R.E. Ladner. The circuit value problem is log space complete for P. SIGACT News, 7:18--20, 1975.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....(x; k) 2 LH ( g(x) 2k) 2 g(L) H ( C n;x ; 2k) 2 W Circ SAT : Thus, the theorem is proved. ut To prove that there is a language in P such that its Hamming closure is W hard, we consider the circuit value problem: CVP = f(C; x) j C is a Boolean circuit that outputs 1 on input xg : Ladner [8] has shown that CVP is P complete (see also Greenlaw et al. 3] For our purposes, we consider the following variant of CVP: CVP (C#C# : C z (n 1) times ; x) j (C; x) 2 CVP and C has n input bits Since CVP is P complete, CVP is P complete as well. Theorem 4.3 CVP H ....

Richard E. Ladner. The circuit value problem is log space complete for P. SIGACT News, 7(1):18--20, 1975.

....to determine the circuit s output. This is the so called circuit value problem: Problem: CVP(B) Input: A B circuit C and an input vector x Question: Does C on input x output 1 In the case that no restrictions on B are given, this problem is known to be P complete under logspace reductions [Lad75] and even under NC reductions, cf. GHR95, Chap. 6] or [Vol99, Chap. 4.6] This means that most researchers in the eld expect that it allows no ecient parallel solution in the sense of an NC algorithm, i.e. no polylog time algorithm using a reasonable amount of hardware (polynomial number of ....

R. E. Ladner. The circuit value problem is log space complete for P. SIGACT News, 7(1):12{ 20, 1975. 18

....NC reducible to calculating the determinant of an integer matrix. DET is not known to be comparable with AC or . Then we have the classical results that, for Boolean gates, Expression Evaluation and Circuit Value are NC complete and P complete respectively, under AC reductions [6, 12]. We will consider circuits and expressions where the sole operation is multiplication in some finite algebra (A; Delta) rather than the usual Boolean operations. Thus our expressions are polynomials like (x 1 Delta x 2 ) Delta (x 2 Delta x 3 ) and our circuits have one kind of node whose ....

R.E. Ladner, "The circuit value problem is LOGSPACEcomplete for P." SIGACT News 7 (1975) 18-20.

.... formulas can express precisely properties in (nonuniform) N( 30,3,7,6] Likewise, because of the ability to use abbreviations for long formulas, the lines in a polynomial size extended Frege proof are essentially polynomial size circuits and thus can express properties that are in nonuniform P [23]. Thus, one can intuitively view polynomial size Frege proofs as proofs with polynomially many steps which can reason with N( properties; whereas polynomial size extended Frege proofs are proofs with polynomially many steps which can reason with properties in nonuniform P. Of course this analogy ....

R. E. LADNER, The circuit value problem is log space complete for P, SIGACT News, 7 (1975), pp. 18 20.

....called NOT, AND and OR gates. On any input x 2 f0; 1g n , the output of each gate is defined in the natural way, including the gate that is the output of the circuit. The circuit value problem, abbreviated CVP , of determining whether a Boolean circuit C outputs 1 on input x was shown by Ladner [Lad75] to be complete for P under logspacecomputable many one reductions. Cook [Coo85] defined the notion of NC 1 reducibility, and notes that this problem is complete for P under NC 1 reductions. This reducibility is somewhat subtle technically, so we refer the reader to [Coo85] for details. ....

R. Ladner. The circuit value problem is log space complete for P. SIGACT News, 7(1):18--20, 1975.

....detailed, it can easily be summarized in a short paragraph. Dwork, Kanellakis, and Mitchell proved that first order unification was complete for PTIME, by showing how unification on first order terms could simulate circuits [15] the proof followed by reduction from the Circuit Value Problem [34]. Type inference for ML programs without let (equivalently, simply typed calculus) is easily reduced to first order unification, where a program is transformed to a unification problem of size linear in the size of the original program [48] and vice versa. As a consequence, type inference for the ....

R. E. Ladner. The Circuit Value Problem is Logspace Complete for P. SIGACT News 7 (1975), pp. 18--20.

....encoding ff of a Boolean circuit together with its inputs x 1 ; x n , and a designated output gate g. 1 A compact encoding of a circuit is polynomial in the circuit size. 14 Problem: Does g evaluate to 1 on input x 1 ; x n Theorem 2. 5 The circuit value problem is P complete [20]. Numerous variants of CVP are P complete [14] In nor CVP the circuit consists entirely of nor gates with fan in and fan out two. nor CVP without fan out restrictions is also P complete for planar circuits; this version is called planar nor CVP. In monotone CVP the circuit is composed of and and ....

