D. Mix Barrington, N. Immerman, and H. Straubing, "On Uniformity Within NC 1 ," JCSS, 41 (1990) 274-306.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....operation similar to Mod j;q which ultimately yields all the nonsolvable group languages. Rather, it is as though each nonsolvable group needs to be indivisibly added to the logic (but see Remark 4. 5) A similar approach was used before, for example when Barrington, Immerman and Straubing [2] added monoidal quantifiers (i.e. a restricted form of Lindstrom quantifiers [8] to first order logic. By group temporal logic (PTL GROUP) we will mean the usual past temporal logic supplemented with group temporal operators Gamma g;G for any finite group G and g 2 G. The operator Gamma g;G ....

D. A. Barrington, N. Immerman, H. Straubing, "On uniformity within NC 1 ," J. Comput. System Sci. 41, 274-306 (1990).

....carry over into the uniform setting, and our uniform depth reduction results are presented in Sections 6 and 7. A summary is found in Section 8. 2 Definitions and Background We assume familiarity with the basics of circuit complexity. For additional background, see Boppana and Sipser (1990) Barrington et al. 1990), and Ruzzo (1981) A family of circuits is a set fC n : n 1g where each C n is a circuit for inputs of length n. In later sections of this paper, we will require that the function n 7 C n be easily computable in some sense (which will be made precise there) Such circuit families are called ....

.... easy to compute. For example, in one of the first papers to consider uniform circuit complexity, Ruzzo (1981) considers a variety of uniformity notions, and P uniform circuit complexity is discussed in Allender (1989a) Even more relevant to this paper are the notions of uniformity discussed in Barrington et al. 1990). In that paper, Barrington et al. consider what version of uniform circuit complexity is most appropriate for use in defining classes of languages accepted by circuits of polynomial size and depth O(1) With some work, it would be possible to adapt the definitions of Barrington et al. 1990) to ....

[Article contains additional citation context not shown here]

Barrington, D. A. M., Immerman, N., and Straubing, H. (1990), On uniformity within NC 1 , Journal of Computer and System Sciences 41, 274--306.

....completeness for which the answer is yes. Namely for every nice complexity class including P, NP, etc, any two sets complete via fops are not only polynomial time isomorphic, they are first order isomorphic. There are additional reasons to be interested in first order computation. It was shown in [BIS] that first order computation corresponds exactly to computation by uniform AC 0 circuits under a natural notion of uniformity. Although it is known that AC 0 is properly contained in NP, knowing that a set A is complete for NP under polynomial time (or logspace) reductions does not currently ....

....a family fC n g of circuits of depth one. The circuits consist entirely of wires connecting input bits or negated input bits to outputs. If the circuit family fC n g is sufficiently uniform, we arrive at the class of first order projections. Recall that first order corresponds to uniform AC 0 [BIS]. We find it useful to work in the framework of first order logic rather than in the circuit model. The rest of this section presents the necessary definitions of first order reductions. The idea of the definition is that the choice of the literals hl 0 ; l 1 ; l n k Gamma1 i in ....

[Article contains additional citation context not shown here]

David Mix Barrington, Neil Immerman, Howard Straubing, "On Uniformity Within NC 1 ," J. Computer Sys. Sci. 41 (1990), 274-306.

....w) where w is a word compatible with x such that for all w 0 that are compatible with x, it is not the case that w 0 defeats w with respect to w. Using the characterizations of AC 0 in terms of alternating Turing machines or in terms of first order logic (as presented, for example in [BIS]) it is easy to see that this computation can be carried out inside AC 0 . As pointed out in [Hu1] the language L considered in the proof of Proposition 3.1 can be accepted by a deterministic two way pushdown automaton. Thus the following corollary is immediate. Corollary 3.1 There is a set L ....

D. A. Mix Barrington, N. Immerman, and H. Straubing, "On uniformity within NC 1 ," JCSS 41: 274--306, 1990.

....carry over into the uniform setting, and our uniform depth reduction results are presented in Sections 6 and 7. A summary is found in Section 8. 2 Definitions and Background We assume familiarity with the basics of circuit complexity. For additional background, see Boppana and Sipser (1990) Barrington et al. 1990), and Ruzzo (1981) A family of circuits is a set fC n : n 1g where each C n is a circuit for inputs of length n. In later sections of this paper, we will require that the function n 7 C n be easily computable in some sense (which will be made precise there) Such circuit families are called ....

