R. Szelepcsenyi. The method of forcing for nondeterministic automata. Bull. European Assoc. Theoret. Comput. Sci., 33:96-100, 1987.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....conjecture for NL. We show that there is a sparse hard set for NL under logspace many one reductions iff NL = L. Our proof builds on [Ogi95, CS95] and uses the algebraic techniques of [CS95] An additional crucial ingredient in the proof is the famous result of Immerman [Imm88] and Szelepcs enyi [Sze87], which shows NL = co NL. Previously, Cai, Naik, and Sivakumar [CNS95] built on the ideas of [CS95] and showed that if there is a sparse set S that is hard for NL under logspace many one reductions, then every NL problem can be solved with one sided bounded error by a probabilistic logspace ....

....s; ti is a valid instance of DAG STCON and W u;v = b, where W = W (G; s; t) belongs to NL. This language is the union of the languages Z 0 = fhG; s; t; u; v; 1i j W u;v = 1g and Z 1 = fhG; s; t; u; v; 0i j W u;v = 0g, which are easily seen, respectively, to be in NL and co NL. Since co NL = NL [Imm88, Sze87], both Z 0 and Z 1 are in NL, and since NL is closed under unions, Z 2 NL. A consequence of this fact is the following: The nondeterministic logspace machines for Z 0 and Z 1 can be used to build a nondeterministic logspace machine MW that, given hG; s; ti and u; v 2 V (G) computes W uv in the ....

R. Szelepcs'enyi. The method of forcing for nondeterministic automata. Bulletin of the EATCS, 33:96--100, 1987.

....are rather powerful. There are already a number of extensions and many additional results. Those results are primarily concerned with various other reducibilities and complexity classes. In [CS95] we combine techniques from this paper with the famous result of Immerman Szelepcs enyi [Imm88, Sze87] to resolve a similar conjecture made by Hartmanis concerning sparse hard sets for nondeterministic logspace. In joint work with A. Naik [CNS95] we use a number of additional techniques, and extend the results to the case of truth table and randomized reductions. For truth table reductions, we ....

R. Szelepcs'enyi. The method of forcing for nondeterministic automata. Bull. of the EATCS, 33:96--100, 1987.

....one hand, we do not know how to solve stcon (which is the complement problem of stcon) in O(log n) space on a nondeterministic JAG. On the other hand, we show that a nondeterministic NNJAG can solve stcon in O(log n) space by simulating Immerman Szelepcs enyi s inductive counting technique [Imm88, Sze88] Using the same technique, we show that a nondeterministic NNJAG is essentially as powerful as a nondeterministic branching program in solving a large class of graph theoretic problems. In Chapter 3, we extend the space lower bound of Cook and Rackoff [CR80] on the JAG model, as well as that of ....

R'obert Szelepcs'enyi. The method of forcing for nondeterministic automata. Acta Informatica, 26:279--284, 1988.

....with nondeterministic logarithmic space (NL) than deterministic logarithmic space (DL) 1 This is also a difficult open problem, but we seem to know more about the power of nondeterministic space than nondeterministic time. For example, it is known that NL is closed under complementation [Imm88, Sze88] while the corresponding question for NP is still open. Savitch [Sav70] shows that nondeterministic space S is contained in deterministic space S 2 for any constructible S = Omega# log n) note: throughout this thesis, log x will be used to denote the logarithm base 2 of x) No such ....

....write only query tape that is erased when a query is answered. 17 For our purposes, many one and Turing reductions are almost always equivalent, since most of our reductions will deal with languages in NL, and Immerman s and Szelepcs enyi s proofs that NL is closed under complementation [Imm88, Sze88] imply that NL is also closed under logspace Turing reductions. Still, for simplicity, we will try to use many one reductions whenever possible. Unless otherwise specified, for the rest of this thesis, logspace reducibility will mean logspace many one reducibility. Intuitively, L 1 log L 2 if ....

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R. Szelepcs'enyi. The method of forcing for nondeterministic automata. Acta Informatica, 26:279--284, 1988.

