Dynamic Topological Logic

Kremer, Philip; Mint︠s︡, Grigori

doi:10.1007/978-1-4020-5587-4_10

Cited by 14 publications

(7 citation statements)

References 16 publications

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“…DTL is a combination of S4 and temporal logic and has been studied recently by several researchers (Artemov et al 1997;Kremer and Mints 2005;Konev et al 2006;Mints 2006) † . DTL is a combination of S4 and temporal logic and has been studied recently by several researchers (Artemov et al 1997;Kremer and Mints 2005;Konev et al 2006;Mints 2006) † .…”

Section: Proposed Embedding Theoremsmentioning

confidence: 99%

“…Some Gentzen-type cut-free sequent calculi and Hilbert-type axiom schemes were introduced in Artemov et al (1997) for S4F and S4C, and the complete topological and Kripke-type semantics were also obtained for S4F and S4C. Trimodal DTLs were formalised semantically in Kremer and Mints (2005) by combining the S4 modal operator ♥ (interior) and the temporal operators X and G. Although sequent calculi for S4F and S4C have been studied, sequent calculi for trimodal DTLs have not been. Trimodal DTLs were formalised semantically in Kremer and Mints (2005) by combining the S4 modal operator ♥ (interior) and the temporal operators X and G. Although sequent calculi for S4F and S4C have been studied, sequent calculi for trimodal DTLs have not been.…”

Section: Proposed Embedding Theoremsmentioning

confidence: 99%

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Embedding theorems for LTL and its variants

Kamide¹

2014

Math. Struct. Comp. Sci.

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In this paper, we prove some embedding theorems for LTL (linear-time temporal logic) and its variants: viz. some generalisations, extensions and fragments of LTL. Using these embedding theorems, we give uniform proofs of the completeness, cut-elimination and/or decidability theorems for LTL and its variants. The proposed embedding theorems clarify the relationships between some LTL-variations (for example, LTL, a dynamic topological logic, a fixpoint logic, a spatial logic, Prior's logic, Davies' logic and an NP-complete LTL) and some traditional logics (for example, classical logic, intuitionistic logic and infinitary logic).We will introduce a Gentzen-type sequent calculus GLT ω for a generalised first-order LTL (GLTL) in Section 4, and we will prove cut-elimination and completeness theorems for GLT ω using some embedding theorems. GLT ω has modal operatorsand it includes (first-order) LT ω as a special case.In the following, we will explain that the proposed new modal operators ♥ i , ♥ G and ♥ F can be regarded as generalisations of X, G and F. These operators are intended to characterise the axiom schemewhere K * is the set of all words of finite length of the alphabetWe now suppose that for any formula α, we have f α is a mapping on the set of formulas such that♥ G α then becomes a fixpoint of f α . This axiom scheme just corresponds to the so-called iterative interpretation of common knowledge. On the other hand, if we takewe can understand ♥ 1 and ♥ G , respectively, as the temporal operators X and G in LTL. The corresponding axiom scheme for the singleton case represents the LTL-axiom schemeSo the operator ♥ G can be regarded as a natural generalisation of G. Similarly, ♥ F can be regarded as a generalisation of F. Infinitary extensions of LTLIn Section 5, we will introduce two Gentzen-type sequent calcului L ω and L − ω , which are infinitary extensions of LTL:-L ω includes a variant of dynamic topological logics; -L − ω , which is a subsystem of L ω , is an integration of both LT ω and LK ω . We will then prove cut-elimination theorems for L ω and L − ω and a completeness theorem for L − ω . There is no completeness theorem for L ω since a subsystem S4 ω of L ω is known to be Kripke-incomplete. S4 ω is an extension of both IL and the normal modal logic S4, and has been used as a base logic for formalising game theory (Kaneko and Nagashima 1997).

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Section: Proposed Embedding Theoremsmentioning

confidence: 99%

Section: Proposed Embedding Theoremsmentioning

confidence: 99%

Embedding theorems for LTL and its variants

Kamide¹

2014

Math. Struct. Comp. Sci.

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“…There are two main streams. One of them is dedicated to adding topological operators to temporal logic (see [172,2]). These logics are generally called dynamic topological logics and were introduced in [171,170,173] and in [12].…”

Section: Space-timementioning

confidence: 99%

Analysis and Synthesis of Logics

Gabbay

Carnieli²,

Coniglio³

et al. 2008

Applied Logic Series

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SCOPE OF THE SERIESLogic is applied in an increasingly wide variety of disciplines, from the traditional subjects of philosophy and mathematics to the more recent disciplines of cognitive science, computer science, artificial intelligence, and linguistics, leading to new vigor in this ancient subject. Springer, through its Applied Logic Series, seeks to provide a home for outstanding books and research monographs in applied logic, and in doing so demonstrates the underlying unity and applicability of logic.The titles published in this series are listed at the end of this volume. Library of Congress Control Number: 2008920294ISBN 978-1-4020-6781-5 (HB) ISBN 978-1-4020-6782-2 (e-book)Published by Springer,

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“…Pioneering works in this field are Gödel's paper about provability and those by McKinsey and McKinsey and Tarski about the use of modal logic for topological spaces. In the last decade we can see works about several types of modalities on topology . The use of modal logics in geometry has also been studied in .…”

Section: Introductionmentioning

confidence: 99%

Completeness of a functional system for surjective functions

Burrieza

Ruiz

Guzmán

2017

Mathematical Logic Qtrly

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Combining modalities has proven to have interesting applications and many approaches that combine time with other types of modalities have been developed. One of these approaches uses accessibility functions between flows of time to study the basic properties of the functions, such as being total or partial, injective, surjective, etc. The completeness of certain systems expressing many of these properties, with the exception of surjectivity, has been proven. In this paper we propose a language with nominals to denote the initial and final sets of each function in a model in order to obtain a complete system for reasoning about surjective partial functions.

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Dynamic Topological Logic

Cited by 14 publications

References 16 publications

Embedding theorems for LTL and its variants

Embedding theorems for LTL and its variants

Analysis and Synthesis of Logics

Completeness of a functional system for surjective functions

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