S. Skyum and L. G. Valiant. A complexity theory based on boolean algebra. Journal of the ACM, 32:484--502, 1985.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....and bounded fan in gates. Negations are allowed only at the input level. Clearly NL=poly SAC . The uniform version of SAC is the same as the class LOGCFL of languages logspace reducible to contextfree languages [S] V1] Properties and characterizations of LOGCFL are studied in [C] [SV]. Semi unbounded fan in circuits of larger depths correspond to extensions of context free languages [C, I, Ru, V1] An interesting property of classes defined by semi unbounded fan in circuits, proved by Borodin et al. BCDRT] is that they are closed under complementation for all depths that ....

S. Skyum and L. G. Valiant, "A complexity theory based on Boolean algebra," In Proc. of the 22nd FOCS, (1981), pp. 244253.

....following branching program value problem BPV k is in : Given a width k cylindrical branching program and a truth assignment to its variables, decide if the program accepts. As any function computed by width k cylindrical polynomial size branching program clearly is a Skyum Valiant projection [6] of BPV k , we will be done. We shall prove that BPV k is in ACC by showing that it reduces, by an AC problem of the monoid M k we define next. Then, we show that the monoid M k is solvable, and since this implies, by the result of Barrington and Therien [4] that the word problem for M k is ....

....it allows us, by induction, to just consider depth 1 circuits. 8 As in [3] it will be convenient to identify input vectors in with its set of maximal 1 intervals, only here we consider cyclic intervals. For example is the vector 1010011011 identified with the set of intervals [3, 3] [6, 7], 9, 1] We will only do this identification for inputs which contain at least one interval, that is, we disregard the vectors 0 and 1 . Lemma 9 Let x contain m interval and f N k . Then f(x) contains at most m intervals. If f(x) in fact contains m intervals and m 1 then f(x) ....

S. Skyum and L. G. Valiant. A complexity theory based on boolean algebra. Journal of the ACM (JACM), 32(2):484--502, 1985.

....3.4 Problem reductions We may deduce similar lower bounds for other boolean functions by the standard technique of problem reduction. In order to preserve read once complexity, we will consider a very restrictive type of problem reduction. We begin with the notion of projection reductions [SV81], as defined in [CSV84] Definition 5 A function f = ff n g n2N is projection reducible to a function g = fg n g n2N , written f proj g, if there is a mapping oe : fy 1 ; y p(n) g f0; 1; x 1 ; x n ; x 1 ; x n g such that f n (x 1 ; x n ) g p(n) oe(y 1 ) ....

S. Skyum and L. Valiant. A complexity theory based on Boolean algebra. Proceedings of the 22nd Annual Symposium on Foundations of Computer Science, (1981) pp. 244-253,

....all nontrivial classes C it holds O 1 C O 2 C We will also simply write (A; B) instead of (A; B) Delta ) Delta . We will see that there is a close connection of this order on dot operators to the following definition. Definition 4. 2 (monotone projections, Skyum Valiant [24]) A function f : Sigma Gamma Sigma is a monotone projection if there exist two functions l : N Gamma N and oe : N Theta N Gamma N such that for every word x 1 Delta Delta Delta xm it holds f(x 1 Delta Delta Delta xm ) m; oe(m; 1) m; oe(m; 2) Delta Delta Delta [m; ....

S. Skyum, L. G. Valiant. A complexity theory based on Boolean algebra, Journal of the ACM 32, 1985, pp. 484--505.

....2:4 we show how to use this spanning forest to find the number of connected components of a graph, and how we solve the st non connectivity problem with it. 2.2 Projections to USTCON . In this paper we will use only the simplest kind of reductions, i.e. LogSpace uniform projection reductions [SV85] Moreover, we will be interested only in reductions to USTCON . In this subsection we define this kind of reduction and we show some of its basic properties. Notation 2.1 Given f : f0; 1g denote by f n : f0; 1g the restriction of f to inputs of length n. Denote by f n;k the k th bit ....

Skyum and Valiant. A complexity theory based on boolean algebra. Journal of the ACM, 1985.

....C it holds O 1 C O 2 C We will also simply write (A; B) op (A 0 ; B 0 ) instead of (A; B) Delta op (A 0 ; B 0 ) Delta . We will see that there is a close connection of this order on dot operators to the following definition. Definition 4. 2 (monotone projections, Skyum Valiant [24]) A function f : Sigma Gamma Sigma is a monotone projection if there exist two functions l : N Gamma N and oe : N Theta N Gamma N such that for every word x 1 Delta Delta Delta xm it holds f(x 1 Delta Delta Delta xm ) m; oe(m; 1) m; oe(m; 2) Delta Delta Delta [m; ....

