M. Otto. Bounded Variable Logics and Counting, volume 9 of Lecture Notes in Logic. Springer, 1997.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....We also denote it by I k . Blurring the distinction between an ordered structure I k (A) and its codeword over f0; 1g again, we can consider I k as an encoding of the complete L k theories. For a thorough presentation of the material of this section we refer the reader to [5] or [16]. 2.5. Non uniform polynomial time. I assume that the reader is familiar with the basics of complexity theory. A complexity class that is not so well known, but very important for us, is P=poly, non uniform P, which has been introduced by Karp and Lipton [12] A P=poly computation is a ....

M. Otto. Bounded Variable Logics and Counting, volume 9 of Lecture Notes in Logic. Springer-Verlag, 1997.

....aim to be comprehensive in its coverage of the work on finite variable logics in finite model theory as several strands of this work are omitted for lack of space. Significant among these is the work on finite variable logics and counting which has been covered in the excellent work by Otto [50], and the relation of finite variable logics to modal and temporal logics for which a good starting point is the survey by Hodkinson [34] One of the central concerns of finite model theory is to study the limits of the expressive power of logical languages on finite structures. It is in this ....

M. Otto, Bounded Variable Logics and Counting, volume 9 of Lecture Notes in Logic, Springer, 1997.

....in particular it implies a collapse of the polynomial hierarchy. The question for such a canonization function came up in the context of the problem of whether there is a logic for P. Slight modifications of our result yield answers to questions of Dawar, Lindell, and Weinstein [4] and Otto [16] concerning the inversion of the so called L k invariants. 1 Introduction Membership in a class of ordered finite structures can be tested in polynomial time if, and only if, the class can be defined in least fixed point logic. This is a fundamental result of Immerman [12] and Vardi [19] In ....

....logic that captures polynomial time on all finite structures, or to prove that no such logic exists. A simple argument shows that the latter would imply P 6= NP. Although at the moment, either solution of the problem seems to be out of reach, various partial results are known. Martin Otto [16] proposed to capture those P prob lems that are invariant under some equivalence relation. To understand this, we first note that a capturing result such as membership in a class of structures can be tested in P if, and only if, the class is definable in a logic L silently assumes that all ....

[Article contains additional citation context not shown here]

M. Otto. Bounded Variable Logics and Counting, volume 9 of Lecture Notes in Logic. Springer-Verlag, 1997.

....We say that a formula in a counting logic only uses unary counting if it contains at most unary counting subformulas. It is easy to see that each IFP C formula is equivalent to an IFP C formula that only uses unary counting (cf. 6] However, the analogous statement for FO C is not true (cf. [16]) It is easy to see that on ordered structures every IFP C formula is equivalent to an IFP formula. Gr adel and Otto [6] derived a normal form result analogous to Theorem 1 for IFP C. We need the following lemma. which can be proved similarly to Theorem 1. Lemma 4 Every IFP C formula ....

M. Otto. Bounded Variable Logics and Counting, volume 9 of Lecture Notes in Logic. Springer-Verlag, 1997.

....related the counting logics to the so called Lehman Weisfeiler algorithm, a combinatorial algorithm for graph isomorphism. Using Ehrenfeucht Frasse games for the logic, they could refute a long standing conjecture that this algorithm decides isomorphism at least for graphs of bounded valence. Otto [63, 64] embarked on a systematic study of the finite variable logics with counting and lifted most results for the logics without counting to the counting extensions. More recently, the 2 variable logic L 2 and its variants such as, for example, 2 variable transitive closure logic, have received much ....

....of the finite variable logics. The main developments in the theory of finite variable logics are presented in a rather condensed form in Sections 3 and 4. Most proofs are only sketched, if given at all. The reader who worries about the details here is referred to the literature (for example, [12, 19, 64]) Section 5 is concerned with the question of whether we can compute a model for a given L k theory. More precisely, we study the complexity of inverting the L k invariants. Furthermore, we discuss the abovementioned canonization problem here. The right definition of a canonization ....

