Proof Complexity of Modal Resolution Systems

In this thesis we initiate the study of the proof complexity of modal resolution systems. To our knowledge there is no previous work on the proof complexity of such systems. This is in sharp contrast to the situation for propositional logic where resolution is the most studied proof system, in part due to its close links with satisfiability solving. We focus primarily on the proof complexity of two recently proposed modal resolution systems of Nalon, Hustadt and Dixon, one of which forms the basis of an existing modal theorem prover. We begin by showing that not only are these two proof systems equivalent in terms of their proof complexity, they are also equivalent to a number of natural refinements. We further compare the proof complexity of these systems with an older, more complicated modal resolution system of Enjalbert and Farinas del Cerro, showing that this older system p-simulates the more streamlined calculi. We then investigate lower bound techniques for modal resolution. Here we see that whilst some propositional lower bound techniques (i.e. feasible interpolation) can be lifted to the modal setting with only minor modifications, other propositional techniques (i.e. size-width) fail completely. We further develop a new lower bound technique for modal resolution using Prover-Delayer games. This technique can be used to establish "genuine" modal lower bounds (i.e lower bounds on the number of modal inferences) for the size of tree-like modal resolution proofs. We apply this technique to a new family of modal formulas, called the modal pigeonhole principle to demonstrate that these formulas require exponential size modal resolution proofs. Finally we compare the proof complexity of tree-like modal resolution systems with that of modal Frege systems, using our modal pigeonhole principle to obtain a "genuinely" modal separation between them

Dugundji’s theorem revisited

sem informaçãoIn 1940 Dugundji proved that no system between S1 and S5 can be characterized by finite matrices. Dugundji’s result forced the development of alternative semantics, in particular Kripke’s relational semantics. The success of this semantics allowed the creation of a huge family of modal systems. With few adaptations, this semantics can characterize almost the totality of the modal systems developed in the last five decades. This semantics however has some limits. Two results of incompleteness (for the systems KH and VB) showed that not every modal logic can be characterized by Kripke frames. Besides, the creation of non-classical modal logics puts the problem of characterization of finite matrices very far away from the original scope of Dugundji’s result. In this sense, we will show how to update Dugundji’s result in order to make precise the scope and the limits of many-valued matrices as semantic of modal systems. A brief comparison with the useful Chagrov and Zakharyaschev’s criterion of tabularity for modal logics is provided.In 1940 Dugundji proved that no system between S1 and S5 can be characterized by finite matrices. Dugundji’s result forced the development of alternative semantics, in particular Kripke’s relational semantics. The success of this semantics allowed the creation of a huge family of modal systems. With few adaptations, this semantics can characterize almost the totality of the modal systems developed in the last five decades. This semantics however has some limits. Two results of incompleteness (for the systems KH and VB) showed that not every modal logic can be characterized by Kripke frames. Besides, the creation of non-classical modal logics puts the problem of characterization of finite matrices very far away from the original scope of Dugundji’s result. In this sense, we will show how to update Dugundji’s result in order to make precise the scope and the limits of many-valued matrices as semantic of modal systems. A brief comparison with the useful Chagrov and Zakharyaschev’s criterion of tabularity for modal logics is provided.8407422sem informaçãosem informaçãosem informaçãohttp://plato.stanford.edu/archives/win2010/entries/logic-modal-origins/, Ballarin, R.: Modern origins of modal logic. In: The Stanford Encyclopedia of Philosophy, Winter 2010 edition. (2010)Béziau, J.Y., A new four-valued approach to modal logic (2011) Log. Anal, 54 (213), pp. 109-121Bueno-Soler, J., (2009) Multimodalidades anódicas e catódicas: a negação controlada em lógicas multimodais e seu poder expressivo (Anhodic and cathodic multimodalities: controlled negation in multimodal logics and their expressive power, in Portuguese). PhD thesis, Instituto de Filosofia e Ciências Humanas (IFCH), Universidade Estadual de Campinas, , Unicamp, Campinas:Carnielli, W.A., Pizzi, C., Modalities and multimodalities (2008) Logic, Epistemology, and the Unity of Science, vol, p. 12. , Springer-Verlag, New York:Chagrov, A.V., Zakharyaschev, M., Modal logic (1997) Oxford Logic Guides, vol, p. 35. , Oxford University Press, Oxford:Creswell, M.J., Hughes, G.E., (1996) A New Introduction to Modal Logic, , Routledge, London:Dugundji, J., Note on a property of matrices for Lewis and Langford’s calculi of propositions (1940) J. Symb. Log, 5 (4), pp. 150-151Esakia, L., Meskhi, V., Five critical modal systems (1977) Theoria, 43 (1), pp. 52-60Gödel, K.: Eine intepretation des intionistischen Aussagenkalkül. Ergebnisse eines mathematischen Kolloquiums 4, 6–7 (1933) (English translation in [13], pp. 300–303)Gödel, K.: Zur intuitionistischen arithmetik und zahlentheorie. Ergebnisse eines mathematischen Kolloquiums 4, 34–38 (1933) (English translation in [13], pp. 222–225)Gödel, K., Kurt Godel, Collected Works: Publications 1929–1936. Oxford University Press (1986) CaryHenkin, L., Fragments of the proposicional calculus (1949) J. Symb. Log, 14 (1), pp. 42-48Lewis, C.I., Langford, C.H., (1932) Symbolic Logic, , Century, New York:Lemmon, E.J., New foundations for Lewis modal systems (1957) J. Symb. Log, 22 (2), pp. 176-186Lemmon, E.J., Algebraic semantics for modal logics I (1966) J. Symb. Log, 31 (1), pp. 44-65Łukasiewicz,J.: O logice trójwartościowej. Ruch Filozoficzny 5, 170–171 (1920) (English translation in [19] pp. 87–88)Łukasiewicz, J., (1970) Selected Works. Studies in Logic, , North-Holland Publishing Company, London:McKinsey, J.C.C., A reduction in number of the postulates for C. I. Lewis’ system of strict implication (1934) Bull. (New Ser.) Am. Math. Soc, 40, pp. 425-427Magari, R., Representation and duality theory for diagonalizable algebras (1975) Stud. Log, 34 (4), pp. 305-313Scroggs, S.J., Extensions of the Lewis system S5 (1951) J. Symb. Log, 16 (2), pp. 112-120Sobociński, B., Family K of the non-Lewis modal systens. Notre Dame (1964) J. Formal Log, V (4), pp. 313-318Zeman, J.J., Modal Logic: The Lewis Systems. Clarendon Press (1973) U

