J. Simon and M. Szegedy. A new lower bound theorem for read-only-once branching programs and its applications. In J.-Y. Cai, editor, Advances in Computational Complexity, pages 183-193. AMS-DIMACS Series, 1993.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....comparison, there is a simple read twice program 2 CLIQUE ONLY of size O(n ) Dunne [Du85] proved a lower bound of 2 Omega Gamma the problems of determining whether a graph on n nodes contains a hamiltonian cycle and determining whether it contains a perfect matching. Simon and Szegedy [SS93], in order to demonstrate their lower bound technique, proved a lower bound of 2 Omega Gamma for the problem of determining whether a graph on n nodes is (n=2) regular. Note that none of these bounds is fully exponential, since the number of input variables, one for each edge, is . Babai, ....

....is fully exponential, since the number of input variables, one for each edge, is . Babai, Hajnal, Szemeredi and Turan [BHST87] proved an asymptotically optimal lower bound of 2 Omega Gamma for computing the parity of the number of triangles in a graph on n nodes; Simon and Szegedy [SS93] simplify and refine their analysis, improving the constant in the exponent. 2.3.3 For nondeterministic programs, read once and read k times Exponential lower bounds for explicit functions have also been proved for nondeterministic read once branching programs. Krause, Meinel, and Waack [KMW91] ....

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J. Simon and M. Szegedy. A new lower bound theorem for read-only-once branching programs and its applications. Advances in Computational Com- plexity Theory (J. Cai, editor), DIMACS Series, Vol. 13, AMS (1993) pp. 183--193.

....input bit be read at most some fixed number of times. This led to the definition of read k branching programs [Weg87] in which each input can be read at most k times. Many lower bounds have been shown for several functions on read once branching programs (for example, see [Weg88, Weg86, Raz91, SS93] Another branching program restriction that has also been considered is that of oblivious branching programs which test the same variable at each time step along any path. For oblivious branching programs, linear length and read k for some constant k are essentially the same and several ....

Janos Simon and Mario Szegedy. A new lower bound theorem for read only once branching programs and its applications. In Advances in Computational Complexity (J. Cai, editor), volume 13 of DIMACS Series in Discrete Mathematics, pages 183-- 193. AMS, 1993. 26

....require that each input bit be read at most some fixed number of times. This led to the definition of read k branching programs [Weg87] in which each input can be read at most k times. Many lower bounds have been shown for several functions on read once branching programs (for example, see [Raz91, SS93] Another branching program restriction that has also been considered is that of oblivious branching programs which test the same variable at each time step along any path. For oblivious branching programs, linear length and read k for some constant k are essentially the same and several ....

Janos Simon and Mario Szegedy. A new lower bound theorem for read only once branching programs and its applications. In Advances in Computational Complexity (J. Cai, editor), volume 13 of DIMACS Series in Discrete Mathematics, pages 183--193. AMS, 1993. 21

....that each input bit be read at most some fixed number of times. This led to the definition of read k branching programs [Weg87] in which each input can be read at most k times. Many lower bounds have been shown for several functions on read once branching programs (for example, see [Weg88, Raz91, SS93] Another branching program restriction that has also been considered is that of oblivious branching programs which test the same variable at each time step along any path. For oblivious branching programs, linear length and read k for some constant k are essentially the same and several ....

Janos Simon and Mario Szegedy. A new lower bound theorem for read only once branching programs and its applications. In Advances in Computational Complexity (J. Cai, editor), volume 13 of DIMACS Series in Discrete Mathematics, pages 183--193. AMS, 1993.

....read once programs have been considered by some researchers for possible use in hardware verification [GM94] Very little was known about the complexity of multiplication. There has been great success in proving lower bounds on the size of read once programs, even for some very simple functions [Ma76, Du85, Za84, We87, SS93]. For example, it was proved in [Za84] see also [We87] that determining whether a graph on n nodes is an n 2 clique (with no further edges) requires size 2## n) Until the time of this writing, the only asymptotically optimal lower bound was found in [BHST87] which proves a bound of 2 ## n ....