....of languages defined by machines that accept their inputs if there is exactly one accepting path. Thus, the existence of two or more accepting computations is not forbidden, but simply leads to rejection. In the polynomial time case we have Co NP 1NP. In the logspace case inductive counting ([11,19]) shows 1NSPACE(logn) NSPACE(logn) 3.1 Space bounded unambiguous classes The concept of unambiguity of space bounded computations is not as uniform as that for time bounded classes. Instead we are confronted with a variety of probably different concepts of unambiguity. In the following we ....

....T d (v 1 ) was assumed to be a tree. To make this more precise, with each STOP state the relevant variables are explicitly listed. Thus the computation is reach unambiguous. 2 It should be remarked that we couldn t replace NEXT and GUESSUNCLE by something using the Immerman Szelepcs eyi procedure ([11,19]) since this inherently admits an exponential number of possible computation before reaching an ACCEPT, REJECT, or STOP state. Thus it could only be unambiguous but never reach unambiguous. The advantage of NEXT and GUESSUNCLE is that they stop early enough such that the complete computational ....

R. Szelepcs'enyi. The method of forcing for nondeterministic automata. Acta Informatica, 26:279--284, 1988.

....imply BP (f) S(f) O(1) 5) It is worth noting that no constructive proof of (5) is known. It follows from the discussion above that Boolean simulations (1) actually reflect class inclusions P NLOGSPACE LOGSPACE NC 1 : It is also worth noting that the famous result of Szeleps enyi [40] and Immerman [20] translates to the following very interesting simulation: RS( f) RS(f) O(1) 6) At the end of this section we discuss possible modifications of our basic measures. Usually the size of branching programs is measured as the total number of nodes. However this changes the ....

R. Szelepcs'enyi. The method of forcing for nondeterministic automata. Bulletin of the EATCS,

....In fact, we reduce various versions of Delta language to the language of firstorder logic with a transitive closure operator FO Omega over finite graphs. It was shown by N. Immerman [17] that in the presence of a linear order (OE) this language (even closed under negations [18] cf. also [41]) exactly corresponds to NLOGSPACE. It is also used an analogous description of DLOGSPACE, i.e. deterministic LOGSPACE [17] The description of PTIME computability over HF mentioned above is based on a similar approach to PTIME in terms of recursive global function(al)s in finite segments f0; ....

....in finite linear ordered structures is equivalent to NLOGSPACE computability. Moreover, FO positive Omega has the same expressive power in these structures as the full FO Omega , i.e. as FO Omega [18] This result is equivalent to the statement CoNLOGSPACE = NLOGSPACE (cf. also [41]) which have been widely believed previously as false. The same holds for FO fi and DLOGSPACE (where the equivalence of fi and positive fi is rather trivial) In particular, we have FO DLOGSPACE. Therefore, we may freely interchange the notions FO Omega =fi OE and (N D)LOGSPACE where OE ....

Szelepcs'enyi, R.: 1987, The method of forcing for nondeterministic automata. Bull. European Association Theor. Comp. Sci. (Oct. 1987) 96--100

.... : Thus, although :f n;d 2 NBP 1 ; the function f n;d itself does not belong to NBP k if k k 0 = 1=2 Gamma ffl) ln n= ln ln n: This fact means that co Gamma NBP 1 n NBP k 0 6= and hence, for all k k 0 NBP k 6= co Gamma NBP k : In particular, this shows that the Immerman Szelepcs enyi [3, 9] constructions, yielding the equality NBP = co Gamma NBP; necessarily require at least logarithmic multiplicity of reading. Let us also mention that we derive our lower bound for f n;d using only the fact that: i) this function accepts sufficiently many vectors, namely, at least 2 n (n 1) ....

R. Szelepc'enyi, The method of forcing for nondeterministic automata, Bull. European Assoc. Theoret. Comput. Sci., 33 (1987), 96-100

....settle the Hartmanis conjecture for NL. We showed that there is a sparse hard set for NL under logspace many one reductions iff NL = L. Our proof uses the algebraic techniques of [CS95] An additional crucial ingredient in the proof is the famous result of Immerman [Imm88] and Szelepcs enyi [Sze87], that NL = co NL. Assuming the existence of a sparse hard set for NL, our proof gives a parallel algorithm for an NLcomplete problem. This parallel algorithm can be implemented by a logspace uniform circuit of polynomial size, log depth circuit that makes polynomially many parallel calls to the ....

