Immerman 1989 Neil Immerman, "Descriptive and Computational Complexity", Proc. of Symposia in Applied Math. 38 (1989), 75--91.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....to the resources needed to express such problems in various logical formalisms. One of the most notable successes of descriptive complexity is the discovery that essentially all major complexity classes have natural characterizations in terms of logical expressibility on nite structures (cf. [Gur88,Imm89]) The prototypical result in this vein is Fagin s theorem [Fag74] which asserts that a class of nite structures is in NP if and only if it is de nable by a formula of existential second order logic. Quite often, certain logical characterizations of other major complexity classes are valid only ....

N. Immerman. Descriptive and computational complexity. In J. Hartmanis, editor, Computational Complexity Theory, Proc. Symp. Applied Math., Vol. 38, pages 75-91, American Mathematical Society, 1989.

....spectrum. It is not clear as to whether the converse is true. Open problem: Are categorical spectra precisely the sets of positive integers in UE The type of characterization of Theorem 5. 1 has been obtained for a number of complexity classes (see Immerman [Imm87] and a survey by Immerman [Imm89]) The most interesting complexity class to consider is P (deterministic polynomial time) Ever since Cobham [Cob64] and Edmonds [Edm65] the class P has been often identified with the class of feasible problems. An ordered structure is one with a built in linear order , that is, a structure ....

....the complexity of describing the class in some logical formalism. For example, we might consider whether a class is firstorder definable, and if so, we might want to ask, say, how many quantifiers are required. Hartmanis suggested the name descriptive complexity to Immerman, who used it in [Imm89]. This section will focus on descriptive complexity, where we are particularly interested in second order quantifiers. By definition, every generalized spectrum is defined by an existential second order sentence 9Q 1 : 9Q k oe(P 1 ; P s ; Q 1 ; Q k ) where oe(P 1 ; P s ....

N. Immerman. Descriptive and computational complexity. In J. Hartmanis, editor, Computational Complexity Theory, Proc. Symp. in Applied Math. 38, pages 75--91. American Mathematical Society, 1989.

....email: fagin almaden.ibm.com. 1 Introduction The computational complexity of a problem is the amount of resources, such as time or space, required by a machine that solves the problem. The descriptive complexity of a problem is the complexity of describing the problem in some logical formalism [Imm89]. The two complexities are sometimes related. This was first discovered by Fagin, who showed [Fag74] that the complexity class NP coincides with the class of properties of finite structures expressible in existential second order logic, otherwise known as Sigma 1 1 . Consequently, NP=co NP if ....

N. Immerman. Descriptive and computational complexity. In J. Hartmanis, editor, Computational Complexity Theory, Proc. Symp. Applied Math., Vol. 38, pages 75-- 91. American Mathematical Society, 1989.

....the simpler version, and then try to extend the tools to move toward an answer to the original question. Such an approach to the question of whether NP equals co NP has spawned an area known as descriptive complexity, which is the complexity of describing problems in some logical formalism [Imm89] This area began in the 1970 s when Fagin [Fag74] showed that the complexity class NP coincides with the class of properties of finite structures expressible in existential secondorder logic, otherwise known as Sigma 1 1 , which allows sentences consisting of a string of existential ....

N. Immerman. Descriptive and computational complexity. In J. Hartmanis, editor, Computational Complexity Theory, Proc. Symp. Applied Math., Vol. 38, pages 75-- 91. American Mathematical Society, 1989.

....orders are definable in fixpoint logic. We call a class having this property polynomially orderable. We investigate this property, and give examples of polynomially orderable classes of graphs and groups. 1 Introduction and summary In the field of descriptive computational complexity theory [11, 6, 1, 5], the complexity of computational problems is investigated in terms of the logics that can express them. Instances of a problem are represented as finite logical structures of some fixed similarity type. A standard result of the theory [10, 17] is that a property of ordered structures is ....

N. Immerman. Descriptive and computational complexity. In J. Hartmanis, editor, Computational Complexity Theory, volume 38 of Proc. Symp. Applied Math., pages 75--91. American Mathematical Society, 1989.

