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This paper presents a self-contained proof of the strong completeness of the labeled tableaux method for partial monoidal Boolean BI: if a formula has no tableau proof then there exists a counter-model for it which is simple. Simple counter-models are those which are generated from the specific constraints that occur during the tableaux proof-search process. As a companion to this paper, we provide a complete formalisation of this result in Coq 1 and discuss some of its implementation details.

We formally prove the undecidability of entailment in intuitionistic linear logic in Coq. We reduce the Post correspondence problem (PCP) via binary stack machines and Minsky machines to intuitionistic linear logic. The reductions rely on several technically involved formalisations, amongst them a binary stack machine simulator for PCP, a verified low-level compiler for instruction-based languages and a soundness proof for intuitionistic linear logic with respect to trivial phase semantics. We exploit the computability of all functions definable in constructive type theory and thus do not have to rely on a concrete model of computation, enabling the reduction proofs to focus on correctness properties.

We investigate first-order separation logic with one record field restricted to a unique quantified variable (1SL1). Undecidability is known when the number of quantified variables is unbounded and the satisfiability problem is pspace-complete for the propositional fragment. We show that the satisfiability problem for 1SL1 is pspace-complete and we characterize its expressive power by showing that every formula is equivalent to a Boolean combination of atomic properties. This contributes to our understanding of fragments of first-order separation logic that can specify properties about the memory heap

We solve the open problem of the decidability of Boolean BI logic (BBI), which can be considered as the core of separation and spatial logics. For this, we define a complete phase semantics for BBI and characterize it as trivial phase semantics. We deduce an embedding between trivial phase semantics for intuitionistic linear logic (ILL) and Kripke semantics for BBI. We single out a fragment of ILL which is both undecidable and complete for trivial phase semantics. Therefore, we obtain the undecidability of BBI.

Abstract. We present a graph-based decision procedure for Gödel-Dummett logics and an algorithm to compute counter-models. A formula is transformed into a conditional bi-colored graph in which we detect some specific cycles and alternating chains using matrix computations. From an instance graph containing no such cycle (resp. no (n + 1)-alternating chain) we extract a counter-model in LC (resp. LCn).

We investigate first-order separation logic with one record field restricted to a unique quantified variable (1SL1). Undecidability is known when the number of quantified variables is unbounded and the satisfiability problem is pspace-complete for the propositional fragment. We show that the satisfiability problem for 1SL1 is pspace-complete and we characterize its expressive power by showing that every formula is equivalent to a Boolean combination of atomic properties. This contributes to our understanding of fragments of first-order separation logic that can specify properties about the memory heap

The logic of Bunched Implications, through both its intuitionistic version (BI) and one of its classical versions, called Boolean BI (BBI), serves as a logical basis to spatial or separation logic frameworks. In BI, the logical implication is interpreted intuitionistically whereas it is generally interpreted classically in spatial or separation logics, as in BBI. In this paper, we aim to give some new insights into the semantic relations between BI and BBI. Then we propose a sound and complete syntactic constraints based framework for the Kripke semantics of both BI and BBI, a sound labelled tableau proof system for BBI, and a representation theorem relating the syntactic models of BI to those of BBI. Finally, we deduce as our main, and unexpected, result, a sound and faithful embedding of BI into BBI.