88 research outputs found
On Sub-Propositional Fragments of Modal Logic
In this paper, we consider the well-known modal logics ,
, , and , and we study some of their
sub-propositional fragments, namely the classical Horn fragment, the Krom
fragment, the so-called core fragment, defined as the intersection of the Horn
and the Krom fragments, plus their sub-fragments obtained by limiting the use
of boxes and diamonds in clauses. We focus, first, on the relative expressive
power of such languages: we introduce a suitable measure of expressive power,
and we obtain a complex hierarchy that encompasses all fragments of the
considered logics. Then, after observing the low expressive power, in
particular, of the Horn fragments without diamonds, we study the computational
complexity of their satisfiability problem, proving that, in general, it
becomes polynomial
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
Goal-directed proof theory
This report is the draft of a book about goal directed proof theoretical formulations of non-classical logics. It evolved from a response to the existence of two camps in the applied logic (computer science/artificial intelligence) community. There are those members who believe that the new non-classical logics are the most important ones for applications and that classical logic itself is now no longer the main workhorse of applied logic, and there are those who maintain that classical logic is the only logic worth considering and that within classical logic the Horn clause fragment is the most important one. The book presents a uniform Prolog-like formulation of the landscape of classical and non-classical logics, done in such away that the distinctions and movements from one logic to another seem simple and natural; and within it classical logic becomes just one among many. This should please the non-classical logic camp. It will also please the classical logic camp since the goal directed formulation makes it all look like an algorithmic extension of Logic Programming. The approach also seems to provide very good compuational complexity bounds across its landscape
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
Complexity results for modal logic with recursion via translations and tableaux
This paper studies the complexity of classical modal logics and of their
extension with fixed-point operators, using translations to transfer results
across logics. In particular, we show several complexity results for
multi-agent logics via translations to and from the -calculus and modal
logic, which allow us to transfer known upper and lower bounds. We also use
these translations to introduce a terminating tableau system for the logics we
study, based on Kozen's tableau for the -calculus, and the one of Fitting
and Massacci for modal logic. Finally, we show how to encode the tableaux we
introduced into -calculus formulas. This encoding provides upper bounds
for the satisfiability checking of the few logics we previously did not have
algorithms for.Comment: 43 pages. arXiv admin note: substantial text overlap with
arXiv:2209.1037
Refutation Systems : An Overview and Some Applications to Philosophical Logics
Refutation systems are systems of formal, syntactic derivations, designed to derive the non-valid formulas or logical consequences of a given logic. Here we provide an overview with comprehensive references on the historical development of the theory of refutation systems and discuss some of their applications to philosophical logics
Self-referentiality in Constructive Semantics of Intuitionistic and Modal Logics
This thesis explores self-referentiality in the framework of justification logic. In this framework initialed by Artemov, the language has formulas of the form t:F, which means the term t is a justification of the formula F. Moreover, terms can occur inside formulas and hence it is legal to have t:F(t), which means the term t is a justification of the formula F about t itself. Expressions like this is not only interesting in the semantics of justification logic, but also, as we will see, necessary in applications of justification logic in formalizing constructive contents implicitly carried by modal and intuitionistic logics.
Works initialed by Artemov and followed by Brezhnev and others have successfully extracted constructive contents packaged by modality in many modal logics. Roughly speaking, they offer methods of substituting modalities by terms in various justification logics, and then computing the exact structure of each term. After performing these methods, each formula prefixed by a modality becomes a formula prefixed by a term, which intuitively stands for the justification of the formula being prefixed. In terminology of this framework, we say that modal logics are realized in justification logics.
Within the family of justification logics, the Logic of Proofs LP is perhaps the most important member. As Artemov showed, this logic is not only complete w.r.t. to arithmetical semantics about proofs, but also accommodates the modal logic S4 via realization. Combined with Godel\u27s modal embedding from intuitionistic propositional logic IPC to S4, the Logic of Proofs LP serves as an intermedium via which IPC receives its provability semantics, also known as Brouwer-Heyting-Kolmogorov semantics, or BHK semantics.
This thesis presents the candidate\u27s works in two directions. (1) Following Kuznets\u27result that self-referentiality is necessary for the realization of several modal logics including S4, we show that it is also necessary for BHK semantics. (2) We find a necessary condition for a modal theorem to require self-referentiality in its realization, and using this condition to derive many interesting properties about self-referentiality
Automated proof search in non-classical logics : efficient matrix proof methods for modal and intuitionistic logics
In this thesis we develop efficient methods for automated proof search within
an important class of mathematical logics. The logics considered are the varying,
cumulative and constant domain versions of the first-order modal logics
K, K4, D, D4, T, S4 and S5, and first-order intuitionistic logic. The use of
these non-classical logics is commonplace within Computing Science and Artificial
Intelligence in applications in which efficient machine assisted proof search
is essential.
Traditional techniques for the design of efficient proof methods for classical
logic prove to be of limited use in this context due to their dependence on
properties of classical logic not shared by most of the logics under consideration.
One major contribution of this thesis is to reformulate and abstract some of these
classical techniques to facilitate their application to a wider class of mathematical
logics.
We begin with Bibel's Connection Calculus: a matrix proof method for classical
logic comparable in efficiency with most machine orientated proof methods
for that logic. We reformulate this method to support its decomposition into
a collection of individual techniques for improving the efficiency of proof search
within a standard cut-free sequent calculus for classical logic. Each technique
is presented as a means of alleviating a particular form of redundancy manifest
within sequent-based proof search. One important result that arises from this
anaylsis is an appreciation of the role of unification as a tool for removing certain
proof-theoretic complexities of specific sequent rules; in the case of classical
logic: the interaction of the quantifier rules.
All of the non-classical logics under consideration admit complete sequent
calculi. We anaylse the search spaces induced by these sequent proof systems
and apply the techniques identified previously to remove specific redundancies
found therein. Significantly, our proof-theoretic analysis of the role of unification
renders it useful even within the propositional fragments of modal and
intuitionistic logic
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