.... easy to compute. For example, in one of the first papers to consider uniform circuit complexity, Ruzzo (1981) considers a variety of uniformity notions, and P uniform circuit complexity is discussed in Allender (1989a) Even more relevant to this paper are the notions of uniformity discussed in Barrington et al. 1990). In that paper, Barrington et al. consider what version of uniform circuit complexity is most appropriate for use in defining classes of languages accepted by circuits of polynomial size and depth O(1) With some work, it would be possible to adapt the definitions of Barrington et al. 1990) to ....

[Article contains additional citation context not shown here]

Barrington, D. A. M., Immerman, N., and Straubing, H. (1990), On uniformity within NC 1 , Journal of Computer and System Sciences 41, 274--306.

.... we separate the languages (FO TC COUNT) and (FO COUNT) logn] Corollary 6 Tree Isomorphism 2 (FO COUNT) logn] Gamma (FO TC COUNT) Recall that (FO COUNT ) log n] is equal to ThC 1 , the set of problems recognized by uniform sequences of polynomial size, log depth threshold circuits, [BIS90]. Furthermore, for the transitive closure logic, in the presence of ordering, counting quantifiers give no extra power, i.e. FO TC COUNT ) FO TC ) NL Thus, Corollary 6 separates the unordered versions of the languages for NL and ThC 1 . Interestingly, the lower bound of Theorem 3 is ....

D. Mix Barrington, N. Immerman, and H. Straubing, "On Uniformity Within NC 1 ," JCSS, 41 (1990) 274-306.

....indicates the class P of languages decided by polynomialtime Turing machines. This claim is true if the circuits are P uniform (computable by a poly time Turing machine) or if they are DLOGTIME uniform (direct connection language decidable by a random access Turing machine in time O(log n) see [BIS]) or any uniformity condition in between. However, if we allow ourselves more than polynomial time to compute the circuit, we may be able to decide more languages (and if we allow ourselves more than exponential time, we can definitely do so as we can then decide the unary version of the ....

....out, a second robust region where a wide range of definitions give the same class, and a third region where additional non uniformity gives steadily larger classes. The distinction can be quite important, as we see in the case of the class NC 1 (which appears twice in our chart) As shown in [BIS], we can define a very restrictive uniformity notion under which NC 1 becomes the class of regular languages. If our non uniformity resource is between DLOGTIME and NC 1 itself, we get a robust class, equal to ALOGTIME. And if we allow polynomial time to build our circuits, we can then do ....

[Article contains additional citation context not shown here]

D. Barrington, N. Immerman, H. Straubing, "On uniformity within NC 1 ," JCSS 41, No. 3 (1990), 274 - 306.

....Amherst MA 01003, U.S.A. June 21, 1993 1. Abstract We review the existing results showing languages to be outside of complexity classes defined by tight constraints on boolean circuits, using the new characterizations of these classes in terms of automata theory [BT88] and formal logic [BIS88]. We outline the methods and results in the case of three types of classes defined by circuits of constant depth, polynomial size, and unbounded fan in, for three different types of gates. These are AND and OR gates [FSS84] AND, OR, and MOD p gates [Ra87, Sm87] and modular gates alone with ....

....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 ....

[Article contains additional citation context not shown here]

D. A. M. Barrington, N. Immerman, and H. Straubing, "On uniformity within NC 1 ," J. Comp. Syst. Sci., in press. Preliminary version Structure in Complexity Theory: Third Annual Conference (Washington: IEEE Computer Society Press, 1988), 47-59.

....complexity class AC 0 , which we will define below in terms of boolean circuits. Researchers in circuit complexity were at the same time searching for a natural uniform subclass of AC 0 (e.g. Bu87] and Immerman proposed FO BIT as a candidate. 4 Barrington, Immerman and Straubing [BIS88] then showed that a wide variety of possible definitions of uniform AC 0 , including FO BIT , coincide. They extended the first order framework to describe a variety of other circuit complexity classes in two ways. New atomic predicates in the formalism (such as BIT ) correspond to relaxing ....

