L. Goldschlager, I. Parberry, On the construction of parallel computers from various bases of Boolean functions. In Theoretical Computer Science 21, (1986), 43-58.  Home/Search   Document Not in Database   Summary   Related Articles   Check

.... [23] learnability of quantified formulas [9] and Constraint Satisfaction Problems [13, 14, 15, 16, 12] Finally we mention some similar results: a dichotomy result for satisfiability, tautology and some counting problems over closed sets of boolean functions [23] the circuit value problem [11, 10], the satisfiability of generalized formulas [24] the inverse generalized satisfiability problem [17] the generalized satisfiability counting problem [7] the approximability of minimization and maximization problems [6, 19, 20] the optimal assignments of Generalized Propositional Formulas [22] ....

L.M. Goldschlager and I. Parberry. On the Construction of Parallel Computers from various bases of Boolean Circuits. Theoretical Computer Science, 43:43--58, 1986.

.... that the computations of recurrent networks of size s that terminate within time t can be unwound into circuits of size st and depth t [107] which, for example, implies the computational equivalence of feedforward and convergent recurrent networks up to a factor of t in size [35]. A.3.a.ii Symmetric Recurrent Networks. The well known fundamental property of symmetric (Hop eld) networks is that their dynamics are constrained by a Liapunov, or energy function E which is a bounded function de ned on their state space decreasing along any productive computation. It follows ....

L.M. Goldschlager and I. Parberry, On the construction of parallel computers from various bases of Boolean functions, Theoretical Computer Science 43 (1986) 43-48.

.... [54] Note also that the computations of recurrent networks of size s that terminate within time t can be unwound into circuits of size st and depth t by standard technique [105] which, for example, implies the computational equivalence of feedforward and convergent recurrent networks [34]. 2.1.3.1.2 Symmetric Recurrent Networks The well known fundamental property of symmetric (Hop eld) networks is that their dynamics are constrained by a Liapunov, or energy function E which is a bounded function de ned on their state space decreasing along any productive computation. It follows ....

L.M. Goldschlager and I. Parberry, On the construction of parallel computers from various bases of Boolean functions, Theoretical Computer Science 43 (1986) 43-48.

....[Lad75] Hint: A proof is given in Section 5 of this paper. Remarks: For the two input basis of Boolean functions, it is known that CVP is P complete except when the basis consists solely of or, consists solely of and, or consists of any or all of the following: xor, equivalence, and not [GP86, Par87]. A.1.2 Topologically Ordered Circuit Value Problem (TopCVP) Given: An encoding ff of a Boolean circuit ff plus inputs x 1 ; x n , with the additional assumption that the vertices in the circuit are numbered and listed in topological order. 48 ffl A Compendium of Problems Complete for P ....

L. M. Goldschlager and I. Parberry. On the construction of parallel computers from various bases of Boolean functions. Theoretical Computer Science, 43(1):43--58, 1986.

....example) However, usually these problems were defined using formulae or circuits with a complete base of boolean operations or gates, mostly with the base f ; g. But what about the complexity of such problems when a different base is used There are several special results of this kind (e.g. [Sim75, Gol77, Lew78, GP86]) In particular, there are very detailed investigations for the special case of boolean formulae in conjunctive normal form in [Sch78, Cre95, CH96, CH97, KST97, KSW97, RV98] But there are no results answering this question for unresricted circuits in full generality. In this paper we will give ....

....following facts on the complexity of VAL(B) can be found in or easily be derived from the literature. Theorem 9 Let B be a finite set of boolean functions. 1. Lad77] If [B] BF then VAL(B) is log m complete for P. 2. Gol77] If fet; velg [B] then VAL(B) is log m complete for P. 3. [GP86] Let B be a set of binary boolean functions. If fet; velg [B] or (B 6 L and B 6 M) then VAL(B) is log m complete for P, otherwise VAL(B) is acceptable in log 2 n space. We start the investigation of the complexity of VAL(B) by strengthening Proposition 3 for this particular case. ....

L.M. Goldschlager and I. Parberry. On the construction of parallel computers from various bases of boolean functions. Theoretical Computer Science, 43:43--58, 1986.

....= 0. 4 3. 45,55] C=P is the class of sets L for which there exist functions f 2 #P and g 2 PF such that for every x 2 , x 2 L ( f(x) g(x) 4. 45,19] PP is the class of sets L for which there exist functions f 2 #P and g 2 PF such that for every x 2 , x 2 L ( f(x) g(x) 5. [36,20] P is the class of sets L for which there exists a function f 2 #P such that for every x 2 , x 2 L ( f(x) is odd. Next, we review the de nitions of some operators that we will use in this paper. De nition 2.3 1. For a class K, 9 K is the class de ned in the following way: A set L is in ....