....prove that any read once branching programs for MULT have size 2 ## # n) This is the first superpolynomial lower bound for multiplication on nonoblivious branching programs. We begin by describing in section 2 a paradigm for read once lower bounds, recognized and distilled by Simon and Szegedy [SS93], in Lemma 1. For ease of presentation we first prove a lower bound of 2 ## 3 # n) in section 3 and then extend the proof to achieve 2 ## # n) in section 4. In section 5, we define read once reductions 3 in order to deduce similar lower bounds for other arithmetic functions. 2. A paradigm ....

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J. Simon and M. Szegedy, A new lower bound theorem for read-only-once branching programs and its applications, in Advances in Computational Complexity Theory, J. Cai, ed., DIMACS Series, Vol. 13, AMS, Providence, RI, 1993, pp. 183--193. A READ-ONCE LOWER BOUND FOR MULTIPLICATION 815

....BP where each variable may appear at most once on each path from the source to a sink. This restricted type of BPs has been the first one for which exponential lower bounds on the size could be proven [55, 61] and it has been extensively studied. A general technique presented by Simon and Szegedy [50] summarizes what is behind most of the known proofs of lower bounds for deterministic read once BPs. More recently, G al [19] and Bollig and Wegener [15] have obtained exponential lower bounds on the size of deterministic read once BPs even for very simple functions, i.e. functions with ....

J. Simon and M. Szegedy. A new lower bound theorem for read-only-once branching programs and its applications. In J.-J. Cai, editor, Advances in Computational Complexity Theory, DIMACS Series in Discrete Mathematics and Theoretical Computer Science 13, 183--193. American Mathematical Society, 1993.

.... bounds are of magnitude 2 Omega Gamma p n) The first lower bound of magnitude 2 Omega Gamma n) where n is the input size was 2 n=c lower bound for a large c for the function parity of the number of triangles in a graph , see [2] The constant c in this lower bound was improved, see [15], to a lower bound of magnitude 2 n=2000 for the same function. For a different function, a lower bound 2 n Gammao(n) was proved in [12] In [9] a lower bound 2 c p n is proved for multiplication. Several generalizations of 1 b. p. s are investigated. Recently, the most powerful among ....

J. Simon, M. Szegedy, A New Lower Bound Theorem for Read Only Once Branching Programs and its Applications, Advances in Computational Complexity Theory (J. Cai, editor), DIMACS Series, Vol. 13, AMS (1993) pp. 183--193.

....[88] functions representing codes ( 61] 83] and many more. Most of these proofs are based on variants of the cut and paste argument already applied in the first works of Wegener and Zak. The various approaches have been unified and generalized by the technique of Simon and Szegedy [107]. Read k times branching programs for k 1 are much more difficult to handle than readonce branching programs. In contrast to read once branching programs, read k times branching programs with k 1 may contain inconsistent paths (also called null chains) An inconsistent path is a path which ....

J. Simon and M. Szegedy. A new lower bound theorem for read-only-once branching programs and its applications. In J.-J. Cai, editor, Advances in Computational Complexity Theory, DIMACS Series in Discrete Mathematics and Theoretical Computer Science 13, 183--193. American Mathematical Society, 1993.

.... lower bound could be established ( 21] 23] By now, the theory of deterministic read once branching programs is well understood, and there is a large collection of interesting lower bound results (Razborov [17] gives an overview, for a summary of proof techniques for lower bounds, see [19]) We mention another variant of restricted branching programs which will turn up in the sequel. OBDDs (ordered binary decision diagrams) introduced by Bryant [7] are even further restricted than read once branching programs, but have nevertheless turned out to be extremely 2 useful in ....

J. Simon and M. Szegedy. A new lower bound theorem for read-only-once branching programs and its applications. In J.-J. Cai, editor, Advances in Computational Complexity Theory, volume 13 of DIMACS Series in Discrete Mathematics and Theoretical Computer Science. American Mathematical Society, 1993.