....of inputs which are one is divisible by some number q. He showed that the classes AC 0 [p] for p prime are independent, in the sense that none of them is contained in another. The union of the classes AC 0 [q] for all q (prime and composite) is called ACC 0 [MT89] also called ACC, e.g. in [BT88, BIS88]) Again, all the techniques for proving languages outside of classes work just as well for the non uniform classes. If we include gates which calculate threshold functions (test whether the number of inputs which are one exceeds some number defined for the gate) we define a new class, which has ....

[Article contains additional citation context not shown here]

D. A. M. Barrington, N. Immerman, and H. Straubing, "On uniformity within NC 1 ," Structure in Complexity Theory: Third Annual Conference (Washington: IEEE Computer Society Press, 1988), 47-59. Revised version J. Comp. Syst. Sci., to appear.

....completeness for which the answer is yes. Namely for every nice complexity class including P, NP, etc, any two sets complete via fops are not only polynomial time isomorphic, they are first order isomorphic. There are additional reasons to be interested in first order computation. It was shown in [BIS] that first order computation corresponds exactly to computation by uniform AC 0 circuits under a natural notion of uniformity. Although it is known that AC 0 is properly contained in NP, knowing that a set A is complete for NP under polynomial time (or logspace) reductions does not currently ....

....by a family fCng of circuits with depth one and no gates. That is, the circuits consist entirely of wires connecting inputs to outputs. If the circuit family fCng is sufficiently uniform, we arrive at the class of first order projections. Recall that first order corresponds to uniform AC 0 [BIS]. We find it useful to work in the framework of first order logic rather than in the circuit model. The rest of this section presents the necessary definitions of first order reductions. The idea of our definition is that the choice of the literals hl 0 ; l 1 ; l n k Gamma1 i in ....

[Article contains additional citation context not shown here]

D. Barrington, N. Immerman, H. Straubing, "On Uniformity Within NC 1 ," J. Computer Sys. Sci. 41 (1990), 274-306.

....computable. For the classes above NC 1 this may be taken to be logspace computable. Alternatively, we would say that the circuit class is polynomial time uniform if this map is computable in polynomial time. For NC 1 and below we assume logtime uniformity or equivalently first order uniformity [BIS]. We discuss uniformity in detail in x3. Beame, Cook, and Hoover have shown: Theorem 2.2 ( BCH] Iterated integer multiplication (Problem Q (Z) is computable by polynomial time uniform NC 1 circuits. Beame, Cook, and Hoover proved Theorem 2.2 by using the Chinese Remainder Theorem. Each of ....

....uniform AC 0 , we make use of the logical relation BIT. BIT(x; y) means that the x th bit in the binary expansion of y is a one. Remember that the variables range over a finite universe, f0; 1; n Gamma 1g, for some value of n. Thus they may be thought of as (log n) bit numbers. In [BIS] it is shown that BIT is definable in FOM(wo BIT) if we assume that the majority quantifier may apply to pairs of individual variables. By wo BIT we mean without the logical relation BIT. Thus FOM = FOM(wo BIT) whereas FO 6= FO(wo BIT) To see the latter, note that the parity of the ....

[Article contains additional citation context not shown here]

D. Barrington, N. Immerman, H. Straubing (1990), On Uniformity Within NC 1 , JCSS 41, No. 3, 274 - 306.

....length of bin(A) is polynomially related to n, and in the case where consists of a single unary relation i.e. inputs are binary strings I (n) n. Define the complexity class FO to be the set of all first order expressible problems. FO is a uniform version of the circuit class AC 0 [BIS90] and it is equal to the set of problems acceptable in constant time on a polynomial size concurrent parallel random access machine [I89] 2.1 Inductive Definitions A useful way to increase the power of first order logic without jumping all the way up to second order logic is to add the power to ....

D. Mix Barrington, N. Immerman, H. Straubing, "On Uniformity Within NC 1 ," JCSS 41, No. 3 (1990), 274 - 306.

....inductive logic with counting, but without ordering does not contain all the polynomial time computable graph properties. In fact, it does not even contain all such properties computable by a uniform sequence of bounded depth, polynomial size Boolean circuits that include parity gates, cf. [12]. Proof We have seen in Corollary 6.5 that the graphs X(T n ) and X(T n ) are indistinguishable in C ffln for some constant ffl 0. Suppose for the sake of a contradiction that there were a sentence oe 2 (FO LFP COUNT) that expresses the isomorphism property for graphs from Gamma. That is ....