L. Goldschlager and I. Parberry. On the construction of parallel computers from various bases of boolean functions. Theoretical Computer Science, 43:43-58, 1986.

....are de ned on Boolean computation models, in that they rely on mere existence of accepting computations. If the existential predicate is replaced by the predicate there is an odd number of accepting computations , we obtain the mod 2 versions of these classes: P ; L and LOGCFL (introduced in [6, 9, 5]) More formally: P is the class of sets of type fx j f(x) 6= 0 (mod 2)g for some f 2 #P . L is the class of sets of type fx j f(x) 6= 0 (mod 2)g for some f 2 #L. LOGCFL is the class of sets of type fx j f(x) 6= 0 (mod 2)g for some f 2 #LOGCFL. As usual, the nonuniform versions of these ....

Leslie Goldschlager and Ian Parberry. On the construction of parallel computers from various basis of Boolean functions. Theoretical Computer Science, 43:43-58, 1986.

....our framework and, in particular, study guarded access to unambiguous computations and robustly unambiguous computations. 2 Locally Definable Acceptance Types Goldschlager and Parberry introduced so called extended Turing machines in order to generalize the concept of alternating Turing machines [11]. They studied the power of nondeterministic, time bounded Turing machines with an altered manner of acceptance. Instead of only labeling the states (and thus also the configurations) of the machine just with AND, OR, NOT, Accept or Reject, as it is done in alternating Turing machines, they ....

L. Goldschlager and I. Parberry. On the construction of parallel computers from various bases of boolean functions. Theoretical Computer Science, 43:43--58, 1986.

....[Lad75] Hint: A proof is given in Section 5 of this paper. Remarks: For the two input basis of Boolean functions, it is known that CVP is P complete except when the basis consists solely of or, consists solely of and, or consists of any or all of the following: xor, equivalence, and not [GP86, Par87]. A.1.2 Topologically Ordered Circuit Value Problem (TopCVP) Given: An encoding ff of a Boolean circuit ff plus inputs x 1 ; x n , with the additional assumption that the vertices in the circuit are numbered and listed in topological order. 48 ffl A Compendium of Problems Complete for P ....

L. M. Goldschlager and I. Parberry. On the construction of parallel computers from various bases of Boolean functions. Theoretical Computer Science, 43(1):43--58, 1986.

....MOD n is O(1) by a folklore theorem (Beigel and Gill 1992, Hertrampf 1990, Beigel and Tarui 1991, Barrington 1992a, Smolensky 1987) 4. An oracle for the conjectured relations among MODmP classes The class MODmP is a generalization of the counting class PhiP (Papadimitriou and Zachos 1983, Goldschlager and Parberry 1986). First developed by Cai and Hemachandra (1990) these classes have since been studied by many others (Beigel 1991, Beigel and Gill 1992, Hertrampf 1990, Babai and Fortnow 1990, Toda and Ogiwara 1992, Tarui 1993) It is known that MODmP = MODm 0 P where m 0 is the product of all distinct prime ....

L. Goldschlager and I. Parberry, On the construction of parallel computers from various bases of Boolean functions. Theoret. Comput. Sci. 43 (1986), 43--58.

....j 2 p(jxj) Gamma2 : 9. CH90,Her90,BG92] For any fixed k 2, MOD k P is the class of languages L for which there exist a set A in P and a polynomial p such that for all strings x 2 Sigma , x 2 L ( j fy j jyj = p(jxj) and hx; yi 2 Ag j 6j 0 mod k: If k = 2, we write PhiP (introduced in [PZ83,GP86]) instead of MOD 2 P. 10. OH93,FFK94] SPP is the class of languages L for which there exist a set A in P and a polynomial p such that for all strings x 2 Sigma , x 2 L = j fy j jyj = p(jxj) and hx; yi 2 Ag j = 2 p(jxj) Gamma1 1; and x 62 L = j fy j jyj = p(jxj) and hx; yi 2 Ag j = ....

L. Goldschlager and I. Parberry. On the construction of parallel computers from various bases of boolean functions. Theoretical Computer Science, 43:43-- 58, 1986.

....In these levels the results of the nondeterministic computations are evaluated according to the languages g 1 ; g k . The bottom part of the computation tree consists out of polynomially long deterministic paths. M is a generalized Turing machine in the sense of Goldschlager and Parberry [GP86] Now we just have to consider the circuit family C that immediately reflects the evaluation scheme of this machine M. This family is certainly uniform since to answer questions about the direction connection language we have to follow polynomially long computation paths of M which requires time ....

