel-Ramsey Separation Logic [BSR(SL k)]. In contrast with existing work on Separation Logic, in which the universe of possible locations is assumed to be infinite, we consider both finite and infinite universes in the present paper. We show that, unlike in first-order logic, the (in)finite satisfiability problem is undecidable for BSR(SL k). Then we define two non-trivial subsets thereof, for which the finite and infinite satisfiability problems are PSPACE-complete, respectively, assuming that the maximum arity of the uninterpreted predicate symbols does not depend on the input. These fragments are defined by controlling the polarity of the occurrences of separating implications, as well as the occurrences of universally quantified variables within their scope. These decidability results have natural applications in program verification, as they allow to automatically prove lemmas that occur in e.g. entailment checking between inductively defined predicates and validity checking of Hoare triples expressing partial correctness conditions

Automated Deduction – CADE 28

This open access book constitutes the proceeding of the 28th International Conference on Automated Deduction, CADE 28, held virtually in July 2021. The 29 full papers and 7 system descriptions presented together with 2 invited papers were carefully reviewed and selected from 76 submissions. CADE is the major forum for the presentation of research in all aspects of automated deduction, including foundations, applications, implementations, and practical experience. The papers are organized in the following topics: Logical foundations; theory and principles; implementation and application; ATP and AI; and system descriptions