....size of a branching program is the number of the nodes. A branching program is read once if every variable occurs at most once along each source sink path. A number of papers have presented lower bounds for read once branching programs. Exponential lower bounds are given for explicit functions in [2, 14, 29, 45, 55, 72, 73, 84, 97, 101]. Results for the more general case of read k times branching programs appear in [22, 46, 77] We list some functions which have previously been shown to require exponential size read once branching programs. This is not intended to be a complete list. 29] considers the Hamiltonian Circuit and ....

....case of read k times branching programs appear in [22, 46, 77] We list some functions which have previously been shown to require exponential size read once branching programs. This is not intended to be a complete list. 29] considers the Hamiltonian Circuit and the Perfect Matching problems. [84] proves an exponential lower bound for the function taking value 1 if and only if the 10 input graph on m vertices is m=2 regular. 72] presents an exponential lower bound for integer multiplication. The Clique Only function is defined as having value 1 if the input represents the edges of a ....

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J. Simon and M. Szegedy, "A new lower bound theorem for read-only-once branching programs and its applications", In Advances in Computational Complexity Theory, DIMACS Series in Discrete Mathematics and Theoretical Computer Science, Volume 13, 1993, pp. 183-193.

....lower bounds for certain weak automata are known, this yields a contradiction. For example, the proof system regular resolution corresponds directly to the class of read once branching programs [Kra95] Thus, the known technology for proving lower bounds on read once branching programs ( AM86, SS93] can be brought to bear on lower bounds for resolution [RWY97] Our idea is to reverse this correspondence: proof system # automata class. In our case we plan to look at the reverse correspondence: automata class # proof system. Roughly, in a natural way given an class of automata we can ....

.... (see also [BT88, BP96] Many lower bounds and general techniques are known for restricted forms of branching programs and mostly for oblivious and read once branching programs: exponential lower bounds are known for computing the parity of the number of triangles in a graph [ABH 86, BHST87, SS93] size of regular resolution proofs for the pigeonhole principle [Hak85, RWY97] integer multiplication [Pon95] for the clique only function [BRS93, Weg87] the k regularity problem for a graph [BHST87, SS93] and many other explicit functions [Gal97, ABH 86, BHST87, Dun85] Recent work of ....

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Janos Simon and Mario Szegedy. A new Lower Bound Theorem for Read-Only-Once Branching Programs and its Applications. DIMACS Series in Discrete Mathematics and Theoretical Computer Science, 13, 1993.

....branching programs (1 b.p. s) where, on every path from the source to a sink, no two branching nodes are labeled by the same input index exponential lower bounds of the form 2 Omega Gamma p n) were given for explicit n variable Boolean functions in [Weg88, Zak84, Dun85, Juk88, KMW91, SS93, Pon99, Gal97, BW98] among others. Moreover, Juk88, KMW91, Gal97, BW98] exhibited Boolean functions in AC 0 that require 1 b.p. s of size at least 2 Omega Gamma p n) 1 After lower bounds of the form 2 Omega Gamma p n) were obtained for 1 b.p. s, the natural problem was to find an ....

.... the natural problem was to find an explicit Boolean function with the truly exponential lower bound 2 Omega Gamma n) The first such bound was proved in [ABH 86] for the Boolean function computing the parity of the number of triangles in a graph; the constant factor was later improved in [SS93] With the objective to improve this lower bound, Savick y and Z ak [SZ96] constructed a Boolean function in P that requires a 1 b.p. of size at least 2 n Gamma3 p n , and gave a probabilistic construction of a Boolean function requiring a 1 b.p. of size at least 2 n GammaO(log n) ....

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J. Simon and M. Szegedy. A new lower bound theorem for read-only-once branching programs and its applications. In J.-Y. Cai, editor, Advances in Computational Complexity, pages 183--193. AMS-DIMACS Series, 1993.

....a survey) In particular, for readonce branching programs (1 b.p. s) where, on every path from the source to a sink, no two branching nodes are labeled by the same input index exponential lower bounds of the form 2 Omega ( p n) were given for explicit n variable Boolean functions in [17, 18, 5, 7, 8, 16, 10, 6, 4] among others. Moreover, 7, 8, 6, 4] exhibited Boolean functions in AC 0 that require 1 b.p. s of size at least 2 Omega ( p n) c flSpringer Verlag. To appear in LATIN 2000. After lower bounds of the form 2 Omega ( p n) were obtained for 1 b.p. s, the natural problem was to find ....