....of Dyn FO essentially as given by [PI97] Since we use Dyn (FO COUNT) at an intermediate step, we will use their more general definition of Dyn C for an arbitrary logical lan guage C. We will assume some familiarity with the basics of first order logic and descriptive complexity (see [Imm98, Imm89a]) For simplicity, let us focus on the setting where our input vocabulary oe is the vocabulary of graphs with one binary relation and two constants representing vertices, oe = hE 2 ; c 1 ; c 2 i. All our definitions easily generalize to an arbitrary input vocabulary oe. We will be maintaining ....

N. Immerman. Descriptive and computational complexity. In J. Hartmanis, editor, Computational Complexity Theory, volume 38, pages 75--91. Proc. Symp. in Applied Math., American Mathematical Society, 1989.

....The motivation for syntactically defined classes of optimization problems lies in Fagin s result [Fag74] that, roughly speaking, any problem of NP is equivalent to a sentence in existential second order logic. The problem instances and solutions are encoded in finite structures G and S (see e.g. [Imm89] for a discussion of the relationship between logic and computational complexity) In [PY88, PR90] an existential second order formula 9S Phi(x; G; S) is interpreted as a maximization problem. Here we ask for a structure S that maximizes the number of values u of the tuple x of the free ....

Neil Immerman. Descriptive and computational complexity. Proceedings of Symposia in Applied Mathematics, 38:75--91, 1989.

....class, e.g. Ptime, find a logic or language L such that any PTime query can be defined in L, and vice versa. Since a logic is a specification of descriptive resources in much the same way a complexity class specifies computational resources, this issue is referred to as descriptive complexity, cf. [13, 15]. Logical equivalents of all major complexity classes have been found in the case of linearly ordered structures. In particular, fixed point logic allows to define exactly those classes of linearly ordered structures that can be recognized in PTime, 12, 21] In the general case of not necessarily ....

N. Immerman, Descriptive and Computational Complexity, in: J. Hartmanis (ed.), Computational Complexity Theory, Proc. AMS Symp. in Appl. Math. 38 (1989), 75-91

....exists an existential second order sentence such that L is precisely the class of finite models of . Some years later, Immerman and Vardi [17, 28] proved that, on ordered structures, the problems solvable in polynomial time are exactly those definable in least fixed point logic. Immerman [17] [20] systematically studied the problem of capturing complexity classes by logical languages and came up with logical characterizations for most of the major complexity classes. The most important results in this field are surveyed in [16, 20] for a textbook on finite model theory we refer to [9] ....

....those definable in least fixed point logic. Immerman [17] 20] systematically studied the problem of capturing complexity classes by logical languages and came up with logical characterizations for most of the major complexity classes. The most important results in this field are surveyed in [16, 20]; for a textbook on finite model theory we refer to [9] These logical characterizations of complexity are model theoretic and based either on fragments of second order logic [10, 12] or on extensions of first order logic by additional means to construct new relations (such as fixed point ....

N. Immerman, Descriptive and Computational Complexity, in: J. Hartmanis (Ed.), Computational Complexity Theory, Proc. of AMS Symposia in Appl. Math. 38 (1989), 75--91.

....digraph defined by E is strongly connected. The formula : TC x;y (9z) Exz Ezy) x; y) says that there is no walk of even length from x to y. Immerman was interested in this logic because of its significance for his long term project to provide logical descriptions of complexity classes (see [16, 23] for surveys on this topic) He proved that on ordered structures, transitive closure logic can express precisely the queries that are computable with nondeterministic logarithmic space [21] More precisely: Let O be the class of finite successor structures, i.e. structures whose vocabulary ....

....it said that on finite ordered structures, Nlogspace is captured by the logic (FO pos TC) the restriction of (FO TC) where the operator TC can occur only positively. However, the closure of Nlogspace under complementation [22, 34] implies the equivalence of (FO pos TC) with (FO TC) See [23] for a direct proof of this fact. In this paper we show that this is not the case on arbitrary finite structures. Quantifier classes in transitive closure logic. Note that the TC operator in a formula j [TC x;y (x; y) u; v) where x and y are k tuples, can be considered as a sort of ....

N. Immerman, Descriptive and Computational Complexity, in: J. Hartmanis (Ed.), Computational Complexity Theory, Proc. of AMS Symposia in Appl. Math. 38 (1989), 75--91.

