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Expressive equivalence of least and inflationary fixed-point logic

2004
*
Annals of Pure and Applied Logic
*

We study the relationship between

doi:10.1016/j.apal.2004.02.001
fatcat:fvmnfo6luvajnk6v57vi5jyfxa
*least**and**inflationary**fixed*-*point**logic*. ... We also show that the alternation hierarchy for IFP collapses to the first level, i.e. the complement*of*any*inflationary**fixed**point*is itself an*inflationary**fixed**point*. ... I am deeply grateful for his patience*and*constructive remarks. Many thanks to Heinz-Dieter Ebbinghaus, Jörg Flum,*and*Michael Möllerfeld for their helpful comments on Section 4. ...##
###
Expressive equivalence of least and inflationary fixed-point logic

*
Proceedings 17th Annual IEEE Symposium on Logic in Computer Science
*

We study the relationship between

doi:10.1109/lics.2002.1029848
dblp:conf/lics/Kreutzer02
fatcat:aofpod6vofhqpbqp3mjkhs3asq
*least**and**inflationary**fixed*-*point**logic*. ... We also show that the alternation hierarchy for IFP collapses to the first level, i.e. the complement*of*any*inflationary**fixed**point*is itself an*inflationary**fixed**point*. ... I am deeply grateful for his patience*and*constructive remarks. Many thanks to Heinz-Dieter Ebbinghaus, Jörg Flum,*and*Michael Möllerfeld for their helpful comments on Section 4. ...##
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Fixed-point extensions of first-order logic

1986
*
Annals of Pure and Applied Logic
*

We prove that the three extensions

doi:10.1016/0168-0072(86)90055-2
fatcat:xpnv7cnvabbhhivlund3hu6iom
*of*first-order*logic*by means*of*positive inductions, monotone inductions,*and*so-called non-monotone (in our terminology,*inflationary*) inductions respectively, all ... have the same*expressive*power in the case*of*finite structures. ... Then the FO + LFP formula :r(~, x)*expresses*the inductive*fixed**point**of*the operator P~ {x :P(x) or ~/,(P, -P, x)}.*fixed**point**of*f if fx = x,*and*a*fixed**point*x*of*f is a*least**fixed**point**of*f if ...##
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Fixed-point extensions of first-order logic

1985
*
26th Annual Symposium on Foundations of Computer Science (sfcs 1985)
*

We prove that the three extensions

doi:10.1109/sfcs.1985.27
dblp:conf/focs/GurevichS85
fatcat:7equt4od7vh6bg22uipcbhu2kq
*of*first-order*logic*by means*of*positive inductions, monotone inductions,*and*so-called non-monotone (in our terminology,*inflationary*) inductions respectively, all ... have the same*expressive*power in the case*of*finite structures. ... Then the FO + LFP formula :r(~, x)*expresses*the inductive*fixed**point**of*the operator P~ {x :P(x) or ~/,(P, -P, x)}.*fixed**point**of*f if fx = x,*and*a*fixed**point*x*of*f is a*least**fixed**point**of*f if ...##
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A restricted second order logic for finite structures
[chapter]

1995
*
Lecture Notes in Computer Science
*

classes are

doi:10.1007/3-540-60178-3_94
fatcat:nttuhzelnzfa7kdq256lr7pnnq
*equivalent*to separations*of*corresponding*fixed**point**logics*. ... We explore the relationship*of*SO | with*fixed**point**logics*, showing that its inclusion relations with these*logics*are*equivalent*to problems in complexity theory. ... I am also grateful to Lauri Hella*and*Victor Vianu for a careful reading*of*a draft*of*this paper. Final manuscript received December 23, 1997 ...##
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Generalising Automaticity to Modal Properties of Finite Structures
[chapter]

2002
*
Lecture Notes in Computer Science
*

We demonstrate that this is a useful

doi:10.1007/3-540-36206-1_11
fatcat:tgw2ps565rditmiphtx7uxomta
*and*fine-grained measure*of*complexity*and*show that it is especially well suited to characterise the*expressive*power*of*modal*fixed*-*point**logics*. ... From this we derive several separation results*of*modal*and*non-modal*fixed*-*point**logics*, some*of*which are already known whereas others are new. ...*least**fixed*-*point**logic*. ...##
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A logic with temporally accessible iteration
[article]

2008
*
arXiv
*
pre-print

Deficiency in

arXiv:0806.2802v1
fatcat:yr4yegpysfhcfgj4dlsd6wcr5m
*expressive*power*of*the first-order*logic*has led to developing its numerous extensions by*fixed**point*operators, such as*Least**Fixed*-*Point*(LFP),*inflationary**fixed*-*point*(IFP), partial ... We show that proposed*logic*FO+TAI subsumes all mentioned*fixed**point*extensions as well as many other*fixed**point**logics*as natural fragments. ... Concluding remarks We proposed in this paper the*logic*with temporally accessible iteration which provides the simple unifying framework for studying*logics*with inductive*fixed**point*operators. ...##
###
Why not negation by fixpoint?