[Article contains additional citation context not shown here]

M. Otto. Bounded Variable Logics and Counting, volume 9 of Lecture Notes in Logic. Springer-Verlag, 1997.

....are PTIME invertible, in a sense that is made precise in (5.3) then there is a logic for the intersection of PTIME with the infinitary k variable logic L k 1 , that is, a logic for PTIME in the world of k variables . See (5. 3) for a discussion of this topic; for more details see [11]. On the other hand, it is known that on certain classes of structures, such as ordered structures, trees, or graphs of bounded degree, we have logics for PTIME. The observation that the restrictions of I k to all these classes are invertible supports the feeling that the two questions are ....

M. Otto. Bounded Variable Logics and Counting, volume 9 of Lecture Notes in Logic. Springer-Verlag, 1997.

....in particular it implies a collapse of the polynomial hierarchy. The question for such a canonization function came up in the context of the problem of whether there is a logic for P. Slight modifications of our result yield answers to questions of Dawar, Lindell, and Weinstein [4] and Otto [16] concerning the inversion of the so called L k invariants. 1 Introduction Membership in a class of ordered finite structures can be tested in polynomial time if, and only if, the class can be defined in least fixed point logic. This is a fundamental result of Immerman [12] and Vardi [19] In ....

....logic that captures polynomial time on all finite structures, or to prove that no such logic exists. A simple argument shows that the latter would imply P 6= NP. Although at the moment, either solution of the problem seems to be out of reach, various partial results are known. Martin Otto [16] proposed to capture those P prob lems that are invariant under some equivalence relation. To understand this, we first note that a capturing result such as membership in a class of structures can be tested in P if, and only if, the class is definable in a logic L silently assumes that all ....

[Article contains additional citation context not shown here]

M. Otto. Bounded Variable Logics and Counting, volume 9 of Lecture Notes in Logic. Springer-Verlag, 1997.

....sorts of variables) along with the following additional rule: if is a formula and x is a variable of the first sort, then #x is a term. The intended semantics is that #x denotes the number (i.e. the member of the number sort) of elements that satisfy the formula . It can now be proved (see [32]) that any formula of LFP(Count) which has no free variables of the second sort can be interpreted in a unique way in a structure (A; R; c) Thus, a sentence of LFP(Count) does define a class of structures, and we can say that LFP(Count) meets our criteria for being a logic. What s more, this ....

....would introduce infinitely many variables into the formula, which is not permitted. The logic L 1 (C) therefore, has expressive power greater than L 1 . In fact, it can be shown that it also subsumes the expressive power of LFP(Count) or the similarly defined PFP(Count) see [32]) Hence, the following result from [5] shows that counting, along with LFP, is not sufficient to express all polynomial time sufficient to Theorem 4 There is a polynomial time decidable class of graphs that is not definable in L 1 (C) 4.3 Generalised Quantifiers The counting quantifiers ....

M. Otto. Bounded Variable Logics and Counting, volume 9 of Lecture Notes in Logic. Springer, 1997.

....We also denote it by I k . Blurring the distinction between an ordered structure I k (A) and its codeword over f0; 1g again, we can consider I k as an encoding of the complete L k theories. For a thorough presentation of the material of this section we refer the reader to [5] or [16]. 2.5. Non uniform polynomial time. I assume that the reader is familiar with the basics of complexity theory. A complexity class that is not so well known, but very important for us, is P=poly, non uniform P, which has been introduced by Karp and Lipton [12] A P=poly computation is a ....

M. Otto. Bounded Variable Logics and Counting, volume 9 of Lecture Notes in Logic. Springer-Verlag, 1997.

....We say that a formula in a counting logic only uses unary counting if it contains at most unary counting subformulas. It is easy to see that each IFP C formula is equivalent to an IFP C formula that only uses unary counting (cf. 6] However, the analogous statement for FO C is not true (cf. [16]) It is easy to see that on ordered structures every IFP C formula is equivalent to an IFP formula. Gradel and Otto [6] derived a normal form result analogous to Theorem 1 for IFP C. 1.3 Infinitary Logics. L1 denotes the usual infinitary logic which is obtained by allowing disjunctions and ....

M. Otto. Bounded Variable Logics and Counting, volume 9 of Lecture Notes in Logic. Springer-Verlag, 1997.

No context found.

