58 research outputs found
Grafting Hypersequents onto Nested Sequents
We introduce a new Gentzen-style framework of grafted hypersequents that
combines the formalism of nested sequents with that of hypersequents. To
illustrate the potential of the framework, we present novel calculi for the
modal logics and , as well as for extensions of the
modal logics and with the axiom for shift
reflexivity. The latter of these extensions is also known as
in the context of deontic logic. All our calculi enjoy syntactic cut
elimination and can be used in backwards proof search procedures of optimal
complexity. The tableaufication of the calculi for and
yields simplified prefixed tableau calculi for these logic
reminiscent of the simplified tableau system for , which might be
of independent interest
Self-Referential Justifications in Epistemic Logic
This paper is devoted to the study of self-referential proofs and/or justifications, i.e.,valid proofs that prove statements about these same proofs. The goal is to investigate whether such self-referential justifications are present in the reasoning described by standard modal epistemic logics such as . We argue that the modal language by itself is too coarse to capture this concept of self-referentiality and that the language of justification logic can serve as an adequate refinement. We consider well-known modal logics of knowledge/belief and show, using explicit justifications, that , , , and with their respective justification counterparts , , , and describe knowledge that is self-referential in some strong sense. We also demonstrate that self-referentiality can be avoided for and . In order to prove the former result, we develop a machinery of minimal evidence functions used to effectively build models for justification logics. We observe that the calculus used to construct the minimal functions axiomatizes the reflected fragments of justification logics. We also discuss difficulties that result from an introduction of negative introspectio
Complexity Issues in Justification Logic
Justification Logic is an emerging field that studies provability, knowledge, and belief via explicit proofs or justifications that are part of the language. There exist many justification logics closely related to modal epistemic logics of knowledge and belief. Instead of modality â–¡ in pure justification logics, or in addition to modality â–¡ in hybrid logics, which has an existential epistemic reading \u27there exists a proof of F,\u27 all justification logics use constructs t:F, where a justification term t represents a blueprint of a Hilbert-style proof of F. The first justification logic, LP, introduced by Sergei Artemov, was shown to be a justification counterpart of modal logic S4 and serves as a missing link between S4 and Peano arithmetic, thereby solving a long-standing problem of provability semantics for S4 and Int.
The machinery of explicit justifications can be used to analyze well-known epistemic paradoxes, e.g. Gettier\u27s examples of justified true belief that can hardly be considered knowledge, and to find new approaches to the concept of common knowledge. Yet another possible application is the Logical Omniscience Problem, which reflects an undesirable property of knowledge as described by modality when an agent knows all the logical consequences of his/her knowledge. The language of justification logic opens new ways to tackle this problem.
This thesis focuses on quantitative analysis of justification logics. We explore their decidability and complexity of Validity Problem for them. A closer analysis of the realization phenomenon in general and of one procedure in particular enables us to deduce interesting corollaries about self-referentiality for several modal logics. A framework for proving decidability of various justification logics is developed by generalizing the Finite Model Property. Limitations of the method are demonstrated through an example of an undecidable justification logic. We study reflected fragments of justification logics and provide them with an axiomatization and a decision procedure whose complexity (the upper bound) turns out to be uniform for all justification logics, both pure and hybrid. For many justification logics, we also present lower and upper complexity bounds
Uniform interpolation via nested sequents
A modular proof-theoretic framework was recently developed to prove Craig interpolation for normal modal logics based on generalizations of sequent calculi (e.g., nested sequents, hypersequents, and labelled sequents). In this paper, we turn to uniform interpolation, which is stronger than Craig interpolation. We develop a constructive method for proving uniform interpolation via nested sequents and apply it to reprove the uniform interpolation property for normal modal logics K, D, and T. While our method is proof-theoretic, the definition of uniform interpolation for nested sequents also uses semantic notions, including bisimulation modulo an atomic proposition
Extensions of K5: Proof Theory and Uniform Lyndon Interpolation
We introduce a Gentzen-style framework, called layered sequent calculi, for
modal logic K5 and its extensions KD5, K45, KD45, KB5, and S5 with the goal to
investigate the uniform Lyndon interpolation property (ULIP), which implies
both the uniform interpolation property and the Lyndon interpolation property.
We obtain complexity-optimal decision procedures for all logics and present a
constructive proof of the ULIP for K5, which to the best of our knowledge, is
the first such syntactic proof. To prove that the interpolant is correct, we
use model-theoretic methods, especially bisimulation modulo literals.Comment: 20-page conference paper + 5-page appendix with examples and proof
Impure Simplicial Complexes: Complete Axiomatization
Combinatorial topology is used in distributed computing to model concurrency
and asynchrony. The basic structure in combinatorial topology is the simplicial
complex, a collection of subsets called simplices of a set of vertices, closed
under containment. Pure simplicial complexes describe message passing in
asynchronous systems where all processes (agents) are alive, whereas impure
simplicial complexes describe message passing in synchronous systems where
processes may be dead (have crashed). Properties of impure simplicial complexes
can be described in a three-valued multi-agent epistemic logic where the third
value represents formulas that are undefined, e.g., the knowledge and local
propositions of dead agents. In this work we present the axiomatization called
and show that it is sound and complete for the class of
impure complexes. The completeness proof involves the novel construction of the
canonical simplicial model and requires a careful manipulation of undefined
formulas
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