.... the class PSPACE and noninflationary fixpoint logic [Var82] cf. AV89] 1 The tight connection between descriptive and computational complexity, typically referred to as the connection between logic and complexity , was then proclaimed by Immerman [Imm87b] and studied by many researchers [Com88, Goe89, Gra84, Gra85, Gur83, Gur84, Gur88, HP84, Imm89, Lei89a, Liv82, Liv83, Lyn82, Saz80b, Saz80a, TU88]. 2 See [Imm89] for a survey. 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 directly to ....

.... and computational complexity, typically referred to as the connection between logic and complexity , was then proclaimed by Immerman [Imm87b] and studied by many researchers [Com88, Goe89, Gra84, Gra85, Gur83, Gur84, Gur88, HP84, Imm89, Lei89a, Liv82, Liv83, Lyn82, Saz80b, Saz80a, TU88] 2 See [Imm89] for a survey. 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 directly to the underlying ....

N. Immerman. Descriptive and computational complexity. In J. Hartmanis, editor, Computational Complexity Theory, Proc. Symp. Applied Math., Vol. 38, pages 75--91. American Mathematical Society, 1989.

....pspace coincides with the class of properties expressible in noninflationary fixpoint logic [Var82] 3 The focus here is on the connection between finite model theory and complexity. The connection between logic and complexity has also a proof theoretic aspect; see [Bus86, GSS90, Lei91] See [Imm89] for a survey. We shall present in this paper various devices operating directly on structures, without encoding. The motivation and benefits for doing this are manyfold. On a fundamental level, encodings of structures seem to be a technical device rather than an intrinsic feature. This point has ....

N. Immerman. Descriptive and computational complexity. In J. Hartmanis, editor, Computational Complexity Theory, Proc. Symp. Applied Math., Vol. 38, pages 75--91. American Mathematical Society, 1989.

....sentence of a countable model belongs to this language. 95 Show that the Scott sentence of a model axiomatizes its infinitary theory. Bibliographic Remarks Ehrenfeucht s game is from Ehrenfeucht 1961, Fraiss e s formulation is from Fraiss e 1954. Proposition 3. 24 and its corollary are due to Immerman and Kozen 1989. Theorem 3.34 is due to Fagin 1974. Immerman (see Immerman 1987a, 1987b, 1989) has made a detailed study of the relations between complexity levels and definability in logical formalisms of several type. Proposition 3.36 is from Fagin 1975. Also, see Gaifman and Vardi 1985, Fagin et al. 1992 ....

....Scott sentence of a model axiomatizes its infinitary theory. Bibliographic Remarks Ehrenfeucht s game is from Ehrenfeucht 1961, Fraiss e s formulation is from Fraiss e 1954. Proposition 3.24 and its corollary are due to Immerman and Kozen 1989. Theorem 3.34 is due to Fagin 1974. Immerman (see Immerman 1987a, 1987b, 1989) has made a detailed study of the relations between complexity levels and definability in logical formalisms of several type. Proposition 3.36 is from Fagin 1975. Also, see Gaifman and Vardi 1985, Fagin et al. 1992 and Ajtai and Fagin 1990. For the development of finitemodel theory, see Fagin ....

Immerman, N. 1989. Descriptive and computational complexity. In Computational Complexity Theory, Proc. Symp. Applied Math., Vol. 38, ed. J. Hartmanis. 75--91. American Mathematical Society.

.... [12] Vardi [14] Note that although the combined problem of finding out whether an implicit query Theta(Q) to a database exists, checking that the query is coherent, and explicitly computing the answer is in general NP complete (in the size of the database) as was shown by Fagin (see Immerman [8]) our method which applies quantifier elimination techniques to semi Horn theories makes the problem solvable in polynomial time for this special case. Most importantly, SHQL is a highly expressive language which covers all PTIME queries and is at the same time purely declarative. Querying with ....

