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Expressive equivalence of least and inflationary fixed-point logic

Stephan Kreutzer
2004 Annals of Pure and Applied Logic  
We study the relationship between 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.  ... 
doi:10.1016/j.apal.2004.02.001 fatcat:fvmnfo6luvajnk6v57vi5jyfxa

Expressive equivalence of least and inflationary fixed-point logic

S. Kreutzer
Proceedings 17th Annual IEEE Symposium on Logic in Computer Science  
We study the relationship between 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.  ... 
doi:10.1109/lics.2002.1029848 dblp:conf/lics/Kreutzer02 fatcat:aofpod6vofhqpbqp3mjkhs3asq

Fixed-point extensions of first-order logic

Yuri Gurevich, Saharon Shelah
1986 Annals of Pure and Applied Logic  
We prove that the three extensions 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  ... 
doi:10.1016/0168-0072(86)90055-2 fatcat:xpnv7cnvabbhhivlund3hu6iom

Fixed-point extensions of first-order logic

Yuri Gurevich, Saharon Shelah
1985 26th Annual Symposium on Foundations of Computer Science (sfcs 1985)  
We prove that the three extensions 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  ... 
doi:10.1109/sfcs.1985.27 dblp:conf/focs/GurevichS85 fatcat:7equt4od7vh6bg22uipcbhu2kq

A restricted second order logic for finite structures [chapter]

Anuj Dawar
1995 Lecture Notes in Computer Science  
classes are 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  ... 
doi:10.1007/3-540-60178-3_94 fatcat:nttuhzelnzfa7kdq256lr7pnnq

Generalising Automaticity to Modal Properties of Finite Structures [chapter]

A. Dawar, S. Kreutzer
2002 Lecture Notes in Computer Science  
We demonstrate that this is a useful 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.  ... 
doi:10.1007/3-540-36206-1_11 fatcat:tgw2ps565rditmiphtx7uxomta

A logic with temporally accessible iteration [article]

Alexei Lisitsa
2008 arXiv   pre-print
Deficiency in 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.  ... 
arXiv:0806.2802v1 fatcat:yr4yegpysfhcfgj4dlsd6wcr5m

Why not negation by fixpoint?

Phokion G. Kolaitis, Christos H. Papadimitriou
1988 Proceedings of the seventh ACM SIGACT-SIGMOD-SIGART symposium on Principles of database systems - PODS '88  
We also establish that existence 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.  ... 
doi:10.1145/308386.308446 dblp:conf/pods/KolaitisP88 fatcat:y2dkzkogt5fqxh6pset5zgs7uu

Why not negation by fixpoint?

Phokion G. Kolaitis, Christos H. Papadimitriou
1991 Journal of computer and system sciences (Print)  
We also establish that existence 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.  ... 
doi:10.1016/0022-0000(91)90033-2 fatcat:2uvhfl25ifglpi72et2wijpclq

Definable fixed points in modal and temporal logics : A survey

Sergey Mardaev
2007 Journal of Applied Non-Classical Logics  
The paper presents a survey 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.  ... 
doi:10.3166/jancl.17.317-346 fatcat:p4zgyz5thzgtfo4qz6tn6wd7y4

On Core XPath with Inflationary Fixed Points

Loredana Afanasiev, Balder ten Cate
2012 RAIRO - Theoretical Informatics and Applications  
In this report, we prove the undecidability 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.  ... 
doi:10.1051/ita/2012027 fatcat:qwgccmxocbgxhlxi2qo7t2mgae

Backtracking games and inflationary fixed points

Anuj Dawar, Erich Grädel, Stephan Kreutzer
2006 Theoretical Computer Science  
The corresponding increase in 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).  ... 
doi:10.1016/j.tcs.2005.10.030 fatcat:5jyecmxhmbgnvh2ys3nn546ipu

The Expressive Power of Two-Variable Least Fixed-Point Logics [chapter]

Martin Grohe, Stephan Kreutzer, Nicole Schweikardt
2005 Lecture Notes in Computer Science  
The two-variable fragment 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.  ... 
doi:10.1007/11549345_37 fatcat:ixiptx4ryrcinj626g4nuovyyu

Backtracking Games and Inflationary Fixed Points [chapter]

Anuj Dawar, Erich Grädel, Stephan Kreutzer
2004 Lecture Notes in Computer Science  
The corresponding increase in 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).  ... 
doi:10.1007/978-3-540-27836-8_37 fatcat:nfbe6j4qebhf5jtphd5gtljfda

Two Forms of One Useful Logic: Existential Fixed Point Logic and Liberal Datalog [article]

Andreas Blass, Yuri Gurevich (Microsoft Research)
2008 arXiv   pre-print
We show that the 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.  ... 
arXiv:0811.0977v1 fatcat:kps6shu77naxxamqcmpicmkikq
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