L. M. Goldschlager and I. Parberry. On the construction of parallel computers from various bases of boolean functions. Theoretical Computer Science, 43:43--58, 1986.

....2)g]g. 6. SP Delta K : fL : 9f 2 GAP Delta K) 8x) L (x) f(x) g. 7. BP Delta K : n L : 9A 2 K) 9p 2 Pol) 8x) Pr p(jxj) y j L (x) A (x; y) 3 4 ] o . Clearly, for K = P, the classes NP, coNP, Sigma p k , Pi p k , k 0, PH [MS72, Sto77] PP, C=P [Sim75, Wag86] PhiP [PZ83, GP86], SPP ( FFK91] independently defined in [OH90] where it was called XP) and BPP [Gil77] respectively, are obtained. Fact 2.4 [FFK91] 1. co Delta SP Delta K = SP Delta co Delta K: 2. O Delta SP Delta K = O Delta K ( SP Delta O Delta K) for O 2 fC; C=g. 6 3. SP Delta SP Delta K = ....

L. Goldschlager and I. Parberry. On the construction of parallel computers from various bases of boolean functions. Theoretical Computer Science 43, 1986, 43--58.

....circuits of size 2 (log n) c with a MidBit gate at the root and AND gates of fan in (log n) c at the leaves. This result improves the known upper bounds for the class ACC. 1 Introduction The complexity classes PP (probabilistic polynomial time [Gi 77] and Phi P (parity P, PaZa 83, GoPa 86] have received much attention since the well known result by Toda [Tod 89] proving that the polynomial time hierarchy (PH) is Turing reducible to PP. These classes are closely related to the class of counting functions #P [Va 79] that count the number of accepting paths on nondeterministic ....

L. Goldschlager, I. Parberry, On the construction of parallel computers from various bases of Boolean functions. In Theoretical Computer Science 21 (1986), 43-58.

....function of A is in # Delta C 6. A 2 X Delta C if the characteristic function of A can be written as the difference of two # Delta C functions. It is well known that the above operators capture, as special cases, the standard complexity classes NP, PP [Sim75] C P [Sim75,Wag86] Phi P [PZ83,GP86] UP [Val76] and SPP [OH93,FFK94] respectively, via 9 Delta P = NP, C Delta P = PP, C Delta P = C P, Phi Delta P = Phi P, U Delta P = UP, and X Delta P = SPP. A careful look at the definitions and a little thought (or a look at [Vol94a] reveal that for classes C and C 0 that ....

L. Goldschlager and I. Parberry. On the construction of parallel computers from various bases of boolean functions. Theoretical Computer Science, 43:43-- 58, 1986.

....3 If every set in P is many one reducible to a 2 polylog sparse set via a function that is O(polylog) space computable, then P DSPACE[polylog] Let us add a few words about the proof. It involves space efficient recognition of class PhiL [3] a logarithmic space bounded version of PhiP [11,4]. Assuming that P has such a sparse hard set, we reduce the topologically sorted circuit value problem to a problem in PhiL, which is a natural analog of the circuit value problem. Since PhiL DSPACE[log 2 n] see [3] and [1] the problem is in DSPACE[log 2 n] The paper is organized ....

L. Goldschlager and I. Parberry. On the construction of parallel computers from various bases of boolean functions. Theoretical Computer Science, 43:43--58, 1986.

....O(1) at the leaves. This result improves the known upper bounds for the class ACC. 1 Introduction The celebrated results of Toda [Tod 91] have sparked renewed interest in the complexity classes #P [Va 79] PP (probabilistic polynomial time [Gi 77] and Phi P (parity polynomial time [PaZa 83, GoPa 86] One relationship among these classes is that Phi P comprises exactly those languages which are decided with the information in the rightmost bit of a #P function f , and PP, those decided with the information in the leftmost bit. For the latter statement we arrange, as described later, that ....

L. Goldschlager, I. Parberry, On the construction of parallel computers from various bases of Boolean functions. In Theoretical Computer Science 21, (1986), 43-58.

....number of accepting paths of machine M on input w. All further special notations will be introduced before they are used. 2 Locally Definable Acceptance Types Goldschlager and Parberry introduced so called extended Turing machines in order to generalize the concept of alternating Turing machines [14]. They studied the power of nondeterministic, time bounded Turing machines with an altered manner of acceptance. Instead of only labeling the states (and thus also the configurations) of the machine just with AND, OR, NOT, Accept or Reject, as it is done in alternating Turing machines, they ....

L. Goldschlager and I. Parberry. On the construction of parallel computers from various bases of boolean functions. Theoretical Computer Science, 43:43--58, 1986.