S. Skyum, L. G. Valiant. A complexity theory based on Boolean algebra, Journal of the ACM 32, 1985, pp. 484--505.

....any number of processors and Omega Gammad 8 n= log log n) steps on PRIORITY using a polynomial number of processors. EXERCISE 21.45 [FSS84] Prove that there is a projection mapping the PARITY of n bits to the MULTIPLICATION of two n bit numbers. Many other examples of projections appear in [SV85] and [CSV84] An analogue of Turing reducibility is also useful for proving lower bounds. Let f be a problem with n inputs and let fg i g be a family of problems where g i has i inputs. Suppose there is a PRAM that can solve f using t time steps and p processors, given access to an oracle that ....

S. Skyum and L. Valiant. A complexity theory based on boolean algebra. Journal of the ACM, 32:484--502, 1985.

....of f for the variables of g. This substitution need not be onto the entire set of g s variables some of them may be fixed to 0 or 1. This is exactly a one to one projection reduction, which we shall call a read once reduction as discussed in the following definitions. Definition 6 (see [SV81]) A function f is projection reducible to a function g, written f # proj g, if for all n there is a polynomially bounded function p(n) and a mapping # n : y 1 , y p(n) # 0, 1, x 1 , x n , x 1 , x n such that f n (x 1 , x n ) g p(n) #(y 1 ) ....

S. Skyum and L. G. Valiant, A complexity theory based on Boolean algebra, J. Assoc. Comput. Mach., 32 (1985), pp. 484--502.

....of the proof is to obtain a lower bound by computing a spanning forest of the graph, which is done in subsection 3.1.2. In subsection 3.1.3 everything is put together. 3.1.1 Projections to USTCON . We will use only the simplest kind of reductions, i.e. LogSpace uniform projection reductions [SV85] Moreover, we will only be interested in reductions to USTCON. In this subsection we define this kind of reduction and we show some of its basic properties. Notation 3.1.1 Given f : f0; 1g 7 f0; 1g denote by f n : f0; 1g n 7 f0; 1g the restriction of f to inputs of length n. Denote by f ....

Skyum and Valiant. A complexity theory based on boolean algebra. Journal of the ACM, 1985.

....y . For an n bit number x, we refer to the ith bit by x (i) where x (n) is the most significant bit. 3 Problem statement Papadimitriou and Yannakakis [PY86] show that any NP complete graph property # to which satisfiability is reducible by a projection, in the sense of Skyum and Valiant [SV82] becomes NEXP complete when problem instances are encoded as circuits. They do this by first constructing a circuit 5 that computes the clause literal incidence matrix of a formula F (x) i.e. given a clause and a literal, the circuit decides whether the literal occurs in the clause in F (x) ....

S. Skyum and L. Valiant. A complexity theory based on Boolean algebra. In Proceedings of the Twenty-third IEEE Symposium on the Foundations of Computer Science, pages 244--253. IEEE Computer Society Press, Los Alamitos, CA, 1982.

....many one [Jon] one way logspace many one [HIM] and first order projections (fops) Dah] These last reductions, defined in Section 3, are provably weaker than logspace reductions. It has been observed that natural complete problems for various complexity classes remain complete via fops, cf. [I87, IL, SV, Ste]. On the other hand, Joseph and Young, JY] have pointed out that polynomialtime, many one reductions are so powerful as to allow unnatural NP complete sets. Most researchers now believe that the isomorphism conjecture as originally stated by Berman and Hartmanis is false. 1 We feel on the ....

S. Skyum and L.G. Valiant, "A Complexity Theory Based on Boolean Algebra," JACM, 32, No. 2, April, 1985, (484-502).

....languages accepted by polynomial size, O( log n) k ) depth semi unbounded fan in Boolean circuits. The uniform version of SAC 1 is the same as the class LOGCFL of languages logspace reducible to context free languages [89] 93] Properties and characterizations of LOGCFL are studied in [24] [85]. Semi unbounded fan in circuits of larger depths correspond to 56 extensions of context free languages. For d = Omega Gamma336 n) the class of languages accepted by polynomial size, depth O(d) semi unbounded fan in circuits is identical to the class of languages accepted by nondeterministic ....

S. Skyum and L. G. Valiant, "A complexity theory based on Boolean algebra," In Proc. of the 22nd FOCS, 1981, pp. 244-253.

....nondeterministic circuit c has two kinds of input gates: in addition to the actual inputs x 1 ; x n , c has a series of distinguished guess inputs y 1 ; ym . The value computed by c on input x 2 Sigma n is 1 if there exists a y 2 Sigma m such that c(xy) 1, and 0 otherwise [SV85]. Nondeterministic oracle circuits and their size are defined exactly as for deterministic oracle circuits. We next define boolean functions that are hard to approximate and related notions. For a real number s and an oracle set A Sigma , CIR A (n; s) NCIR A (n; s) denotes the class of ....

