N. Immerman, Expressibility as a Complexity Measure: Results and Directions, Proc. of 2nd Conf. on Structure in Complexity Theory (1987), 194--202.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....models for other kinds of structures like trees or graphs. In descriptive complexity, since Fagin [Fag74] showed that the complexity class NP coincides with the sets of models of existential second order ( 1 ) sentences, many complexity classes have been characterized by extensions of FO logic [Var82, Imm86, Imm87, AV89, Gr92] and there is still hope that separations of complexity classes might be possible by separating the expressive power of the respective logics. For a recent result in this direction see the paper of Libkin and Wong [LW98] Despite its importance as an ingredient for more expressive logics, it is ....

N. Immerman. Expressibility as a complexity measure: results and directions. In Second Structure in Complexity Conference, pages 194--202, 1987.

....non deterministic Turing machines can be axiomatised by first order logic plus transitive closure. Thus on finite ordered structures first order logic with transitive closure and LOGSPACEbounded non deterministic Turing machines define exactly the same classes of structures, as Immerman showed [3]. In this paper, we start from classes of finite structures definable in RSRL and search for the computing device that is required to decide these classes. To do so, we will systematically distinguish between two types of RSRL: RSRL as defined by Richter and so called chainless RSRL, which is a ....

....is decidable, while King and Simov [7] proved the stronger notion of grammaticality of an SRL formula to be undecidable. It is well known that satisfiability of classical first order logic is undecidable [1] On the other hand, first order logic plus transitive closure is in LOGSPACE, as Immerman [3] showed. That is to say, given a formula of first order logic with transitive closure and a finite first order structure, the complexity to calculate whether the formula holds true in the given structure is LOGSPACE bounded by the size of the structure. It turns out that the corresponding question ....

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Immerman, N., Expressibility as a complexity measure: Results and directions, in: Second Structure in Complexity Theory Conference (1987), pp. 194--202.

....power of our finite variable logics is somehow unbalanced: On the one hand they can express all fixed point definable queries, they can even express non recursive queries, whereas on the other hand they fail to say that a set has even cardinality. To overcome this mismatch, Immerman and Lander [45, 47, 49] augmented the logics by counting quantifiers such as 9 m . Recall the definition of the logic C k (see page 5) We also consider the infinitary k variable logic with counting quantifiers C k 1 whose formulas are obtained by adding the rule W where is a set of formulas to the ....

.... ; where = 0 and = 2 are abbreviations with the obvious meaning. Using the pebble game (on the two sorted structure A # ) it can easily be seen that this query is not definable in IFP # . Thus we have IFP # IFP C. Fixed point logic with counting has been introduced by Immerman [47]. In the form presented here it is due to Gradel and Otto [25] Though it may seem artificial at first sight, the logic has turned out to be quite robust. Gradel and Otto [25] studied several different formalizations of the concept inductive definability with counting and proved them all to be ....

N. Immerman. Expressibility as a complexity measure: results and directions. In Proceedings of the 2nd IEEE Symposium on Structure in Complexity Theory, pages 194--202, 1987.

....graph in C, we obtain a quite surprising corollary: Corollary 1. Let C be a class of planar graphs that is closed under taking minors. Then C is definable in fixed point logic. As another by product of our main results, we obtain a theorem that continues a study initiated by Immerman and Lander [17, 18]. C k denotes the k variable (first order) logic with counting quantifiers. Theorem 3. Let k 1. For each database D of tree width at most k there is a C k 3 sentence that characterizes D up to isomorphism. On the other hand, Cai, Furer, and Immerman [6] proved that for each k 1 there ....

N. Immerman. Expressibility as a complexity measure: results and directions. In Proceedings of the 2nd IEEE Symposium on Structure in Complexity Theory, pages 194--202, 1987.

....one circuit has quite small depth, while the other implements the naive procedure and has very large depth, but is activated only for a very small fraction of inputs. To get used to the circuits, we prove the following fact, which will be useful in the future as well. Theorem 6. 14 (Immerman [13]) Every FO k query is computable in AC 0 : Proof. Let us remind that according to the definition, a structure A 2 Sigma will be given to the circuit as enc(A) and therefore for all A s of the same cardinality the encodings will be of the same length. This makes our circuits to work on ....

N. Immerman, `Expressibility as a complexity measure: Results and directions', Technical Report DCS-TR-538, Yale Universsity, New Haven, CT, USA, 1987.