M. Otto. Bounded Variable Logics and Counting, volume 9 of Lecture Notes in Logic. Springer, 1997.

....1 2 MARTIN GROHE logics to the so called Lehman Weisfeiler algorithm, a combinatorial algorithm for graph isomorphism. Using Ehrenfeucht Frasse games for the logic, they could refute a long standing conjecture that this algorithm decides isomorphism at least for graphs of bounded valence. Otto [63, 64] embarked on a systematic study of the finite variable logics with counting and lifted most results for the logics without counting to the counting extensions. More recently, the 2 variable logic L 2 and its variants such as, for example, 2 variable transitive closure logic, have received much ....

....of the finite variable logics. The main developments in the theory of finite variable logics are presented in a rather condensed form in Sections 3 and 4. Most proofs are only sketched, if given at all. The reader who worries about the details here is referred to the literature (for example, [12, 19, 64]) Section 5 is concerned with the question of whether we can compute a model for a given L k theory. More precisely, we study the complexity of inverting the L k invariants. Furthermore, we discuss the abovementioned canonization problem here. The right definition of a canonization ....

[Article contains additional citation context not shown here]

M. Otto. Bounded Variable Logics and Counting, volume 9 of Lecture Notes in Logic. Springer-Verlag, 1997.

....discussion in Section 5. 2 Preliminaries Throughout the paper we use the terminology and notation of mathematical logic [EFT94] For background on database theory we refer to Abiteboul, Hull, and Vianu [AHV95] and for nite model theory to Ebbinghaus and Flum [EF95] Immerman [Imm98] and Otto [Ott97]. A relational vocabulary is what in the eld of databases is known as a relational schema; a structure over is what is known as an instance of that schema with an explicit domain. Structures are always assumed to be nite in this paper. We denote the domain of a structure A by A, and the ....

....that R is a chain, and first(x) and last(z) de ne the rst and the last element of the chain, respectively. This yields the following proposition. Proposition 3.14 FO(IFOR) lies strictly between FO(IFP) and FO(FOR) 3. 5 A comparison with logics that count In ationary xpoint logic with counting [GO93, Ott96, Ott97], here denoted by FO(IFP; #) is a two sorted logic. With any structure A with universe A, we associate the two sorted structure A : A [ hf0; ng; i with 22 n = jAj and where is the canonical ordering on f0; ng. The two sorts are related by counting terms: if (x; y) is a ....

[Article contains additional citation context not shown here]

M. Otto. Bounded Variable Logics and Counting, volume 9 of Lecture Notes in Logic. Springer, 1997.

.... BQL and FO(FOR) 2 Preliminaries Throughout the paper we will use the terminology and notation of mathematical logic [EFT94] For background on database theory we refer to Abiteboul, Hull, and Vianu [AHV95] and for nite model theory to Ebbinghaus and Flum [EF95] Immerman [Imm98] and Otto [Ott97]. A relational vocabulary is what in the eld of databases is known as a relational schema; a structure over is what is known as an instance of that schema with an explicit domain. Structures are always assumed to be nite in this paper. We denote the domain of a structure A by A, and the ....

....sentence saying that R is a chain, and first(x) and last(z) de ne the rst and the last element of the chain, respectively. The above implies: Proposition 3.14 FO(IFOR) lies strictly between FO(IFP) and FO(FOR) 3. 5 A comparison with logics that count In ationary xpoint logic with counting [GO93, Ott96, Ott97], denoted by FO(IFP; #) is a two sorted logic. With any structure A with universe A, we associate the two sorted structure A : A [ hf0; ng; i with n = jAj 22 and where is the canonical ordering on f0; ng. The two sorts are related by counting terms: if (x; y) is a ....

[Article contains additional citation context not shown here]

M. Otto. Bounded Variable Logics and Counting, volume 9 of Lecture Notes in Logic. Springer, 1997.

No context found.

M. Otto. Bounded Variable Logics and Counting, volume 9 of Lecture Notes in Logic. Springer, 1997.

No context found.

M. Otto. Bounded Variable Logics and Counting, volume 9 of Lecture Notes in Logic. Springer, 1997.

Online articles have much greater impact   More about CiteSeer.IST   Add search form to your site   Submit documents   Feedback  

CiteSeer.IST - Copyright Penn State and NEC