R. Szelepcs'enyi. The method of forcing for nondeterministic automata. Bull. of the EATCS, 33:96--100, 1987.

....is NL complete. Proof: We first describe a nondeterministic logspace Turing machine for deciding, given a two colored undirected graph G and a start node s, whether G is not covered with probability one on all infinite sequences. It follows that the stated problem is in NL since NL = coNL [10] [16]. We use the following equivalence, the proof of which is implicit in the proof of the exponential upper bound of Theorem 2.2. A walk from s visits node t with probability strictly less than one on some infinite color sequence if and only if there exists a node v such that: 1) v is reachable from ....

....a nondeterministic polynomial space bounded Turing machine for deciding, given a colored undirected graph G and a start node s, whether G is not covered with probability one on all infinite sequences. It follows that the original problem is in PSPACE since PSPACE is closed under complement [10] [16] and under the addition of nondeterminism [14] We use the following equivalence, the proof of which is implicit in the proof of the doublyexponential upper bound of Theorem 2.1. A walk from s visits node t with probability strictly less than one on some infinite color sequence if and only if ....

[Article contains additional citation context not shown here]

R. Szelepcs'enyi. The method of forcing for nondeterministic automata. In Bulletin of the European Association for Theoretical Computer Science, 1987.

....are of equal complexity. For nondeterministic or one sided error probabilistic algorithms, however, the complexities may differ. In particular, if a problem L is solvable nondeterministically in O(log n) space, then the complement of L is, too, by the result of Immerman [33] and Szelepcs enyi [47]. For the problem of undirected st connectivity, this also follows from the result of Nisan and Ta Shma [43] However, their algorithms are rather slow. For example, a logarithmic space nondeterministic RAM can solve st connectivity in time O(n) but to solve the complementary ....

....easy to see that a nondeterministic two pebble WAG can recognize nonbipartite graphs (guess and verify an odd cycle) but not so easy to see a direct way to recognize bipartite graphs. In fact this is also possible, by the following corollary to Theorem 7 and Immerman and Szelepcs enyi s Theorem [33, 47]. Corollary 10: Let H be an undirected graph problem, and let S(n) Omega Gamma421 n) If H is solvable using space O(S(n) by a nondeterministic JAG or WAG J , then so is its complement G Gamma H. Proof: Simulate J by a nondeterministic, S(n) space bounded general machine M . By a ....

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R. Szelepcs'enyi. The method of forcing for nondeterministic automata. Acta Informatica, 26:279--284, 1988.

....corresponding map of general (i.e. nonmonotone) classes. Note that all of this occurs well below the more familiar classes of P and NP . First we list the previously known equalities and separations of these maps. NL = co NL: By the inductive counting technique of Immerman [21] and Szelepcs enyi [42], the class of functions computable by nondeterministic logspace Turing machines is closed under complementation. mBWBP mL mNL co mNL mAC 0 mAC 1 mNC 1 BWBP L NL co NL AC 0 AC 1 NC 1 Figure 1 1: Maps of low level monotone and general complexity classes. In terms of directed graphs, ....

....3 1 (the threshold k function is 1 iff at least k of its inputs are 1) Our main goal in this chapter is to show that this is not true in general for all of mNL, i.e. that mNL 6= co mNL. In particular this implies, as discussed in Chapter 1, that Immerman and Szelepcs enyi s NL = co NL simulation [21, 42] has no analogue in the monotone case. Our argument is an extension of that of Karchmer and Wigderson [23] for the undirected st connectivity function. Before the main argument, however, we review some known properties of monotone nondeterministic branching programs, in particular related to the ....

R. Szelepcs'enyi. The method of forcing for nondeterministic automata. Bull. Europ. Ass. Theoretical Computer Sci., 33:96--100, 1987.

....of languages defined by machines that accept their inputs if there is exactly one accepting path. Thus, the existence of two or more accepting computations is not forbidden, but simply leads to rejection. In the polynomial time case we have Co NP 1NP [3] In the logspace case inductive counting [13, 25] shows 1NSPACE(logn) NSPACE(logn) A more restrictive form of unambiguity is Strong Unambiguity. A nondeterministic machine is said to be strongly unambiguous, if for every pair of configurations there exits at most one computational path connecting these configurations. An ordinary unambiguous ....