....k P [CH90] is the class of languages L for which there exists a polynomial time bounded nondeterministic Turing machine M such that for every x, it holds that x 2 L if and only if the number of accepting computation paths of M on x is not a multiple of k. Especially, MOD 2 P is denoted by PhiP [GP86, PZ83] We will also be specially interested in MOD 3 P. Purely as a convention, we will be using Phi 3 P to denote MOD 3 P throughout this paper. For a class of languages C, define Phi Delta C (respectively, Phi 3 Delta C) be the class of languages L for which there exist a polynomial p ....

L. Goldschlager and I. Parberry. On the construction of parallel computers from various bases of boolean functions. Theoretical Computer Science, 43:43-- 58, 1986.

....value problem to Parity CVP, the circuit value problem of a circuit consisting exclusively of parity gates. However, this restricted circuit value problem is known to be in DSPACE[log 2 n] Readers familiar with the class PhiL [BDHM92] a logarithmic space bounded 1 6 version of PhiP [PZ83, GP86] will recognize our reduction to map an instance of the circuit value problem to a problem in PhiL, and recall that PhiL DSPACE[log 2 n] see [BDHM92] and [ AJ93] The paper is organized as follows. In Section 2, I define the basic notation 1 7 and the circuit value problems. In ....

L. Goldschlager and I. Parberry. On the construction of parallel computers from various bases of Boolean functions. Theoretical Computer Science, 43:43--58, 1986.

....to be 8 : co Delta 9 Delta co, e.g. coNP = 8 Delta P. In this paper, we focus on the following counting classes defined and intensely studied in the literature rather than formally defining them, we just refer the reader to the respective references: UP [Val76] FewP [All86] PhiP [PZ83; GP86] PP, C=P [Sim75; Wag86] SPP [OH90; FFK91] and LWPP [FFK91] Below we summarize the known relations among these classes and state some properties to be applied in proving Corollary 3.3. 3 Fact 2.1 1. UP FewP NP coC=P PP. 2. FewP SPP LWPP C=P PP. 3. SPP PhiP. 4. PhiP, SPP, ....

.... Delta K 9 Delta K, 9 Delta NP = NP [Sto77; MS72] and 8 Delta C=P = C=P [Tod91] Thus, Few Delta NP 9 Delta NP = NP and Few Delta coC=P 9 Delta coC=P = co Delta 8 Delta C=P = coC=P. 2 Note that, in the above proof, there is nothing special about the mod 2 defining PhiP [PZ83; GP86] all we need is its self lowness and that FewP PhiP [CH90] Thus, the result holds as well for all classes Mod p P (defined in [CH90; BGH90] for prime p. 4 Conclusions and Open Problems We have presented several new upward separation results contrasting recently discovered results about ....

L. Goldschlager and I. Parberry. On the construction of parallel computers from various bases of boolean functions. Theoretical Computer Science 43, 1986, 43--58.

....has been applied in circuit complexity to establish improved simulations of the class ACC. 1 Introduction The seminal results of Toda [Tod91] have sparked renewed interest in the complexity classes #P (number P [V79] PP (probabilistic polynomial time [G77] and PhiP (parity polynomial time [PZ83, GP86]) as well as in Wagner s class C=P (exact counting; W86] and in the classes Mod k P [CH90, H90, BG92] Toda s Theorem is proved in two parts which are of independent interest. In the first part, using techniques introduced by Valiant and Vazirani [VV86] Toda shows that the polynomial time ....

L. Goldschlager, I. Parberry, On the construction of parallel computers from various bases of Boolean functions. Theoretical Computer Science 21, (1986), 43-58.

....before they are used. Due to lack of space some results appear without proof. Mainly proofs are omitted for results where similar techniques as in Hertrampf s work [11] apply. A full paper containing all proofs is available. 2 Extended locally definable acceptance types Goldschlager and Parberry [7] introduced so called extended Turing machines in order to generalize the concept of alternating Turing machines [4] They studied the power of nondeterministic, time bounded Turing machines with an altered manner of acceptance. Instead of only labeling the states (and thus in a straightforward ....

....to have complete problems. Definition 4 [11] A language L is a member of (F)P, iff there is an F machine M such that the result of M on input w is 1 if w 2 L and the result is different from 1 if w = 2 L. Our definition seems to be a natural extension of Goldschlager s and Parberry s definition [7] for two valued logic. In [7] a word is accepted iff the root of the computation tree evaluates to 1. If in m valued logic the codomain may have m different values, why shouldn t we nevertheless demand that the root of the computation tree comes to a clear answer, i.e. saying yes (resp. ....