....for 1 b.p. s, the natural problem was to find an explicit Boolean function with the truly exponential lower bound 2 Omega (n) The first such bound was proved in [1] for the Boolean function computing the parity of the number of triangles in a graph; the constant factor was later improved in [16]. With the objective to improve this lower bound, Savick y and Z ak [15] constructed a Boolean function in P that requires a 1 b.p. of size at least 2 n Gamma3 p n , and gave a probabilistic construction of a Boolean function requiring a 1 b.p. of size at least 2 n GammaO(log n) Finally, ....

[Article contains additional citation context not shown here]

J. Simon and M. Szegedy. A new lower bound theorem for read-only-once branching programs and its applications. In J.-Y. Cai, editor, Advances in Computational Complexity, pages 183--193. AMS-DIMACS Series, 1993.

....of restricted b. p. s are intensively investigated. An important restriction are the so called 1 b. p. s, where each computation reads each input bit at most once. There is a series of lower bounds for 1 b. p. s. The largest known lower bound was 2 n=2000 for a function of n variables, see [14]. In the present paper, a lower bound of 2 n Gamma3 p n is given for an explicit function. A generalization of the construction is also presented that may in principle lead to a lower bound that almost matches a general upper bound. 1 Introduction A branching program (b. p. is a computation ....

....A lower bound 2 p n=2 GammaO(1) appeared in [5] for testing existence of a Hamiltonian cycle and existence of a perfect matching in a graph. In [2] a lower bound 2 n=c for a large c is proved for the function parity of the number of triangles in a graph . This lower bound was improved in [14] to a lower bound 2 n=2000 for the same function. In [14] a technique for proving lower bounds for 1 b. p. s that generalizes most of the previous results on 1 b. p. s, is described. In [10] a bound 2 Omega Gamma p n) is proved for multiplication. Besides 1 b. p. s more general models are ....

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J. Simon, M. Szegedy, A New Lower Bound Theorem for Read Only Once Branching Programs and its Applications, Advances in Computational Complexity Theory (J. Cai, editor), DIMACS Series, Vol. 13, AMS (1993) pp. 183--193.

....kinds of restricted b. p. s are intensively investigated. An important restriction are so called 1 b. p. s, where each computation reads each input bit at most once. There is a series of lower bounds for 1 b. p. s. The largest known lower bound was 2 n=2000 for a function of n variables, see [11]. In the present paper, a lower bound of 2 n Gammao(n) is given. 1 Introduction A branching program (b. p. is a computation model for representing the Boolean functions. The input of a branching program is a vector consisting of n input bits. The branching program itself is a directed acyclic ....

....in [3] for testing Hamiltonian cycle The research of both authors was supported by GA of the Czech Republic, grant No. 201 95 0976. and perfect matching in a graph. In [1] a lower bound 2 n=c for a large c for the function parity of the number of triangles. This lower bound was improved in [11] to a lower bound 2 n=2000 for the same function. In [11] a technique for proving lower bounds for 1 b. p. s is described that generalizes most of the previous results on 1 b. p. s. In [7] a bound 2 Omega Gamma p n) is proved for multiplication. Besides 1 b. p. s more general models are ....

[Article contains additional citation context not shown here]

J. Simon, M. Szegedy, A New Lower Bound Theorem for Read Only Once Branching Programs and its Applications, Advances in Computational Complexity Theory (J. Cai, editor), DIMACS Series, Vol. 13, AMS (1993) pp. 183--193.

.... been proved for readonce branching programs (i.e. the case k = 1 in the above definition) The theory of this model is especially well understood, and there is a large collection of interesting lower bound results for it (Razborov (1991) gives an overview, for a summary of proof techniques, see Simon and Szegedy (1993)) In the context of this paper, lower bounds for general k are more interesting. For syntactic read k times BPs, exponential lower bounds have been independently proved by Okolnishnikova (1993) for k (log n= log log n) 1, and by Borodin, Razborov, and Smolensky (1993) even for ....