1988
*
Proceedings of the seventh ACM SIGACT-SIGMOD-SIGART symposium on Principles of database systems - PODS '88
*

We also establish that existence

doi:10.1145/308386.308446
dblp:conf/pods/KolaitisP88
fatcat:y2dkzkogt5fqxh6pset5zgs7uu
*of*a*least*fixpoint, for*fixed*DATALOG' programs, is even harder. ... This last result illustrates one more difference between data complexity*and**expression*complexity (cf. Vardi [VA82]). ... ACKNOWLEDGMENTS Many thanks to Stavros Cosmadakis, Alex Schatfer, Larry Stockmeyer, Allen Van Gelder,*and*Moshe Vardi for several useful suggestions*and*comments on an earlier draft*of*this paper. ...##
###
Why not negation by fixpoint?

1991
*
Journal of computer and system sciences (Print)
*

We also establish that existence

doi:10.1016/0022-0000(91)90033-2
fatcat:2uvhfl25ifglpi72et2wijpclq
*of*a*least*fixpoint, for*fixed*DATALOG' programs, is even harder. ... This last result illustrates one more difference between data complexity*and**expression*complexity (cf. Vardi [VA82]). ... ACKNOWLEDGMENTS Many thanks to Stavros Cosmadakis, Alex Schatfer, Larry Stockmeyer, Allen Van Gelder,*and*Moshe Vardi for several useful suggestions*and*comments on an earlier draft*of*this paper. ...##
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Definable fixed points in modal and temporal logics : A survey

2007
*
Journal of Applied Non-Classical Logics
*

The paper presents a survey

doi:10.3166/jancl.17.317-346
fatcat:p4zgyz5thzgtfo4qz6tn6wd7y4
*of*author's results on definable*fixed**points*in modal, temporal,*and*intuitionistic propositional*logics*. ...*least**fixed**points*. ... I wish to*express*my thanks to Larisa Maksimova, Valentin Shehtman, Oleg Kudinov, Jan Zeitz,*and*to the anonymous referee. ...##
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On Core XPath with Inflationary Fixed Points

2012
*
RAIRO - Theoretical Informatics and Applications
*

In this report, we prove the undecidability

doi:10.1051/ita/2012027
fatcat:qwgccmxocbgxhlxi2qo7t2mgae
*of*Core XPath 1.0 (CXP) [6] extended with an*Inflationary**Fixed**Point*(IFP) operator. ... We prove that the satisfiability problem*of*this language is undecidable. In fact, the fragment*of*CXP+IFP containing only the self*and*descendant axes is already undecidable. ... The second author is supported NWO grant 639.021.508*and*by ERC Advanced Grant Webdam on Foundation*of*Web data management. We thank Anuj Dawar for a useful discussion. ...##
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Backtracking games and inflationary fixed points

2006
*
Theoretical Computer Science
*

The corresponding increase in

doi:10.1016/j.tcs.2005.10.030
fatcat:5jyecmxhmbgnvh2ys3nn546ipu
*expressiveness*allows us to use backtracking games as model checking games for*inflationary**fixed*-*point**logics*such as IFP or MIC. ... We identify a natural subclass*of*backtracking games, the simple games,*and*show that these are the "right" model checking games for IFP by a) giving a translation*of*formulae ϕ*and*structures A into simple ... The most important*of*these extensions are*least**fixed**point**logic*(LFP)*and**inflationary**fixed**point**logic*(IFP). ...##
###
The Expressive Power of Two-Variable Least Fixed-Point Logics
[chapter]

2005
*
Lecture Notes in Computer Science
*

The two-variable fragment

doi:10.1007/11549345_37
fatcat:ixiptx4ryrcinj626g4nuovyyu
*of*monadic*least**fixed*-*point**logic*with parameters is as*expressive*as full monadic*least**fixed*-*point**logic*(on binary structures). 2. ... The two-variable fragment*of*monadic*least**fixed*-*point**logic*without parameters is as*expressive*as the two-variable fragment*of*binary*least**fixed*-*point**logic*without parameters. 3. ... Fig. 1 . 1*Expressive*power*of*the two-variable fragments*of**least**fixed*-*point**logic**Least**fixed*-*point**logic*. The*logic*LFP(τ ) is the extension*of*FO(τ ) by*least**fixed*-*point*operators. ...##
###
Backtracking Games and Inflationary Fixed Points
[chapter]

2004
*
Lecture Notes in Computer Science
*

The corresponding increase in

doi:10.1007/978-3-540-27836-8_37
fatcat:nfbe6j4qebhf5jtphd5gtljfda
*expressiveness*allows us to use backtracking games as model checking games for*inflationary**fixed*-*point**logics*such as IFP or MIC. ... We identify a natural subclass*of*backtracking games, the simple games,*and*show that these are the "right" model checking games for IFP by a) giving a translation*of*formulae ϕ*and*structures A into simple ... The most important*of*these extensions are*least**fixed**point**logic*(LFP)*and**inflationary**fixed**point**logic*(IFP). ...##
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Two Forms of One Useful Logic: Existential Fixed Point Logic and Liberal Datalog
[article]

2008
*
arXiv
*
pre-print

We show that the

arXiv:0811.0977v1
fatcat:kps6shu77naxxamqcmpicmkikq
*expressive*power*of*this liberal Datalog is that*of*existential*fixed*-*point**logic*. The exposition is self-contained. ... A natural liberalization*of*Datalog is used in the Distributed Knowledge Authorization Language (DKAL). ... We thank Nikolaj Bjørner for helpful comments on a draft*of*this paper. ...
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