....is a TM which, given on the tape a standard encoding enc(I) of an input I, produces a standard encoding of OE(I) in time polynomial in jenc(I)j. We assume familiarity with the first order queries, FO, expressed by the relational calculus and algebra. Recall also that FO is in (uniform) ac 0 [Imm87], so FO queries can be evaluated in constant parallel time with polynomial resources. There are many useful queries that FO cannot express, such as the transitive closure of a graph. Numerous extensions of FO with recursion have been proposed. Most of them converge towards two central 2 classes ....

N. Immerman. Expressibility as a complexity measure: Results and directions. Technical Report DCS-TR-538, Yale Univ., 1987.

....has 1 focused on the computational complexity of problems. A more recent branch of complexity theory, started by Fagin in [Fag74, Fag75] and developed during the 1980s, focuses on the descriptive complexity of problems, which is the complexity of describing problems in some logical formalism [Imm87a]. One of the exciting developments in complexity theory is the discovery of a very intimate connection between computational and descriptive complexity. This intimate connection was first discovered by Fagin, who showed that the complexity class NP coincides with the class of properties ....

N. Immerman. Expressibility as a complexity measure: results and directions. In Second Structure in Complexity Conference, pages 194--202, 1987.

.... on the descriptive complexity of problems, i.e. the complexity of describing problems in some logical formalism [Imm87b] The tight connection between descriptive and computational complexity, typically referred to as the connection between logic and complexity , was then proclaimed by Immerman [Imm87a], and studied by many researchers 3 . Although the relationship between descriptive and computational complexity is intimate, it is not without its problems, and the partners do have some irreconcilable differences. While computational devices work on encodings of problems, logic is applied ....

.... algebra that we denote here A (see [Ull88] which involves (projection) Theta (cross product) set union) Gamma (set difference) and oe i=j (select from a relation the tuples where the i th and j th co ordinates are equal) It is also important to note that FO is in (uniform) AC 0 [Imm87a], so FO queries can be evaluated in constant parallel time with polynomial resources. The fixpoint queries (fixpoint, also FP ) CH82] are constructed using the firstorder constructors as in FO together with a fixpoint operator ( The fixpoint operator binds a predicate symbol R that is free and ....

N. Immerman. Expressibility as a complexity measure: results and directions. In Second Structure in Complexity Conference, pages 194--202, 1987.

....can be translated into this language of programs over monoids, and those of Barrington, Straubing, and Th erien which were obtained within it. A related view of these complexity classes arises from the logical expressibility theory of Immerman, as extended by Barrington, Immerman and Straubing [Im83, Im87, BIS88]. Consider formulas of first order logic, where variables range over places in the input and atomic formulas are x y, x = y, and a (x) for variables x and y and input letters a. The last means the x th input is an a and is the only way for the formula to access the input. Formulas are built ....

....of that predicate. Secondly, we can add new operators to the logical system. Immerman [Im83] gives logical systems characterizing P , LOGSPACE, and NLOGSPACE in this way. By introducing an operator to sytactically iterate formulas, he characterizes a wide variety of parallel complexity classes [Im87]. In [BIS88] classes up to and including NC 1 are characterized in terms of operators which are new quantifiers in the logical system. Logical definitions can be given for AC 0 (ordinary quantifiers) ACC 0 (modular counting quantifiers) TC 0 (threshold counting quantifiers) and NC 1 ....

N. Immerman, "Expressibility as a complexity measure: Results and directions, " Second Structure in Complexity Theory Conf. (1987), 194-202.

....spent on the problem of capturing polynomial time, that is, describing all polynomial time computable classes of not necessarily ordered nite structures by a logic in a similar way. The most obvious shortcoming of xed point logic itself on unordered structures is that it cannot count. Immerman [13] responded to this by adding counting constructs to xed point logic. Although it has been proved by Cai, F urer, and Immerman [1] that the resulting xed point logic with counting, denoted by IFP C, still does not capture all of polynomial time, it does capture polynomial time on several ....

N. Immerman. Expressibility as a complexity measure: results and directions. In Proceedings of the 2nd IEEE Symposium on Structure in Complexity Theory, pages 194-202, 1987.

.... small circuits [Pip79] and the identification of various forms of interactive proof systems with standard complexity classes as PSPACE , EXP and NP [Sha90, BFL90, ALM 92] Also, the classification of complexity classes by various logical theories is a rapidly growing field of interest [Imm84, Imm87] The class P , of polynomial time decidable sets, was first described by Edmonds [Edm65] as the class of problems for which feasible algorithms exist. Unfortunately, many problems of interest are not known to be in P . Therefore, interest has shifted from P to classes near P , and classes of ....