R. Szelepcs'enyi. The method of forcing for nondeterministic automata. Acta Informatica, 26:279--284, 1988.

....reducibility and exhibit complete problems. This provides a framework for stating existing results and asking new questions. We show that mNL (monotone nondeterministic log space) is not closed under complementation, in contrast to Immerman s and Szelepcs enyi s nonmonotone result [Imm88, Sze87] that NL = co NL; this is a simple extension of the monotone circuit depth lower bound of Karchmer and Wigderson [KW90] for st connectivity. We also consider mBWBP (monotone bounded width branching programs) and study the question of whether mBWBP is properly contained in mNC 1 , motivated by ....

....class containments carry over to their monotone analogues. Many of the obvious containments carry through, e.g. the simulation of Turing machines by circuits. In particular we consider two recent surprising containment results in space bounded complexity: 1. Immerman [Imm88] and Szelepcs enyi [Sze87] showed that NL is closed under complementation, and 2. Barrington [Bar89] showed that BWBP (bounded width branching programs) contains all of NC 1 . These results are interesting because the simulation techniques do not seem to carry over to the corresponding monotone models. We show that the ....

Szelepcs'enyi R., The method of forcing for nondeterministic automata. Bull. European Ass. for Theoretical Computer Science, Vol. 33 (1987), pp. 96--100.

....N. Immerman [22, 23] Theorem 3 Over finite linearly ordered structures FO Omega NLOGSPACE. 2 This implies the following very unexpected 2 In [22, 23] the notation TC is used essentially for Omega . cong fin.tex Date: April 26, 1996 Time: 11:26 6 VLAD I M I R YU . SAZONOV Corollary 4 [23, 48] NLOGSPACE = co NLOGSPACE. 4. DEFINABILITY AND COMPUTABILITY IN BST 4.1. PTIME computability over HF The key notion for our approach to computability over HF is the following definition of PTIME computability of any operation f : HF n HF. Given any encoding : Codes HF of HF sets, we say ....

Szelepcs'enyi, R.: 1987, `The method of forcing for nondeterministic automata', Bull. European Association Theor. Comp. Sci. (Oct. 1987), pp. 96--100.

....Thus, we analyze in detail the consistency problem for monadic and or recursion free method schemas, which happen to be decidable. We also quantify the effect of covariance, which is a widely used constraint on the signature of methods. For the various concepts used from complexity theory, see [15, 19, 20, 29] and from database theory, see [21, 30] We briefly summarize our other results. Let n be the size of method definitions in the input method schema and c the size of the class hierarchy. In the case of monadic schemas, the set of possible computations can be described using a context free ....

....cm 1 m 2 . m l as input. We do not have to write down the complete FSA all at once. Given the base method definitions, we can compute parts of the transition function of the FSA as we need it. Thus, inconsistency is in NLOGSPACE. We now use the fact that NLOGSPACE is closed under complement [19, 29], and conclude that consistency is in NLOGSPACE. Finally, we show that consistency of simple, monadic schemas is hard using a reduction of the reachability problem for graphs of out degree 2 which is known to be logspace complete in NLOGSPACE [20] Let G = V; E) be a graph where each vertex has ....

R. Szelepcs'enyi. The Method of Forcing for Nondeterministic Automata. Bull. EATCS 33, 96--100, 1987.

....t i . This reduction is due to Avi Wigderson [Wig94] and it exploits the isolation lemma of Mulmuley, Vazirani and Vazirani [MVV87] In contrast, the proof given below eliminates the need for the unique witness property; instead, it uses the famous Immerman Szelepcs enyi result that NL = co NL [Imm88, Sze87]. First we note that the s t connectivity problem for directed acyclic graphs (DAGSTCON) is complete for NL under logspace (or even NC 1 ) computable many one reduc Research supported in part by NSF grants CCR 9057486 and CCR 9319093, and by an Alfred P. Sloan Fellowship. y Research ....

R. Szelepcs'enyi. The method of forcing for nondeterministic automata. Bull. of the EATCS, 33:96--100, 1987.

....machines determine a larger class than the deterministic ones. For the remaining class, the LBA s, the problem is still unsolved, as for the polynomially time bounded machines (but in this case we have recently learned that at least the nondeterministic class is closed under complementation [11] [17]. The present note arose out of a discussion between two of the authors provoked by the self evident observation uttered by one of us that in the world of unbounded computation nondeterministic devices are more powerful than deterministic ones as exemplified by the inequality REC 6= RE . ....