R. E. Ladner. The circuit value problem is log space complete for P. SIGACT News, 7:18, 1975. 49

....C s C s t C s t a b a; b a; b a b ab : A n n QR A r R n n A r O n k ; n A A a A A n A n a : T choice adopted here, which is the one made in the library versions of the algorithm improves its numerical accuracy. The Circuit Value Problem (CVP) is well known to be log space complete for P [6]. Given (an encoding of) a boolean circuit and inputs , the problem asks for the output of on input , denoted by ( The NAND CVP is one in which the circuit encoded is composed entirely of (two input) nand gates. NAND CVP is P complete, as reported in [4] In order to simplify the proof of ....

Ladner, R. E., The Circuit Value Problem is Log Space Complete for P, (1975) 18--20.

....machine running in polynomial time. The class NP is 8 defined similarly except that Turing machines are allowed to be nondeterministic. P has many equivalent definitions. In particular, it can be defined as the class of languages recognized by a uniform family of polynomial size Boolean circuits [45, 14]. Moreover, it is also equal to the class of languages recognized by an auxiliary pushdown automaton using only logarithmic space [23] Observe that in the last definition, time is unbounded and the model can be deterministic or not without changing the class of languages defined. NP also has a ....

R. Ladner, The circuit value problem is logspace complete for P, SIGACT News, 7 pp.18-20, 1975.

....to calculating the determinant of an integer matrix. DET is not known to be comparable with AC 1 or ACC 1 . Then we have the classical results that, for Boolean gates, Expression Evaluation and Circuit Value are NC 1 complete and P complete respectively, under NC 0 and LOGSPACE reductions [10, 17]. We will consider circuits and expressions where the sole operation is multiplication in some nite groupoid (A; rather than the usual Boolean operations. Thus our circuits have one kind of node whose output is the product a b of its two inputs, and our expressions are strings like (x 1 ....

R.E. Ladner, \The circuit value problem is LOGSPACE-complete for P." SIGACT News 7 (1975) 18-20.

....stronger result. Theorem: There is a LOGSPACE uniform family of NC 0 permutations which are Pcomplete to invert. Proof: We will reduce the problem of evaluating a straight line program to the problem of inverting an NC 0 permutation. Since the former problem is well known to be P complete [6] the latter will be P hard. We will use the term P complete to mean P complete under LOGSPACE reductions. Thus if a P complete problem is in LOGSPACE every problem in P is in LOGSPACE. In a similar manner a problem is defined to be P hard if it has the above property but is not known to be in P. ....

Ladner, R.E., "The Circuit Value Problem is Log Space Complete for P", SIGACT News, vol. 7, No 1, Jan 1975 (18-20).

....and are particularly useful for proving other problems are P complete. See Section 5 of this paper for more details. A.1.1 Circuit Value Problem (CVP) Given: An encoding ff of a Boolean circuit ff plus inputs x 1 ; x n . Problem: Does ff on input x 1 ; x n output 1 Reference: [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, ....

....given a deterministic finite state automaton M , whether D 2 L(M) If G is acyclic then the problem is complete for SAC 1 = LOGCFL [Ruz79] A.7.8 Two Way DPDA Acceptance (2DPDA) Given: A two way deterministic pushdown automaton M and a string x. Problem: Is x accepted by M Reference: [Coo71a, Gal74, Gal77, Lad75] Hint: See, e.g. HU79] for a definition of 2DPDAs. Cook [Coo71a] gives a direct simulation of a polynomial time Turing machine by a logarithmic space auxiliary pushdown automaton. Galil [Gal74, Gal77] shows existence of a P complete language accepted by a 2DPDA, in effect showing that the ....

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R. E. Ladner. The circuit value problem is log space complete for P. SIGACT News, 7(1):18--20, January 1975.

....arity. Circuits are a standard way of representing Boolean functions. We will just assume that they are encoded as words in some standard way, for the details concerning circuits we refer for example to [22] Remember that a given circuit can be evaluated on a given assignment in polynomial time [20]. Each circuit c describes a Boolean function F (x 1 ; x n ) Note that here we have an example of the classical syntax semantics dichotomy like we have it for programs: The circuits (programs) are the syntactical objects, which we have as nite words at our ngertips, whereas the Boolean ....

....cases. First case: A has a maximum a. Then co NP is p m reducible to Absolute Counting(A) Let a language L in co NP be given and let M be a machine for L in the sense that x 2 L i no path of M(x) is accepting. Let m(x) denote the number of accepting paths of M(x) Cook [11] see also Ladner [20]) established a method to construct in polynomial time a circuit Cook M x with inputs y 1 ; y n (n depends on x and is bounded by a polynomial in the length of x) such that accepting computation paths of M(x) and satisfying assignments of Cook M x (y 1 ; y n ) correspond to ....

