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

2008
*
Electronical Notes in Theoretical Computer Science
*

Some extensions of first-order classical

doi:10.1016/j.entcs.2008.03.067
fatcat:oznbu6jswngrhfaxbqdphlgida
*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 . ...##
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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*. ...##
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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’. ...##
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Stratified least fixpoint logic

1994
*
Theoretical Computer Science
*

A complete sequent calculus

doi:10.1016/0304-3975(94)90091-4
fatcat:jdd4l7j2lbe3fcusdaqtdjmmey
*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. ...##
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A Buchholz Rule for Modal Fixed Point Logics

2011
*
Logica Universalis
*

We introduce a

doi:10.1007/s11787-010-0022-1
fatcat:lwamdhunrbfnhn223fxiqzjkwm
*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

2016
*
Annual Conference for Computer Science Logic
*

In this paper, we consider

doi:10.4230/lipics.csl.2016.42
dblp:conf/csl/BaeldeDS16
fatcat:vpem436twbaarpzosgwkng6kfu
*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*...##
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On the Proof Theory of the Modal mu-Calculus

2008
*
Studia Logica: An International Journal for Symbolic Logic
*

We study

doi:10.1007/s11225-008-9133-6
fatcat:rk4x37l6mbgfxohzv3b675otqi
*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

1997
*
Annals of the Japan Association for Philosophy of Science
*

Therefore,
a

doi:10.4288/jafpos1956.9.69
fatcat:7or3qxowerdhvpvqsws3fn4oqe
*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. ...##
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Extending the Language of Set Theory
[article]

2016
*
arXiv
*
pre-print

We introduce almost self-referential formulas, use them to extend set theory, and relate their expressive power to that of

arXiv:math/0504375v2
fatcat:slhhmaa6erhjjgyc5xp4lnnfcq
*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

2005
*
Annals of Pure and Applied Logic
*

In

doi:10.1016/j.apal.2004.10.004
fatcat:2hb5zd3omvf6fp25jji2m6nvfu
*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. ...##
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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

doi:10.2991/eusflat.2011.140
dblp:conf/eusflat/BelohlavekV11
fatcat:xhbzjt6le5gc7ddnbwv36kh3sq
*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 ∈ ...##
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Canonical completeness of infinitary μ

2008
*
The Journal of Logic and Algebraic Programming
*

This paper also mentions

doi:10.1016/j.jlap.2008.02.005
fatcat:zr34nwn6org2nbytef5efdhfjm
*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. ...##
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A Logical Framework for Convergent Infinite Computations
[article]

2002
*
arXiv
*
pre-print

A class of

arXiv:cs/0105020v3
fatcat:rfe3p54xtnblvb2t44b7g6td4m
*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

1989
*
Journal of symbolic computation
*

Unification theory provides

doi:10.1016/s0747-7171(89)80012-4
fatcat:mjwsvawtirdvbc5dxdflp4n2zi
*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 ...##
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We must know – We shall know
[article]

2020
*
arXiv
*
pre-print

In this article, I focus on

arXiv:2004.09235v1
fatcat:oiympk24rjgejjv5ls4pugiqvm
*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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