[Article contains additional citation context not shown here]

L. Goldschlager and I. Parberry. On the construction of parallel computers from various bases of boolean functions. Theoretical Comput. Sci., 43:43--58, 1986.

....y Department of Computer Science and Information Mathematics, University of ElectroCommunications, Chofu shi, Tokyo 182, Japan. Work done in part while visiting Fakultat fur Informatik, Universitat Ulm; email address: toda cs.uec.ac. jp 1 The class PhiP was independently defined in [GoPa86] where it is called EP. nondeterministic machine. By considering arbitrary (but constant) moduli, PhiP was subsequently generalized to the classes Mod k P, k 2, CH89, Her90, BG92] The class PhiP plays a key role in Toda s recent result [Tod91] that the polynomialtime hierarchy PH is ....

L. Goldschlager, I. Parberry. On the construction of parallel computers from various bases of boolean functions. Theoretical Computer Science 43, 43-58, 1986.

....M it holds that for each x, x 2 L if and only if #accM (x) 2 Z , where #accM (x) represents the number of accepting paths of machine M on input x. Replacing Z with the fixed set f1; 3; 5; Delta Delta Deltag yields a perfectly acceptable definition of the complexity class PhiP [PZ83, GP86] and so on for many standard classes, such as coNP, US, etc. In fact, quite surprisingly, it turns out that there is a single polynomial time computable set, BitIsOne = fhi; ji fi fi the jth bit of i is a oneg, that is universal for PP PH in the sense that this one set serves simultaneously ....

L. Goldschlager and I. Parberry. On the construction of parallel computers from various bases of boolean functions. Theoretical Computer Science, 43:43-- 58, 1986.

.... P if and only if there is a nondeterministic polynomial time Turing machine M such that, for each x, it holds that x 2 L ( acc M (x) 6j 0 (modulo k) As is standard, PhiP will be used to represent Mod 2 P, a class first studied by Papadimitriou and Zachos [PZ83] and Goldschlager and Parberry [GP86] OptP [Kre88] informally stated, is the class of functions computing the maxima of multivalued NP functions. Definition 2.2 [Kre88] For a nondeterministic Turing transducer M , define SM (x) to be the set of all outputs of M on x. A function f belongs to OptP if there exist a polynomial ....

L. Goldschlager and I. Parberry. On the construction of parallel computers from various bases of boolean functions. Theoretical Computer Science, 43:43--58, 1986.

....unsatisfiable formulas (or detecting unique solutions, as in [28] 1. Introduction Valiant [27] defined the class #P of functions whose values equal the number of accepting paths of polynomial time bounded nondeterministic Turing machines. Many interesting classes, such as PP [9, 21] and PhiP [17, 10], are subclasses of P #P[1] the class of languages computable in polynomial time with one query to a function in #P. Since a PP machine accepts when more than half of its paths accept, PP can be considered equivalent to computing the high order bit of a #P function. Since a PhiP machine ....

..... The class CP= was first studied by Russo [19] and by Wagner [29] The class CP was defined by Wagner [29] who showed it is equal to PP. The classes CP =f(x) and CP f(x) were studied by Tor an [25] The class PhiP was defined by Papadimitriou and Zachos [17] and by Goldschlager and Parberry [10]. 4. Thresholds In this section, we consider machines whose acceptance is based on the number of accepting paths reaching some threshold. Thresholds were studied in [21] We present one framework in which computational power is a monotone function of the threshold, and another framework in which ....

L. M. Goldschlager and I. Parberry. On the construction of parallel computers from various bases of Boolean functions. Theoretical Comput. Sci., 43:43--58, 1986.

....if #accM (x) 2 f1; 2; 3; Delta Delta Delta g, where #accM (x) represents the number of accepting paths of machine M on input x. Replacing f1; 2; 3; Delta Delta Delta g with the set f1; 3; 5; Delta Delta Deltag yields a perfectly acceptable definition of the complexity class PhiP [PZ83,GP86] and so on for many standard classes, such as coNP, US, etc. In fact, quite surprisingly, it turns out that there is a single polynomial time computable set, MiddleBit = fi fi fi the b(log 2 i) 2cth bit of i is a oneg, that is universal for PP PH in the sense that this Some of these ....

L. Goldschlager and I. Parberry. On the construction of parallel computers from various bases of boolean functions. Theoretical Computer Science, 43:43--58, 1986.

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L. Goldschlager, I. Parberry, On the construction of parallel computers from various bases of Boolean functions. In Theoretical Computer Science 21, (1986), 43-58.

No context found.

L. Goldschlager and I. Parberry, On the construction of parallel computers from various bases of Boolean functions, Theoretical Computer Science 43, 43--58.

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