Richard Ladner, The circuit value problem is log-space complete for P, SIGACT News 7, 1975, 18-20.

....1 1 Introduction A Boolean circuit is a circuit whose wires do not form directed cycles. The problem of evaluating a Boolean circuit, given the values of all its inputs, is called the circuit value problem (CVP) This is a central problem in the area of algorithms and complexity. Ladner [11] has shown that the CVP is P complete under log space reductions. Some special cases of the CVP have been studied, among which the monotone circuit value problem, where the Boolean circuit has only AND and OR gates, and the planar circuit value problem, where the Boolean circuit has a plane ....

Ladner, R. E. "The Circuit Value Problem is log Space Complete for P " SIGACT News, 1975, p18-20. 13

....and circuit based combinatorial problems was studied through the more than three decades of Complexity Theory. Already in 1971, S.A. COOK [Coo71] proved that the satisfiability problem for boolean formulae is NP complete: This was the first NP complete problem ever discovered. R.E. LADNER [Lad77] proved in 1977 that the circuit value problem is P complete. In many cases when a new complexity class was introduced and investigated, a formula based or circuit based combinatorial problem was the first which was proved to be complete for this class (see [SM73, Gil77] for example) However, ....

....j C is a B circuit, a 2 f0; 1g (C) and f C (a) 1g It is obvious that VAL(B) 2 P for every finite set B of boolean functions. The 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 ....

R. E. Ladner. The circuit value problem is logspace complete for P. SIGACT News, pages 18--20, 1977.

....of the same formula of size p(n) S p simulates T iff the S proof is obtainable as a polynomial time function of the T proof. Open Question: Does F simulate eF This open question is related to the question of whether Boolean circuits have equivalent polynomial size formulas. By Ladner [36] and Buss [5] this is a non uniform version of the open question of whether P is equal to alternating logarithmic time (ALOGTIME) Open Question: Is there a maximal proof system which simulates all other propositional proof systems Kraj icek and Pudl ak [33] have shown that if NEXP ....

R. E. Ladner, The circuit value problem is log space complete for P, SIGACT News, 7 (1975), pp. 18--20.

....output either 0 or 1 regardless of their input) which inputs x 1 ; x n and outputs a description of A with a copy of the input x 1 ; x n . The output is an instance of CVP which is a member of CVP iff x 2 A. Therefore, A l CVP. 2 The uniform version of Theorem 4. 2 is due to Ladner [17]. The proof in that reference is somewhat sketchy; a more detailed proof appears in Parberry [28] P complete problems are 13 P P complete Problems CVP PARITY AC Figure 7: P complete problems (conjectured) interesting since, by Lemma 4.1, if one of them is in AC, then AC = P . If the ....

R. E. Ladner. The circuit value problem is log space complete for P. SIGACT News, 7(1):18--20, 1975.

....the resulting simulation of P is carried out more efficiently (in logspace uniform NC 1 ) 2.1.5 The Circuit Value Problem The Circuit Value Problem, abbreviated CVP, consists of pairs hC;xi where C is a Boolean circuit with k inputs, x 2 f0;1g k , and on input x, C outputs 1. Ladner [Lad75] showed that CVP is complete for P under logspace many one reductions, and Cook [Coo85] observed that CVP is, in fact, complete for P under NC 1 reductions. In fact, Immerman [Imm87] has shown that CVP is complete for P under an extremely weak form of reducibility using the notion of a ....

R. Ladner. The circuit value problem is log space complete for P. SIGACT News, 7(1):18--20, 1975.

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R.E. Ladner. The circuit value problem is log space complete for P. SIGACT News, 7:18--20, 1975.

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R. E. Ladner. The circuit value problem is log space complete for P. SIGACT News, 7(1):18--20, 1975.

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R. E. Ladner, The circuit value problem is log space complete for P, SIGACT News, 7 (1975), pp. 18--20.

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R. E. Ladner, The circuit value problem is log space complete for P, SIGACT News, 7 (1975), pp. 18--20.

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R. E. Ladner, The circuit value problem is log space complete for P, SIGACT News, 7 (1975), pp. 18--20.

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R. E. Ladner, The circuit value problem is log space complete for P, SIGACT News, 7 (1975), pp. 18--20.

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Ladner, R.E.: The Circuit Value problem is logspace complete for P, SIGACT News 7, 1 (1975), 18-20

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