Simon, J. and M. Szegedy (1993). A new lower bound theorem for read-only-once branching programs and its applications. In J.-J. Cai (Ed.), Advances in Computational Complexity Theory, Volume 13 of DIMACS Series in Discrete Mathematics and Theoretical Computer Science. American Mathematical Society.

....Hierarchy Jayram S. Thathachar January 12, 1998 Abstract We obtain an exponential separation between consecutive levels in the hierarchy of classes of functions computable by polynomial size syntactic read k times branching programs, for all k 0, as conjectured by various authors [Weg87, SS93, Pon95] For every k, we exhibit two explicit functions that can be computed by linear sized read (k 1) times branching programs but require size exp n Omega i n 1=k 1 2 Gamma2k k Gamma4 jo to be computed by any read k times branching program. The result actually gives the strongest ....

....(see [Weg87] have the restriction that along each path from source to sink, each variable may be tested at most k times. The case k = 1, i.e. read once branching programs, is well understood and a variety of lower bound and separation results have been shown for it (see [Raz91] for an overview, SS93] for an overview and summary of proof techniques and [Sau97b] for a discussion on the separation results) Wegener [Weg87] conjectured that the hierarchy of classes of functions computable by polynomial size read k times branching programs, for k 1, is proper. Borodin, Razborov and Smolensky ....

[Article contains additional citation context not shown here]

Janos Simon and Mario Szegedy. A new lower bound theorem for read only once branching programs and its applications. In Advances in COmputational Complexity (J. Cai, editor), volume 13 of DIMACS Series in Discrete Mathematics, pages 183--193. AMS, 1993.

....[9] Ordered read once branching program or ordered binary decision diagrams (OBDD) 4, 15] also important for practical computer science. They are used in circuits verifications. But many important functions cannot be computed by determinsitc read once branching programs of polynomial size [4, 13, 8]. In [2] we introduced the model of randomized branching programs and showed that randomized ordered read once branching programs can be more effective than determinstic ones. In [3] we defined exclusive boolean function fn in n variables which can be computed by polynomial size randomized ....

J. Simon and M. Szegedy, A new lower bound theorem for read-only-once branching programs and its applications, Advances in Computational Complexity Theory, ed. Jin-Yi Cai, DIMACS Series, 13, AMS (1993), 183-193.

....of OBDDs, namely Free BDDs (FBDDs) FBDDs have also been considered in complexity theory under the name read once branching programs. There are many papers presenting lower bound methods for FBDDs. The first ones are due to Wegener [20] and Zak [21] and in the paper of Simon and Szegedy [18] most previous approaches are handled in a unified way. Already in the early paper of Fortune, Hopcroft and Schmidt [7] it was shown that FBDDs are exponentially more powerful than OBDDs by presenting an example of a function with polynomial FBDD size but exponential OBDD size. The algorithmic ....

Simon, J. and Szegedy, M. (1993). A new lower bound theorem for read-only-once branching programs and its applications. In Advances in Computational Complexity Theory, JinYi Cai, ed., DIMACS Series in Discrete Mathematics and Theoretical Computer Science 13, American Mathematical Society, 183--193.

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J. Simon and M. Szegedy. A new lower bound theorem for read-only-once branching programs and its applications. In J.-Y. Cai, editor, Advances in Computational Complexity, pages 183-193. AMS-DIMACS Series, 1993.

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J. Simon, M. Szegedy, A New Lower Bound Theorem for Read Only Once Branching Programs and its Applications, Advances in Computational Complexity Theory (J. Cai, editor), DIMACS Series, Vol. 13, AMS (1993) pp. 183--193.

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J. Simon, M. Szegedy, A New Lower Bound Theorem for Read Only Once Branching Programs and its Applications, Advances in Computational Complexity Theory (J. Cai, editor), DIMACS Series, Vol. 13, AMS (1993) pp. 183--193.

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Simon, J., and Szegedy, M. (1993). A new lower bound theorem for read-only-once branching programs and its applications. DIMACS Series in Discrete Mathematics and Theoretical Computer Science 13, 183--193.

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