....in the next section. 4 The Method We first observe that the problem whether a result of an implicit query Theta(Q) to a database B exists reduces to the question whether the second order formula 9Q Theta(Q) is satisfied in B. By Fagin s theorem the problem is NP complete in the size of B (see [8]) In what follows we concentrate on selecting a class of implicit queries for which the problem is in PT IME. Conceptually, the SHQL query method consists of four steps: 1. State a query Theta(Q) to a relational database B in SHQL, where Theta(Q) is a semi Horn formula (for the definition of ....

Immerman, N. (1989) Descriptive and Computational Complexity, Proceedings of Symposia in Applied Mathematics, 38, 75-91.

....computable properties of structures in which there is a built in order are captured exactly by the extension of first order logic with a least fixed point operation. Since then, several further logical characterizations of computational complexity classes have been studied (see, for instance, [Immerman, 1989]) The early work of Chandra and Harel [Chandra and Harel, 1982] and Vardi [Vardi, 1982] linked this to the development of query languages for relational databases. A relational database can be viewed as a finite relational structure and query languages for such databases have generally been based ....

N. Immerman. Descriptive and computational complexity. In J. Hartmanis, editor, Computational Complexity Theory, Proc. of AMS Symposia in Appl. Math., volume 38, pages 75--91, 1989.

....19] proved that the problems solvable in polynomial time are those that are expressible in the least fixpoint logic (FO LFP) i.e. first order logic together with a least fixed point operator. Immerman systematically studied the problem of designing logics that capture complexity classes [12] [14] and came up with logical descriptions for many other important classes. As in the case of fixpoint logic, most of these logical characterisations are obtained by augmenting the syntax of first order logic by operators, e.g. by various forms of transitive closure operators. In the absence of a ....

.... fail to express some very simple properties, such as parity , i.e. the property that the cardinality of a given structure is even [4] Alternative characterisations of complexity classes by fragments of second order logic were given in [8] The most important results in this field are surveyed in [14, 11]. In this paper, we are interested in describing functions rather than decision problems. Therefore, instead of sentences, we need formulae with free variables. Definition 2.4 In an abstract logic L, as given by Definition 2.1, a formula (x) of vocabulary oe with free variables x 1 ; x ....

N. Immerman, Descriptive and Computational Complexity, in: J. Hartmanis (Ed.), Computational Complexity Theory, Proc. of AMS Symposia in Appl. Math. 38 (1989), 75--91.

....logs in parallel, and the related problems of iterated multiplication and division for arbitrary integers. 1.2 The Model of Computation We will be measuring the complexity of problems primarily as the parallel time required to solve them with a polynomial number of processors. This is equivalent [18, 17, 7, 5] to (a) the depth of a boolean circuit of polynomial size and unbounded fan in solving the problem, or (b) the quantifier depth of a formula in first order logic expressing the problem. Consider the multiplication table of a group (or semigroup) to be input as an n by n array of numbers, each in ....

....family of constant depth, unbounded fan in and polynomial size, i.e. if and only if it is in the circuit complexity class (logtime uniform) AC 0 . Thus the three group axioms, as well as other properties such as commutativity, are testable for a multiplication table in AC 0 . Immerman [18, 17, 5] has developed a formalism whereby families of first order formulas may be defined to express properties outside of AC 0 . Given the necessary conventions for reusing variables, we can define a formula with a quantifier block in front which is syntactically iterated d(n) times on input of size n ....

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N. Immerman, Descriptive and computational complexity, in Computational Complexity Theory, ed. J. Hartmanis, Proc. of Symposia in Applied Mathematics 38, (Providence, RI: American Mathematical Society, 1989), pp. 75--91.

....power of different logical formalisms. In the last twenty years, these investigations have also been motivated by a close connection to computational complexity theory most computational complexity classes have been given characterisations as finite model classes of appropriate logics, cf. [Imm98]. In these investigations it became apparent that in order to describe computation over a finite structure, a formula has to be able to refer to some linear order of the elements of this structure. Given such an order, the universe of the structure, i.e. the set of its elements, can be identified ....

....of numerical predicates) is definable in FO[#, mod p p # N ] where mod p (i) is true iff i # 0 mod p. # Although according to Theorem 3.9 the Crane Beach conjecture holds for the set of all unary relations, it is not true for all binary relations, since FO[#, #] FO[#, Bit] c.f. [Imm98]. In fact, it already fails for the set of all unary functions, or for the set of all linear orderings. This follows from the existence of a unary function f : N # N (see the proof of Theorem 3 in [Sch97] and a set O of linear orderings (in fact, four order relations suffice, cf. ScSc] such ....

