Abstract. We consider two styles of proof calculi for a family of tense logics, presented in a formalism based on nested sequents. A nested sequent can be seen as a tree of traditional single-sided sequents. Our first style of calculi is what we call "shallow calculi", where inference rules are only applied at the root node in a nested sequent. Our shallow calculi are extensions of Kashima's calculus for tense logic and share an essential characteristic with display calculi, namely, the presence of structural rules called "display postulates". Shallow calculi enjoy a simple cut elimination procedure, but are unsuitable for proof search due to the presence of display postulates and other structural rules. The second style of calculi uses deep-inference, whereby inference rules can be applied at any node in a nested sequent. We show that, for a range of extensions of tense logic, the two styles of calculi are equivalent, and there is a natural proof theoretic correspondence between display postulates and deep inference. The deep inference calculi enjoy the subformula property and have no display postulates or other structural rules, making them a better framework for proof search.
Abstract. The description logic SHI extends the basic description logic ALC with transitive roles, role hierarchies and inverse roles. The known tableau-based decision procedure  for SHI exhibit (at least) NEXP-TIME behaviour even though SHI is known to be EXPTIME-complete. The automata-based algorithms for SHI often yield optimal worst-case complexity results, but do not behave well in practice since good optimisations for them have yet to be found. We extend our method for global caching in ALC to SHI by adding analytic cut rules, thereby giving the first EXPTIME tableau-based decision procedure for SHI, and showing one way to incorporate global caching and inverse roles.
Abstract. We present a labelled sequent calculus for Boolean BI (BBI), a classical variant of the logic of Bunched Implication. The calculus is simple, sound, complete, and enjoys cut-elimination. We show that all the structural rules in the calculus, i.e., those rules that manipulate labels and ternary relations, can be localised around applications of certain logical rules, thereby localising the handling of these rules in proof search. Based on this, we demonstrate a free variable calculus that deals with the structural rules lazily in a constraint system. We propose a heuristic method to quickly solve certain constraints, and show some experimental results to confirm that our approach is feasible for proof search. Additionally, we show that different semantics for BBI and some axioms in concrete models can be captured by adding extra structural rules.
Substructural logics are traditionally obtained by dropping some or all of the structural rules from Gentzen's sequent calculi LK or LJ. It is well known that the usual logical connectives then split into more than one connective. Alternatively, one can start with the (intuitionistic) Lambek calculus, which contains these multiple connectives, and obtain numerous logics like: exponential-free linear logic, relevant logic, BCK logic, and intuitionistic logic, in an incremental way. Each of these logics also has a classical counterpart, and some also have a "cyclic" counterpart. These logics have been studied extensively and are quite well understood. Generalising further, one can start with intuitionistic Bi-Lambek logic, which contains the dual of every connective from the Lambek calculus. The addition of the structural rules then gives Bi-linear, Bi-relevant, Bi-BCK and Bi-intuitionistic logic, again in an incremental way. Each of these logics also has a classical counterpart, and some even have a "cyclic" counterpart. These (bi-intuitionistic and bi-classical) extensions of Bi-Lambek logic are not so well understood. Cut-elimination for Classical Bi-Lambek logic, for example, is not completely clear since some cut rules have side conditions requiring that certain constituents be empty or non-empty. The Display Logic of Nuel Belnap is a general Gentzen-style proof theoretical framework designed to capture many different logics in one uniform setting. The beauty of display logic is a general cut-elimination theorem, due to Belnap, which applies whenever the rules of the display calculus obey certain, easily checked, conditions. The original display logic, and its various incarnations, are not suitable for capturing bi-intuitionistic and bi-classical logics in a uniform way. We remedy this situation by giving a single (cut-free) Display calculus for the Bi-Lambek Calculus, from which all the well-known (bi-intuitionistic and bi-classical) extensions are obtained by the incremental addition of structural rules to a constant core of logical introduction rules. We highlight the inherent duality and symmetry within this framework obtaining "four proofs for the price of one". We give algebraic semantics for the Bi-Lambek logics and prove that our calculi are sound and complete with respect to these semantics. We show how to define an alternative display calculus for bi-classical substructural logics using negations, instead of implications, as primitives. Borrowing from other display calculi, we show how to extend our display calculus to handle bi-intuitionistic or bi-classical substructural logics containing the forward and backward modalities familiar from tense logic, the exponentials of linear logic, the converse operator familiar from relation algebra, four negations, and two unusual modalities corresponding to the non-classical analogues of Sheffer's "dagger" and "stroke", all in a modular way. Using the Gaggle Theory of Dunn we outline relational semantics for the binary and unary intensional connectives, but ma...
Abstract. We give an optimal (exptime), sound and complete tableaubased algorithm for deciding satisfiability with respect to a TBox in the logic ALCI using global state caching. Global state caching guarantees optimality and termination without dynamic blocking, but in the presence of inverse roles, the proofs of soundness and completeness become significantly harder. We have implemented the algorithm in OCaml, and our initial comparison with FaCT++ indicates that it is a promising method for checking satisfiability with respect to a TBox.
Abstract. We present a sound and complete tableau calculus for the class of regular grammar logics. Our tableau rules use a special feature called automaton-labelled formulae, which are similar to formulae of automaton propositional dynamic logic. Our calculus is cut-free and has the analytic superformula property so it gives a decision procedure. We show that the known EXPTIME upper bound for regular grammar logics can be obtained using our tableau calculus. We also give an effective Craig interpolation lemma for regular grammar logics using our calculus.
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