N. Immerman. Expressibility as a complexity measure: results and directions. In Proc. Structure in Complexity Theory 2nd annual conference, pages 194--202, Ithaca, NY, 1987. IEEE Computer Society Press.

....for other kinds of structures like trees or graphs. In descriptive complexity, since Fagin [Fag74] showed that the complexity class NP coincides with the sets of models of existential second order ( 1 1 ) sentences, many complexity classes have been characterized by extensions of FO logic [Var82, Imm86, Imm87, AV89, Gr92] and there is still hope that separations of complexity classes might be possible by separating the expressive power of the respective logics. For a recent result in this direction see the paper of Libkin and Wong [LW98] Despite its importance as an ingredient for more expressive logics, it is ....

N. Immerman. Expressibility as a complexity measure: results and directions. In Second Structure in Complexity Conference, pages 194--202, 1987.

....models for other kinds of structures like trees or graphs. In descriptive complexity, since Fagin [Fag74] showed that the complexity class NP coincides with the sets of models of existential secondorder (# 1 1 ) sentences, many complexity classes have been characterized by extensions of FO logic [Var82, Imm86, Imm87, AV89, Gra92] and there is still hope that separations of complexity classes might be possible by separating the expressive power of the respective logics. For a recent result in this direction see the paper of Libkin and Wong [LW98] Despite its importance as an ingredient for more expressive logics, it is ....

N. Immerman. Expressibility as a complexity measure: results and directions. In Second Structure in Complexity Conference, pages 194--202, 1987.

....naturally use a single, fixed data interpretation. Finite model theory. Many complexity characterizations have been made of problems involving finite model theory, with (Jones, Selman 74) apparently the first in a field developed quite considerably since then (Gurevich, 1983; Gurevich, 1984; Immerman, 1987; Goerdt, 1992) There is a natural connection between computation by read only programs and in finite model theory as defined by Gurevich and others, close enough that some complexity characterizations from finite model theory imply corresponding results about read only programs. The connection ....

Immerman, N. Expressibility as a complexity measure: results and directions. Proceedings 2. Conference on Structure in Complexity Theory (IEEE Computer Society Press) (1987), 223--257.

....sure theories are identical for every c. 2 Introduction Expressibility is a central question in computer science. Over classes of ordered finite structures, membership in a complexity classes is often equivalent to the expressibility of the desired set in a given logic. For example, Immerman [6] showed that the expressibility in transitive closure logic is equivalent to NLOGSPACE, and Fagin [4] proved Sigma 1 1 captures NPTIME. The characterizations of logics and the limit probabilities of their sentences over ordered structures could shed light on issues in complexity theory. We focus ....

Neil Immerman. Expressibility as a complexity measure: results and directions. In Second Structure in Complexity Conference, pages 194--202. Springer, 1987.

.... holds: A 2 K ( A j= j (A is in K iff A satisfies the sentence j) Following Fagin s 1974 result [9] that the logic S 1 1 (existential second order logic) captures NP, logical characterizations have been found for many other complexity classes such as P [13, 24] PSPACE [1] and LOGSPACE [14]. For the latter characterizations the structures have to be ordered, as otherwise the logics fail to express even simple statements such as the structure has an even size . It is an 2 Here 9 =b xj is used as shorthand for 9 b xj 8y(9 y xj y b) We also write x y = z instead of ....

N. Immerman. Expressibility as a complexity measure: results and directions. Second Structure in Complexity Conference, pages 194--202, 1987.

....for other kinds of structures like trees or graphs. In descriptive complexity, since Fagin [Fag74] showed that the complexity class NP coincides with the sets of models of existential second order ( Sigma 1 1 ) sentences, many complexity classes have been characterized by extensions of FO logic [Var82, Imm86, Imm87, AV89, Gra92] and there is still hope that separations of complexity classes might be possible by separating the expressive power of the respective logics. For a recent result in this direction see the paper of Libkin and Wong [LW97] Despite its importance as an ingredient for more expressive logics, it is ....

N. Immerman. Expressibility as a complexity measure: results and directions. In Second Structure in Complexity Conference, pages 194-- 202, 1987.

....in Ptime or Pspace. However, on arbitrary finite structures they do not, and almost all known examples showing this involve counting. While in the presence of an ordering, the ability to count is inherent e.g. in fixpoint logic, hardly any of it is retained in its absence. Thus, Immerman [15] proposed to add counting quantifiers to fixpoint logic and asked whether this would suffice to capture Ptime. Cai, Furer and Immerman [5] answered this question negatively; in fact (FP C) does not even express all Logspace computable queries. Nevertheless we argue that fixpoint logic with ....