N. Immerman. Descriptive and Computational Complexity. Springer--Verlag, New York, 1998.

.... as fetch and add ) Here we consider these three dimensions and the variety of operations in the framework of descriptive complexity, where we measure the complexity of a language by the syntactic resources (in an particular logical formalism) needed to express the property of membership in it [I87, I89, I89b]. We review this frame work in Section 2 below. It has been known for some time that two parameters in descriptive complexity, number of variables and quantifier depth, correspond exactly to space and parallel time in either the circuit or PRAM models [I89b] More recently (along with ....

....ranging from 1 to n. Though it is not entirely satisfactory, we do develop enough machinery to bring the constant width characterizations of NC 1 [B89] and PSPACE [CF] into the framework. 2 Background: Descriptive Complexity Our notation follows the conventions of Descriptive Complexity. See [I87, I89] for more detail and motivation. For the notions of reductions and operators, we closely follow [IL] We will code all inputs as finite logical structures. For example, consider a binary string of length n, b = b 1 : b n . We associate this string with a finite structure A(b) hf1; 2; ....

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N. Immerman, "Descriptive and computational complexity," Computational Complexity Theory, ed. J. Hartmanis, Proc. Symp. in Applied Math., 38, American Mathematical Society (1989), 75-91.

....algorithms. In Section 5, we describe and investigate reductions honoring dynamic complexity. Finally, we suggest some future directions for the study of dynamic complexity. 2 Descriptive Complexity: Background and Definitions In this section we recall the notation of Descriptive Complexity. See [I89] for a survey and [IL94] for an extensive study of first order reductions. In the development of descriptive complexity it has turned out that natural complexity classes have natural descriptive characterizations. For example, space corresponds to number of variables; and parallel time is ....

N. Immerman, "Descriptive and Computational Complexity," in Computational Complexity Theory, ed. J. Hartmanis, Proc. Symp. in Applied Math., 38, American Mathematical Society (1989), 75-91.

....reachability over one way locally ordered graphs. We also show that DTC of first order formulas does not suffice to express reachability for two way local ordered graphs without numbers. 2 Descriptive Complexity In this paper our notation follows the conventions of Descriptive Complexity. See [I87, I89] for more detail and motivation. We code all inputs as finite logical structures. The typical example in this paper is a graph, G = hf0; 1; n Gamma 1g; E; s; ti The universe of G, V g = fv 0 ; v 1 ; v n Gamma1 g is the set of vertices and the binary relation E is the edge ....

N. Immerman, "Descriptive and Computational Complexity," in Computational Complexity Theory, ed. J. Hartmanis, Proc. Symp. in Applied Math., 38, American Mathematical Society (1989), 75-91.

....is incomparable with the class of first order projections. Other interesting results concerning 1 L reductions may be found in [BH90, HH] 4 3 Descriptive Complexity In this section we recall the notation of Descriptive Complexity which we will need to state and prove our main results. See [I89] for a survey and [IL] for an extensive discussion of the reductions we use here including first order projections. We will code all inputs as finite logical structures. The most basic example is a binary string, w of length n = jwj. We will represent w as a logical structure: A(w) hf0; 1; ....

Neil Immerman, "Descriptive and Computational Complexity," Computational Complexity Theory, ed. J. Hartmanis, Proc. Symp. in Applied Math., 38, American Mathematical Society (1989), 75-91.

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Immerman 1989 Neil Immerman, "Descriptive and Computational Complexity", Proc. of Symposia in Applied Math. 38 (1989), 75--91.

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N. Immerman, Descriptive and Computational Complexity, in: J. Hartmanis (Ed.), Computational Complexity Theory, Proc. of AMS Symposia in Appl. Math. 38 (1989), 75--91.

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N. Immerman, Descriptive and Computational Complexity, in: J. Hartmanis (Ed.), Computational Complexity Theory, Proc. of AMS Symposia in Appl. Math. 38 (1989), 75--91.

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