David Mix Barrington, Neil Immerman, and Howard Straubing, "On Uniformity Within NC 1 ," J. Comput. System Sci. 41, No. 3 (1990), 274-306.

No context found.

D. Barrington, N. Immerman, H. Straubing, "On Uniformity Within NC 1 ," JCSS 41, No. 3 (1990), 274 - 306.

....to the set of properties checkable by a Turing machine in DSPACE [n k ] where n is the size of the universe) This set is also equal to the set of properties describable using an iterative definition for a finite set of relations of arity k. This is a refinement of the theorem PSPACE = VAR[O[1]] 7] We suggest some directions for exploiting this result to derive trade offs between the number of variables and the quantifier depth in descriptive complexity. 1 Introduction In Descriptive Complexity one analyzes the complexity of a language in terms of the complexity of describing the ....

.... Delta Delta n a k rdlog ne, consisting of one bit for each a i tuple, potentially in the relation R i , and dlog ne bits to name each constant, c j . Define the complexity class FO to be the set of all first order expressible problems. FO is a uniform version of the circuit class AC 0 [1] and it is equal to the set of problems acceptable in constant time on a polynomial size concurrent parallel random access machine [10] On notation: We reserve n to indicate the size of the universe of the input structure. We will denote the length of the input string by n = I = jbin(A)j. The ....

[Article contains additional citation context not shown here]

D. Mix Barrington, N. Immerman, H. Straubing, "On Uniformity Within NC 1 ," JCSS 41, No. 3 (1990) 274 - 306.

....of uniformity, still more restrictive, based on the regular languages. Here we give characterizations of subclasses of the regular languages based on their logical expressibility, extending recent work of Straubing, Th erien, and Thomas [STT88] A preliminary version of this work appeared as [BIS88]. 2 Introduction 2.1 Circuit Complexity Computer scientists have long tried to classify problems (defined as Boolean predicates or functions) by the size or depth of Boolean circuits needed to solve them. This effort has 1 Former name David A. Barrington. Supported by NSF grant CCR 8714714. ....

D. A. M. Barrington, N. Immerman, and H. Straubing, "On uniformity within NC 1 ," Structure in Complexity Theory: Third Annual Conference (Washington: IEEE Computer Society Press, 1988), 47-59.

....indicates the class P of languages decided by polynomial time Turing machines. This claim is true if the circuits are P uniform (computable by a poly time Turing machine) or if they are DLOGTIME uniform (direct connection language decidable by a random access Turing machine in time O(log n) see [BIS]) or any uniformity condition in between. However, if we allow ourselves more than polynomial time to compute the circuit, we may be 1. Time, Hardware, and Uniformity 3 able to decide more languages. If we allow ourselves enough extra time, we can definitely do so. For example, if we allow ....

....to other models cannot be carried out, a second robust region where a wide range of definitions give the same class, and a third region where additional nonuniformity gives steadily larger classes. The distinction can be quite important, as we see in the case of the class NC 1 . As shown in [BIS], we can define a very restrictive uniformity notion under which NC 1 becomes the class of regular languages. If our non uniformity resource is between DLOGTIME and NC 1 itself, we get a robust class, equal to ALOGTIME. And if we allow polynomial time to build our circuits, we can then do ....

[Article contains additional citation context not shown here]

D. Barrington, N. Immerman, H. Straubing, "On uniformity within NC 1 ," JCSS 41(3 (1990), 274 - 306.

....= 1 [ k=1 FO( n k ] Gamma VAR[k] 4 The uniformity in question can be purely syntactical, i.e. the n th sentences of a FO[t(n) property consists of a fixed block of quantifiers repeated t(n) times followed by a fixed quantifier free formula. Uniformity is discussed extensively in [6]. In this paper the reader may think of uniform as meaning that the map from n to n is easily computable, e.g. in logspace. 1. Describing Graphs: a First Order Approach to Graph Canonization 5 Example 1.2.3 The monotone circuit value problem is an example of a complete problem for P which we ....

D. Mix Barrington, N. Immerman, and H. Straubing, "On Uniformity Within NC 1 ," Third Annual Structure in Complexity Theory Symp. (1988), 47-59.

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