....provide a polynomial time graph canonization algorithm for a dense class of graphs. It should be clear that the stable colouring of a graph is definable in (FP C) see [16] for more details) Remarks. A slightly different definition of fixpoint logic with counting has been proposed by Immerman [15], who related the two sorts by counting quantifiers rather than counting terms. Counting quantifiers have the form (9i x) for there exist at least i x , where i is a second sort variable. It is obvious, that the two definitions are equivalent. In fact, FP C) is a very robust logic. For ....

N. Immerman, Expressibility as a Complexity Measure: Results and Directions, Proc. of 2nd Conf. on Structure in Complexity Theory (1987), 194--202.

....of connected components, and fixpoint cannot express ptime queries such as: Is there an even number of connected components However, this difficulty can be overcome by adding counting to fixpoint. Indeed, we can show that in this case fixpoint counting (as defined in [GO93] and implicit in [I87]) expresses precisely the ptime queries on topological invariants [Seg] The idea of the proof is to construct using fixpoint counting an isomorphic copy of the invariant over the auxiliary ordered domain provided by fixpoint counting, then use the fact that fixpoint expresses ptime on ....

N. Immerman. Expressibility as a complexity measure: results and directions. In Proc. 2nd Structure in Complexity Conference, pp. 194--202, 1987.

.... small circuits [Pip79] and the identification of various forms of interactive proof systems with standard complexity classes as PSPACE, EXP and NP [Sha90, BFL90, ALM 92] Also, the classification of complexity classes by various logical theories is a rapidly growing field of interest [Imm84, Imm87] The class P , of polynomial time decidable sets, was first described by Edmonds [Edm65] as the class of problems for which feasible algorithms exist. Unfortunately, many problems of interest are not known to be in P . Therefore, interest has shifted from P to classes near P , and classes of ....

N. Immerman. Expressibility as a complexity measure: results and directions. In Proc. Structure in Complexity Theory second annual conference, pages 194--202. IEEE Computer Society Press, 1987.

....power of our finite variable logics is somehow unbalanced: On the one hand they can express all fixed point definable queries, they can even express non recursive queries, whereas on the other hand they fail to say that a set has even cardinality. To overcome this mismatch, Immerman and Lander [45, 47, 49] augmented the logics by counting quantifiers such as 9 m . Recall the definition of the logic C k (see page 5) We also consider the infinitary k variable logic with counting quantifiers C k 1 whose formulas are obtained by adding the rule Phi W Phi where Phi is a set of formulas to ....

....; where = 0 and = 2 are abbreviations with the obvious meaning. Using the pebble game (on the two sorted structure A # ) it can easily be seen that this query is not definable in IFP # . Thus we have IFP # ae IFP C. Fixed point logic with counting has been introduced by Immerman [47]. In the form presented here it is due to Gradel and Otto [25] Though it may seem artificial at first sight, the logic has turned out to be quite robust. Gradel and Otto [25] studied several different formalizations of the concept inductive definability with counting and proved them all to be ....

N. Immerman. Expressibility as a complexity measure: results and directions. In Proceedings of the 2nd IEEE Symposium on Structure in Complexity Theory, pages 194--202, 1987.

....graph in C, we obtain a quite surprising corollary: Corollary 3. Let C be a class of planar graphs that is closed under taking minors. Then C is definable in fixed point logic. As another by product of our main results, we obtain a theorem that continues a study initiated by Immerman and Lander [17, 18]. C k denotes the k variable (first order) logic with counting quantifiers. Theorem 4. Let k 1. For each database D of tree width at most k there is a C k 3 sentence that characterizes D up to isomorphism. On the other hand, Cai, Furer, and Immerman [6] proved that for each k 1 there are ....

N. Immerman. Expressibility as a complexity measure: results and directions. In Proceedings of the 2nd IEEE Symposium on Structure in Complexity Theory, pages 194--202, 1987.

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N. Immerman, Expressibility as a Complexity Measure: Results and Directions, Proc. of 2nd Conf. on Structure in Complexity Theory (1987), 194--202.

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Immerman, N. 1987a. Expressibility as a complexity measure: results and directions. In Second Structure in Complexity Conference, 194--202.

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N. Immerman. Expressibility as a complexity measure: results and directions. In Proc. 2nd Structure in Complexity Conference, 194--202, 1987.

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