Nondeterministic Phase Semantics and the Undecidability of Boolean BI

Dominique Larchey-Wendling, Didier Galmiche
2013 ACM Transactions on Computational Logic  
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 the elementary fragment of ILL which is both undecidable and complete for trivial phase semantics. Thus, we obtain
more » ... e undecidability of BBI. Kripke semantics. Cut-elimination was also derived but, despite the expectations of Brotherston, no decidability result followed. On the other hand, [Larchey-Wendling and Galmiche 2009] proposed a labeled tableaux proof-system for (partial monoidal) BBI and by the study of the relations between the proof-search generated counter-models of BI and BBI, showed that (intuitionistic) BI could be faithfully embedded into BBI. This result, at first counter-intuitive, hinted that BBI, originally thought simpler than BI, could in fact be much more difficult to decide. In this paper, we consider models of BBI belonging to different classes: ND. The class of non-deterministic monoids; PD. The class of partial (deterministic) monoids; TD. The class of total (deterministic) monoids; HM. The class of heaps monoids (i.e. separation logic models); FM. The class of free monoids. Generally, each class of models defines a different notion of (universal Kripke) validity on the formulae of BBI. For instance, we recall the result that the set BBI ND of BBI-formulae valid in every non-deterministic monoid is strictly included in the set BBI PD of BBI-formulae valid in every partial deterministic monoid [Larchey-Wendling and Galmiche 2010]. The classification of these classes of models with respect to BBI Kripke validity is not finished though. The principal result of this paper is the undecidability of universal validity in BBI, whichever class of models is chosen amongst ND, PD, TD, HM and FM. Although these classes of models generally define different notions of universal validity for the whole BBI, we have identified a fragment of BBI on which these semantics collapse to one. This fragment is the direct image of the elementary fragment of ILL (denoted eILL) by an embedding of ILL into BBI. This elementary fragment is different from the minimal fragment of Boolean BI identified in [Brotherston and Kanovich 2010] but has similar properties. In our case, undecidability is obtained by the following steps: -we show that the embedding of eILL into BBI is faithful for trivial phase semantics; -we show that the eILL fragment is complete for trivial phase semantics, whichever class of models is chosen amongst ND, PD, TD, HM and FM; -we show how to encode the computations of two counter Minsky machines in eILL. As a consequence, we derive the undecidability of the eILL fragment, from which we deduce the undecidability of BBI. We complete the pictures with additional results of undecidability on the models based on the free monoid N × N and the models based on the partial monoid P f (N) (which is also the simplest heap monoid). This last result is obtained using bisimulation techniques. Compared to the original LICS'10 paper [Larchey-Wendling and Galmiche 2010], this paper contains a more extensive study of the semantics of the eILL fragment with completeness results for various classes of models and the adaptation of our undecidability result of BBI to heaps models (i.e. to Separation Logic) using a bisimulation between free monoids and heap monoids. CLASSES OF NON-DETERMINISTIC MONOIDS In this section, we define the algebraic notion of non-deterministic (commutative) monoid. We denote algebraic structure by M, N,... classes of structures by C, D,... sets by X, Y,... elements by x, y,... and well known constructs like the powerset by P(X) or the set of (finite) multisets by M f (X). The symbol N = {0, 1, 2, . . .} denotes the set of natural numbers. The symbol ∅ is used either to denote the empty set, the empty multiset or the empty class. Non-deterministic monoids Let us consider a set M and its powerset P(M), i.e. the set of subsets of M. A composition is a binary function • : M × M −→ P(M) which naturally extended to a binary operator on P(M) by X • Y = {x • y | x ∈ X and y ∈ Y} (1)
doi:10.1145/2422085.2422091 fatcat:4lz7blpptvhwvaebbqfdxwlb6i