Generic Modal Cut Elimination Applied to Conditional Logics

We develop a general criterion for cut elimination in sequent calculi for propositional modal logics, which rests on absorption of cut, contraction, weakening and inversion by the purely modal part of the rule system. Our criterion applies also to a wide variety of logics outside the realm of normal modal logic. We give extensive example instantiations of our framework to various conditional logics. For these, we obtain fully internalised calculi which are substantially simpler than those known in the literature, along with leaner proofs of cut elimination and complexity. In one case, conditional logic with modus ponens and conditional excluded middle, cut elimination and complexity were explicitly stated as open in the literature

Deciding regular grammar logics with converse through first-order logic

We provide a simple translation of the satisfiability problem for regular grammar logics with converse into GF2, which is the intersection of the guarded fragment and the 2-variable fragment of first-order logic. This translation is theoretically interesting because it translates modal logics with certain frame conditions into first-order logic, without explicitly expressing the frame conditions. A consequence of the translation is that the general satisfiability problem for regular grammar logics with converse is in EXPTIME. This extends a previous result of the first author for grammar logics without converse. Using the same method, we show how some other modal logics can be naturally translated into GF2, including nominal tense logics and intuitionistic logic. In our view, the results in this paper show that the natural first-order fragment corresponding to regular grammar logics is simply GF2 without extra machinery such as fixed point-operators.Comment: 34 page

Computational Aspects of Proofs in Modal Logic

Various modal logics seem well suited for developing models of knowledge, belief, time, change, causality, and other intensional concepts. Most such systems are related to the classical Lewis systems, and thereby have a substantial body of conventional proof theoretical results. However, most of the applied literature examines modal logics from a semantical point of view, rather than through proof theory. It appears arguments for validity are more clearly stated in terms of a semantical explanation, rather than a classical proof-theoretic one. We feel this is due to the inability of classical proof theories to adequately represent intensional aspects of modal semantics. This thesis develops proof theoretical methods which explicitly represent the underlying semantics of the modal formula in the proof. We initially develop a Gentzen style proof system which contains semantic information in the sequents. This system is, in turn, used to develop natural deduction proofs. Another semantic style proof representation, the modal expansion tree is developed. This structure can be used to derive either Gentzen style or Natural Deduction proofs. We then explore ways of automatically generating MET proofs, and prove sound and complete heuristics for that procedure. These results can be extended to most propositional system using a Kripke style semantics and a fist order theory of the possible worlds relation. Examples are presented for standard T, S4, and S5 systems, systems of knowledge and belief, and common knowledge. A computer program which implements the theory is briefly examined in the appendix

Provability in BI's Sequent Calculus is Decidable

The logic of Bunched Implications (BI) combines both additive and multiplicative connectives, which include two primitive intuitionistic implications. As a consequence, contexts in the sequent presentation are not lists, nor multisets, but rather tree-like structures called bunches. This additional complexity notwithstanding, the logic has a well-behaved metatheory admitting all the familiar forms of semantics and proof systems. However, the presentation of an effective proof-search procedure has been elusive since the logic's debut. We show that one can reduce the proof-search space for any given sequent to a primitive recursive set, the argument generalizing Gentzen's decidability argument for classical propositional logic and combining key features of Dyckhoff's contraction-elimination argument for intuitionistic logic. An effective proof-search procedure, and hence decidability of provability, follows as a corollary.Comment: Submitted to CADE-2

A graph-theoretic account of logics

A graph-theoretic account of logics is explored based on the general notion of m-graph (that is, a graph where each edge can have a finite sequence of nodes as source). Signatures, interpretation structures and deduction systems are seen as m-graphs. After defining a category freely generated by a m-graph, formulas and expressions in general can be seen as morphisms. Moreover, derivations involving rule instantiation are also morphisms. Soundness and completeness theorems are proved. As a consequence of the generality of the approach our results apply to very different logics encompassing, among others, substructural logics as well as logics with nondeterministic semantics, and subsume all logics endowed with an algebraic semantics

Retracing some paths in categorical semantics: From process-propositions-as-types to categorified reals and computers

The logical parallelism of propositional connectives and type constructors extends beyond the static realm of predicates, to the dynamic realm of processes. Understanding the logical parallelism of process propositions and dynamic types was one of the central problems of the semantics of computation, albeit not always clear or explicit. It sprung into clarity through the early work of Samson Abramsky, where the central ideas of denotational semantics and process calculus were brought together and analyzed by categorical tools, e.g. in the structure of interaction categories. While some logical structures borne of dynamics of computation immediately started to emerge, others had to wait, be it because the underlying logical principles (mainly those arising from coinduction) were not yet sufficiently well-understood, or simply because the research community was more interested in other semantical tasks. Looking back, it seems that the process logic uncovered by those early semantical efforts might still be starting to emerge and that the vast field of results that have been obtained in the meantime might be a valley on a tip of an iceberg. In the present paper, I try to provide a logical overview of the gamut of interaction categories and to distinguish those that model computation from those that capture processes in general. The main coinductive constructions turn out to be of this latter kind, as illustrated towards the end of the paper by a compact category of all real numbers as processes, computable and uncomputable, with polarized bisimulations as morphisms. The addition of the reals arises as the biproduct, real vector spaces are the enriched bicompletions, and linear algebra arises from the enriched kan extensions. At the final step, I sketch a structure that characterizes the computable fragment of categorical semantics.Comment: 63 pages, 40 figures; cut two words from the title, tried to improve (without lengthening) Sec.8; rewrote a proof in the Appendi