S. Skyum and L. G. Valiant. A complexity theory based on boolean algebra. Journal of the ACM, 32:484--502, 1985.

....proof is to obtain a lower bound by computing a spanning forest of the graph, which is done in subsection 2.3. In subsection 2.4 everything is put together. 2.2 Projections to USTCON . In this paper we will use only the simplest kind of reductions, i.e. LogSpace uniform projection reductions [SV85] Moreover, we will be interested only in reductions to USTCON. In this subsection we define this kind of reduction and we show some of its basic properties. Notation 2.1 Given f : f0; 1g 7 f0; 1g denote by f n : f0; 1g n 7 f0; 1g the restriction of f to inputs of length n. Denote by f ....

Skyum and Valiant. A complexity theory based on boolean algebra. Journal of the ACM, 1985.

....although a single synthesis step may cause an exponential blow up of the size (Becker, Drechsler and Werchner(1995) 3. REDUCTION CONCEPTS A lot of reduction concepts have been introduced for various types of problems. Projections and monotone projections intensively investigated already by Skyum and Valiant (1985) are perhaps the most restricted ones. Definition 3: The function f = f n ) more precisely the sequence of functions, is a (polynomial) projection of g = g n ) f proj g, if f n (x 1 ; x n ) g p(n) y 1 ; y p(n) for some polynomially bounded function p and y j 2 fx 1 ; ....

....complete problem) For each function f of polynomial circuit size, i.e. f 2 P poly, f mrop CVP (circuit value problem) Sketch of proof: i) The coding of graph problems as Boolean functions is often obvious. Numbers as the desired clique size are given in unary. For SAT we use the coding of Skyum and Valiant (1985). SAT n (X; Y ) a:f1; ng f0;1g 1in 1jn [ x ij a(j) y ij a(j) The function has 2n 2 Boolean variables and can describe all inputs for SAT with n clauses and n variables z 1 ; z n . The variable x ij describes whether z j is contained in the i th clause, y ij does ....

[Article contains additional citation context not shown here]

Skyum, S. and Valiant, L. G. (1985). A complexity theory based on Boolean algebra. Journal of the ACM 32, 484--502.

....P complete problems seem to involve redundancy in some form or the other. We restrict ourselves therefore to P complete problems that are in some sense non redundant. To do that we look at a much stricter definition of P completeness; one in terms of projections, introduced by Skyum and Valiant [15]. Definition 5 A decision problem 1 is projection reducible to another decision problem 2 ( 1 proj 2 ) if there is a function p(n) bounded above by a polynomial in n, and a family of polynomial time computable mappings oe = foe n g n1 where oe n : fy 1 ; y p(n) g fx 1 ; x 1 ; ....

S. Skyum and L. G. Valiant, "A Complexity theory based on Boolean Algebra," Proc. 22nd IEEE Symposium on Foundations of Computer Science (1981), 244--253.

....to be projection equivalent to each other [2] In his survey on connectivity [16] Wigderson mentions that under almost any choice of reducibility, USTCONN is harder than UCONN. For example, in the Boolean decision tree model and under monotone p projections, USTCONN is provably harder than UCONN [3, 12]. The status of knowledge about the complexity of these two functions in the monotone Boolean circuit setting is consistent with this relative hardness. While a Omega Gamma log 2 (n) depth bound is known for USTCONN [7] a similar bound is expected for UCONN [16] but seems elusive. Recently, ....

S. Skyum and L. Valiant, A complexity theory based on Boolean algebra, J. Assoc. Comput. Mach. 32 (1985), 484-502.

....92 00106 and by a Wolfson research award administered by the Israeli Academy of Sciences. level. The uniform version of SAC 1 is the same as the class LOGCFL of languages logspace reducible to context free languages [S] V1] Properties and characterizations of LOGCFL are studied in [C] [SV]. Semi unbounded fan in circuits of larger depths correspond to extensions of context free languages [C, I, Ru, V1] An interesting property of classes defined by semiunbounded fan in circuits, proved by Borodin et al. BCDRT] is that they are closed under complementation for all depths that ....

S. Skyum and L. G. Valiant, "A complexity theory based on Boolean algebra," In Proc. of the 22nd FOCS, (1981), pp. 244253.

No context found.

S. Skyum and L. G. Valiant. A complexity theory based on boolean algebra. Journal of the ACM, 32:484--502, 1985.

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S. Skyum and G. Valiant. A complexity theory based on boolean algebra. In Proceedings of the 22nd Foundations of Computer Science, pages 244-253, 1981.

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S. Skyum and L. Valiant. A complexity theory based on boolean algebra. Journal of the ACM, 1985.

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Skyum and Valiant. A complexity theory based on boolean algebra. Journal of the ACM, 1985. 96

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S. Skyum and L. G. Valiant. A complexity theory based on boolean algebra. Journal of the ACM, 32:484-502, 1985.

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Skyum S., Valiant L.G., A complexity theory based on Boolean algebra. Proceedings of the 22nd IEEE FOCS Symposium (1981), 244-253.

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S. Skyum, L.G. Valiant. A Complexity Theory Based on Boolean Algebra. Journal of the ACM 32(2), pp. 484-502 (1985).

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