The topology of justification

Justification Logic is a family of epistemic logical systems obtained from modal logics of knowledge by adding a new type of formula t:F, which is read t is a justification for F. The principal epistemic modal logic S4 includes Tarski’s well-known topological interpretation, according to which the modality 2X is read the Interior of X in a topological space (the topological equivalent of the ‘knowable part of X’). In this paper, we extend Tarski’s topological interpretation from S4 to Justification Logic systems with both modality and justification assertions. The topological semantics interprets t:X as a reachable subset of X (the topological equivalent of ‘test t confirms X’). We establish a number of soundness and completeness results with respect to Kripke topology and the real topology for S4-based systems of Justification Logic

Stone-Type Dualities for Separation Logics

Stone-type duality theorems, which relate algebraic and relational/topological models, are important tools in logic because -- in addition to elegant abstraction -- they strengthen soundness and completeness to a categorical equivalence, yielding a framework through which both algebraic and topological methods can be brought to bear on a logic. We give a systematic treatment of Stone-type duality for the structures that interpret bunched logics, starting with the weakest systems, recovering the familiar BI and Boolean BI (BBI), and extending to both classical and intuitionistic Separation Logic. We demonstrate the uniformity and modularity of this analysis by additionally capturing the bunched logics obtained by extending BI and BBI with modalities and multiplicative connectives corresponding to disjunction, negation and falsum. This includes the logic of separating modalities (LSM), De Morgan BI (DMBI), Classical BI (CBI), and the sub-classical family of logics extending Bi-intuitionistic (B)BI (Bi(B)BI). We additionally obtain as corollaries soundness and completeness theorems for the specific Kripke-style models of these logics as presented in the literature: for DMBI, the sub-classical logics extending BiBI and a new bunched logic, Concurrent Kleene BI (connecting our work to Concurrent Separation Logic), this is the first time soundness and completeness theorems have been proved. We thus obtain a comprehensive semantic account of the multiplicative variants of all standard propositional connectives in the bunched logic setting. This approach synthesises a variety of techniques from modal, substructural and categorical logic and contextualizes the "resource semantics" interpretation underpinning Separation Logic amongst them

The Contextual Character of Modal Interpretations of Quantum Mechanics

In this article we discuss the contextual character of quantum mechanics in the framework of modal interpretations. We investigate its historical origin and relate contemporary modal interpretations to those proposed by M. Born and W. Heisenberg. We present then a general characterization of what we consider to be a modal interpretation. Following previous papers in which we have introduced modalities in the Kochen-Specker theorem, we investigate the consequences of these theorems in relation to the modal interpretations of quantum mechanics.Comment: 21 pages, no figures, preprint submitted to SHPM

Towards an I/O Conformance Testing Theory for Software Product Lines based on Modal Interface Automata

We present an adaptation of input/output conformance (ioco) testing principles to families of similar implementation variants as appearing in product line engineering. Our proposed product line testing theory relies on Modal Interface Automata (MIA) as behavioral specification formalism. MIA enrich I/O-labeled transition systems with may/must modalities to distinguish mandatory from optional behavior, thus providing a semantic notion of intrinsic behavioral variability. In particular, MIA constitute a restricted, yet fully expressive subclass of I/O-labeled modal transition systems, guaranteeing desirable refinement and compositionality properties. The resulting modal-ioco relation defined on MIA is preserved under MIA refinement, which serves as variant derivation mechanism in our product line testing theory. As a result, modal-ioco is proven correct in the sense that it coincides with traditional ioco to hold for every derivable implementation variant. Based on this result, a family-based product line conformance testing framework can be established.Comment: In Proceedings FMSPLE 2015, arXiv:1504.0301

Weighted Modal Transition Systems

Specification theories as a tool in model-driven development processes of component-based software systems have recently attracted a considerable attention. Current specification theories are however qualitative in nature, and therefore fragile in the sense that the inevitable approximation of systems by models, combined with the fundamental unpredictability of hardware platforms, makes it difficult to transfer conclusions about the behavior, based on models, to the actual system. Hence this approach is arguably unsuited for modern software systems. We propose here the first specification theory which allows to capture quantitative aspects during the refinement and implementation process, thus leveraging the problems of the qualitative setting. Our proposed quantitative specification framework uses weighted modal transition systems as a formal model of specifications. These are labeled transition systems with the additional feature that they can model optional behavior which may or may not be implemented by the system. Satisfaction and refinement is lifted from the well-known qualitative to our quantitative setting, by introducing a notion of distances between weighted modal transition systems. We show that quantitative versions of parallel composition as well as quotient (the dual to parallel composition) inherit the properties from the Boolean setting.Comment: Submitted to Formal Methods in System Desig