An Alternative Direct Simulation of Minsky Machines into Classical Bunched Logics via Group Semantics

Dominique Larchey-Wendling
2010 Electronical Notes in Theoretical Computer Science  
Recently, Brotherston & Kanovich, and independently Larchey-Wendling & Galmiche, proved the undecidability of the bunched implication logic BBI. Moreover, Brotherston & Kanovich also proved the undecidability of the related logic CBI, as well as its neighbours. All of the above results are based on encodings of two-counter Minsky machines, but are derived using different techniques. Here, we show that the technique of Larchey-Wendling & Galmiche can also be extended, via group Kripke semantics,
more » ... to prove the undecidability of CBI. Hence, we propose an alternative direct simulation of Minsky machines into both BBI and CBI. We identify a fragment called elementary Boolean BI (eBBI) which is common to the BBI/CBI families of logics and we show that the problem of Minsky machine acceptance can be encoded into eBBI. The soundness of the encoding is derived from the soundness of a goal directed sequent calculus designed for eBBI. The faithfulness of the encoding is obtained from a Kripke model based on the free commutative group Z n . Open access under CC BY-NC-ND license. of [14] can also be adapted, via group Kripke semantics, to simultaneously prove the undecidability of both BBI and CBI. Recall that the logic BI of bunched implications [16] is a sub-structural logic which freely combines additive connectives ∧, ∨, → and multiplicative connectives * , − * . In BI, both the multiplicatives and the additives behave intuitionistically. From its inception, BI was given a nice bunched sequent proof-system enjoying cutelimination [17] . Later, Galmiche et al. [8] gave BI a sound and complete labeled tableaux system from which decidability was derived. The logic BI is sometimes called intuitionistic BI to distinguish it with other variants where either the multiplicatives or the additives include a negation and thus behave classically. From a proof-theoretical perspective, Boolean BI (or simply BBI) can be considered as the first investigated variant of BI which contained a negation: BBI combines intuitionistic multiplicatives with Boolean additives. This focus on BBI is the consequence of the natural links between BBI and separation or spatial logics. Indeed, for instance, the pure part of separation logic is essentially obtained by considering a particular model of BBI, based on a (partial) monoid of heaps [11] (see [13] for a more general discussion on these links). The Hilbert proof-system of BBI was proved complete w.r.t. relational (or non-deterministic) Kripke semantics [7] . However, the proof-theory of BBI was rather poorly developed because it was difficult to conceive how the bunched sequent calculus of (intuitionistic) BI could be extended to BBI without losing key properties such as e.g. cut-elimination. Then Classical BI (CBI) was introduced [3] as a bunched logic which contained both a multiplicative negation and an additive negation. It could be used as a basis for resource models which contain a dualizing operator. For this logic, Brotherston and Calcagno [4] provided a Display calculusà la Belnap and established its soundness and completeness both w.r.t. the Hilbert proof-system and (dualizing) relational Kripke semantics. They proved cut-elimination as a by product of their Display proof-system and described a substantial part of the model theory of CBI, including the proof of the incompleteness of CBI w.r.t. the (dualizing) partial monoidal Kripke semantics. However, no decidability result followed from these achievements. Then, back to BBI, two main families of results emerged giving a contrasted view of its proof-theory. On the one hand, Brotherston [2] adapted the Display proofsystem of CBI to BBI, circumventing the difficulty of the multiplicatives of BBI lacking a negation. This system was proved sound and complete w.r.t. relational 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 [13] proposed a labeled tableaux proof-system for (partial monoidal) BBI and by the study of the relations between the proof-search generated countermodels 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. To complete the picture, Larchey-Wendling and Galmiche [14] recently established that relational Kripke semantics and partial monoidal Kripke semantics define different notions D. Larchey-Wendling /
doi:10.1016/j.entcs.2010.08.022 fatcat:jyzey2fzc5djtpkgu3bmvihska