M. Grohe. Finite variable logics in descriptive complexity theory. Bulletin of Symbolic Logic, 4:345--398, 1998.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....and let A and B be L structures. Then, the Duplicator wins the existential k pebble game on A and B if and only if every 9L 1 sentence that holds in A also holds in B. The k variable fragments of infinitary logics have played a crucial role in the development of finite model theory (see [17] for a good survey) 3 Combinatorial Characterization as Games It is well known that r CNF formulas may be encoded as finite relational structures. Indeed, let L = fP 0 ; P 1 ; P r g be the finite relational language that consists of r 1 relations of arity r each. An r CNF formula F ....

M. Grohe. Finite variable logics in descriptive complexity theory. Bulletin of Symbolic Logic, 4:345--398, 1998.

....and let A and B be L structures. Then, the Duplicator wins the existential k pebble game on A and B if and only if every 9L 1 sentence that holds in A also holds in B. The k variable fragments of infinitary logics have played a crucial role in the development of finite model theory (see [Gro98] for a good survey) 3 Combinatorial characterization as games It is well know that r CNF formulas may be encoded as finite relational structures. Indeed, let L = fP 0 ; P 1 ; P r g be the finite relational language that consists of r 1 relations of arity r each. An r CNF formula F ....

M. Grohe. Finite variable logics in descriptive complexity theory. Bulletin of Symbolic Logic, 4:345--398, 1998.

....is not expressible in first order logic with the addition of the fixed point operator and counting quantifiers. Some beautiful work by Martin Grohe and his students has shown that there are large and important classes of graphs on which the language FO(wo) LFP; C) does capture order independent P [11, 8, 10]. For these classes of graphs, isomorphism is testable in polynomial time using the natural algorithm for equivalence in a sublanguage of FO(wo) LFP; C) These results include natural classes of graphs for which no polynomial time graph isomorphism algorithm had been previously known. Most ....

M. Grohe, "Finite Variable Logics in Descriptive Complexity Theory," Bulletin of Symbolic Logic , 4(4) (1998), 345- 398.

....variable fragments of C1 : For k 1, we let C k 1 be the set of all C1 formulas that contain at most k variables. Finite variable logics play an important role in descriptive complexity theory; for background material on such logics I refer the reader to the monographs [9; 20] and the survey [13]. We give three typical examples to illustrate the expressive power of our logics: Example 1. For all r 1, the C 2 1 sentence 8x9 =r y E(x; y) says that a graph is r regular. Similarly, for all r; s; t 2 N there is a C 3 1 sentence saying that a graph is strongly regular with ....

M. Grohe. Finite-variable logics in descriptive complexity theory. Bulletin of Symbolic Logic, 4:345-399, 1998.

....proved by Balcazar, Gabarro, and Santha [4] that bisimilarity is complete for polynomial time. My original motivation for this work came from a different direction. Logics with finitely many variables have always been playing an important role in descriptive complexity theory (see, for example, [8, 11, 18]) One of the most important results in this area is the Abiteboul Vianu Theorem [2] which translates the question of whether PTIME equals PSPACE to the purely logical question of whether least fixed point logic and partial fixedpoint logic have the same expressive power. Abiteboul, Vardi, and ....

M. Grohe. Finite-variable logics in descriptive complexity theory. Bulletin of Symbolic Logic, 4:345--399, 1998.

....out to capture polynomial time on several classes of databases. It follows easily from the Immerman Vardi Theorem mentioned earlier that IFP C captures polynomial time on the class of all ordered databases. The following Lemma, which is based on the notion of definable canonization introduced in [13, 14], can be used to extend this result to further classes of databases. The straightforward proof can be found in [13] For an IFP C formula ( w) and a database D we let ( w) D # = f d 2 D # j D # j= d)g. Note, in particular, that if w is an l tuple of num variables, then ....

....earlier that IFP C captures polynomial time on the class of all ordered databases. The following Lemma, which is based on the notion of definable canonization introduced in [13, 14] can be used to extend this result to further classes of databases. The straightforward proof can be found in [13]. For an IFP C formula ( w) and a database D we let ( w) D # = f d 2 D # j D # j= d)g. Note, in particular, that if w is an l tuple of num variables, then ( w) D # is an l ary relation on num. Lemma 1. Let = fR 1 ; Rn g be a database schema, where R i is ....

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M. Grohe. Finite-variable logics in descriptive complexity theory, 1998.

....out to capture polynomial time on several classes of databases. It follows easily from the Immerman Vardi Theorem mentioned earlier that IFP C captures polynomial time on the class of all ordered databases. The following Lemma, which is based on the notion of definable canonization introduced in [13, 14], can be used to extend this result to further classes of databases. The straightforward proof can be found in [13] For an IFP C formula ( w) and a database D we let ( w) D # = f d 2 D # j D # j= d)g. Note, in particular, that if w is an l tuple of num variables, then ....

....earlier that IFP C captures polynomial time on the class of all ordered databases. The following Lemma, which is based on the notion of definable canonization introduced in [13, 14] can be used to extend this result to further classes of databases. The straightforward proof can be found in [13]. For an IFP C formula ( w) and a database D we let ( w) D # = f d 2 D # j D # j= d)g. Note, in particular, that if w is an l tuple of num variables, then ( w) D # is an l ary relation on num. Lemma 5. Let = fR 1 ; Rn g be a database schema, where R i ....

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M. Grohe. Finite-variable logics in descriptive complexity theory, 1998.

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Grohe, M. (1998). Finite variable logics in descriptive complexity theory. Bulletin of Symbolic Logic, 4:345--398.

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