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An Infinitary System for the Least Fixed-Point Logic restricted to Finite Models

Alexandre Matos Arruda, Ana Teresa Martins
2008 Electronical Notes in Theoretical Computer Science
Some extensions of first-order classical logic (F OL) with fixed-point operators, as the least fixed-point logic (LF P ), were proposed to deal with problems related to the expressivity of F OL.  ...  The notion of the least fixed-point of an operator is widely applied in computer science as, for instance, in the context of query languages for relational databases.  ...  Conclusions In this paper, we introduced an infinitary natural deduction system to the Least Fixed-Point logic restricted to finite models, LF P fin .  ...

Page 4871 of Mathematical Reviews Vol. , Issue 2004f [page]

2004 Mathematical Reviews
In contrast to the traditional least fixed point semantics of Datalog, in inf-Datalog some recursive rules (corresponding to least fixed points) are allowed to unfold only finitely many times, whereas  ...  The labels can be of an arbitrary semantic nature.  ...

Page 1569 of Mathematical Reviews Vol. , Issue 2003C [page]

2003 Mathematical Reviews
Summary: “We consider an extension of modal logic with an operator for constructing inflationary fixed points, just as the modal u-calculus extends basic modal logic with an operator for least fixed points  ...  The proposed deductive procedure consists of three separate decid- able deductive procedures replacing the infinitary omega-type rule for the operator ‘always’.  ...

Stratified least fixpoint logic

Kevin J. Compton
1994 Theoretical Computer Science
A complete sequent calculus with one infinitary rule is given for SLFP. It is argued that SLFP is the most appropriate assertion language for program verification.  ...  Stratified least jixpoint logic, or SLFP, characterizes the expressibility of stratified logic programs and, in a different formulation, has been used as a logic of imperative programs.  ...  Partial correctness is a logical notion with respect to first-order logic, but total correctness is not. The infinitary nature of the deductive system for SLFP cannot be avoided.  ...

A Buchholz Rule for Modal Fixed Point Logics

Gerhard Jäger, Thomas Studer
2011 Logica Universalis
We introduce a deductive system based on an Ω-rule tailored for modal fixed point logic and develop the basic techniques for establishing soundness and completeness of the corresponding system.  ...  The aim of this paper is to put this approach into the new context of modal fixed point logic.  ...  The article Jäger, Kretz, and Studer [7] presents and studies an infinitary version of the full propositional modal µ-calculus which treats greatest fixed points by an infinitary rule reminiscent of  ...

Infinitary Proof Theory: the Multiplicative Additive Case

David Baelde, Amina Doumane, Alexis Saurin, Marc Herbstritt
2016 Annual Conference for Computer Science Logic
In this paper, we consider the infinitary proof system µMALL ∞ for multiplicative and additive linear logic extended with least and greatest fixed points, and prove these two key results.  ...  for lattice logic (purely additive linear logic with least and greatest fixed points) and showed that certain cut reductions converge to a limit cut-free derivation.  ...  µMALL and its infinitary proof system µMALL ∞ In this section we introduce multiplicative additive linear logic extended with least and greatest fixed point operators, and an infinitary proof system for  ...

On the Proof Theory of the Modal mu-Calculus

Thomas Studer
2008 Studia Logica: An International Journal for Symbolic Logic
We study the proof theoretic relationship between several deductive systems for the modal mu-calculus.  ...  Moreover, this provides a new proof theoretic proof for the finite model property of the mu-calculus.  ...  They also make use of an infinitary deduction rule which derives the validity of a greatest fixed point from the validity of all its (infinitely many) finite approximations.  ...

Existence and Consistence

Yuzuru KAKUDA
1997 Annals of the Japan Association for Philosophy of Science
Therefore, a natural system with the above view is desired, which can be expressed and developed in the ordinary natural language as possible for pedagogical view point.  ...  with ‡™"¥ƒ©, C and C, "C"¥ƒ¦ initial hyperseqents is an admissible rule in the cut-free system for hyperseqents.  ...

Extending the Language of Set Theory [article]

Dmytro Taranovsky
2016 arXiv   pre-print
We introduce almost self-referential formulas, use them to extend set theory, and relate their expressive power to that of infinitary logic. We discuss the nature of proper classes.  ...  We discuss the problems of incompleteness and inexpressibility.  ...  A less expressive extension is the least fixed point logic.  ...

About cut elimination for logics of common knowledge

Luca Alberucci, Gerhard Jäger
2005 Annals of Pure and Applied Logic
In the following we will present several deductive systems for common knowledge above epistemic logics -such as K, T, S4 and S5 -with a fixed number of agents.  ...  , psychology and many other fields which deal with the interaction within a group of "agents", agreement or coordinated actions.  ...  Acknowledgement Alberucci's research was supported by the Swiss National Science Foundation.  ...

Fuzzy attribute logic with model constraints

2011 Proceedings of the 7th conference of the European Society for Fuzzy Logic and Technology (EUSFLAT-2011)
Presented are preliminary results on ordinary-style and graded-style completeness results for fuzzy attribute logic with models constrained by fuzzy closure operators with hedges.  ...  P103/11/1456 of the Czech Science Foundation and by grant no. MSM 6198959214.  ...  On the other hand, we can take an approach described in [9] and extend the deductive system by an additional infinitary deduction rule: (Add ω C ) from E ⇒ F i (i ∈ I) such that E, F i ∈ fix(C) (i ∈  ...

Canonical completeness of infinitary μ

Gerhard Jäger, Mathis Kretz, Thomas Studer
2008 The Journal of Logic and Algebraic Programming
This paper also mentions an infinitary derivation rule, similar to those which we will introduce later, and proves soundness as well as completeness of a deduction system incorporating this rule.  ...  This paper presents a new model construction for a natural cut-free infinitary version K + ω (µ) of the propositional modal µ-calculus.  ...  The µ-heights and finite sequences of ordinals play an important role in the context of so-called signed truth sets.  ...

A Logical Framework for Convergent Infinite Computations [article]

Wei Li, Shilong Ma, Yuefei Sui, Ke Xu
2002 arXiv   pre-print
A class of fixed points characterizing the logical properties of the limits can be represented by means of infinite-length terms defined by Cauchy sequences.  ...  On the basis of infinitary terms, a computation model for convergent infinite computations is proposed.  ...  For example, being finite can be expressed in the infinite logic, and a fix point of a monotonic function in a complete lattice can be expressed in the logic for the infinite length terms.  ...

Unification theory

Jörg H. Siekmann
1989 Journal of symbolic computation
Unification theory provides the formal framework for investigations into the properties of this operation.  ...  This operation, called unification in work on deduction, is the "addition-and-multiplication" of AI-systems and is consequently often supported by special purpose hardware or by a fast instruction set  ...  A standard set of inference rules for equational logic is the following: An equation s = t can be derived orproved from an axiomatization E, E t-s = t, if it can be obtained in finitely many steps from  ...

We must know – We shall know [article]

Jacob Zimbarg Sobrinho
2020 arXiv   pre-print
In this article, I focus on the resiliency of the P=?NP problem. The main point to deal with is the change of the underlying logic from first to second-order logic.  ...  In this manner, after developing the initial steps of this change, I can hint that the solution goes in the direction of the coincidence of both classes, i.e., P=NP.  ...  A problem is in polynomial time if and only if it is describable in first-order logic with the addition of the least-fixed-point operator.  ...
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