Szelepcs'enyi, R., The method of forcing for nondeterministic automata, Bull. EATCS 33 (1987) 96-100

....To the authors knowledge, all known deterministic or probabilistic algorithms for directed stcon are implementable on a JAG. However, it is not clear how a nondeterministic JAG can simulate Immerman s and Szelepcs enyi s O(log n) space algorithm for directed st nonconnectivity (stcon) Imm88, Sze88] This motivated Poon [Poo93] to introduce the more general Node Named JAG (NNJAG) model, an extension of the JAG where the computation is allowed to depend on the names of the nodes on which the pebbles are located. Using this added power, Poon [Poo93] showed how to simulate the Immerman ....

R'obert Szelepcs'enyi. The method of forcing for nondeterministic automata. Acta Informatica, 26:279--284, 1988.

....of languages defined by machines that accept their inputs if there is exactly one accepting path. Thus, the existence of two or more accepting computations is not forbidden, but simply leads to rejection. In the polynomial time case we have Co NP 1NP [2] In the logspace case inductive counting [11,22] shows 1NSPACE(logn) NSPACE(logn) A more restrictive form of unambiguity is Strong Unambiguity. A nondeterministic machine is said to be strongly unambiguous, if for every pair of configurations there exits at most one computational path connecting these configurations. An ordinary unambiguous ....

R. Szelepcs'enyi. The method of forcing for nondeterministic automata. Acta Informatica, 26:279--284, 1988.

....disjunctive, bounded truth table and truth table completeness differ on nondeterministic space classes bigger than logspace. The methods used are similar to those of Watanabe [15] but the key new idea is an application of the recent theorem of Immerman [8] and (independently) of Szelepsc enyi [14] showing closure of these classes under complement. 2 Preliminaries 2.1 Machines and languages Let Sigma = f0; 1g. Strings are elements of Sigma , and are denoted by small letters x; y; u; v; For any string x the length of a string is denoted by jxj. Languages are subsets of ....

....xi 2 LA , the simulation of M A jhi; xij i uses S(jhi; xij) tape cells and hi; xi 62 L(M i ; A jhi; xij ) CLAIM 7.1 LA 2 NSPACE (S(n) Proof: Since A 2 NSPACE (S(n) there exists a nondeterministic S(n) space bounded Turing machine MA , which accepts A. Immerman [8] and Szelepcs eny [14] showed independently that nondeterministic space is closed under complementation. Therefore the complement of A (A) is also in NSPACE (S(n) and is recognized by a nondeterministic S(n) space bounded Turing machine M A . We are first going to construct a machine that recognizes LA . Note that ....

Szelepcs'enyi, R. The method of forcing for nondeterministic automata. Bulletin of the EATCS 33 (1987) pp. 96--100.

....containments still go through, because the simulations used to prove those containments are monotonicity preserving. Differences in the structure of the two schemes highlight simulations which do not preserve monotonicity. For example the inductive counting technique used to prove that NL = co NL [6, 12] cannot be replaced with a monotone simulation since mNL 6= co mNL (below) We see the further elucidation of the monotone classification scheme as an important step in the development of circuit complexity. In this paper we show that monotone logarithmic depth circuits are strictly weaker than ....

R. Szelepcs' enyi, The method of forcing for nondeterministic automata, Bull. Europ. Ass. Theoretical Computer Sci. 33 (1987), 96--100.

....Thus, we analyze in detail the consistency problem for monadic and or recursion free method schemas, which happen to be decidable. We also quantify the effect of covariance, which is a widely used constraint on the signature of methods. For the various concepts used from complexity theory, see [15, 19, 20, 30] and from database theory, see [21, 31] We briefly summarize our other results. Let n be the size of method definitions in the input method schema and c the size of the class hierarchy. In the case of monadic schemas, the set of possible computations can be described using a context free ....

....cm 1 m 2 : m l as input. We do not have to write down the complete FSA all at once. Given the base method definitions, we can compute parts of the transition function of the FSA as we need it. Thus, inconsistency is in NLOGSPACE. We now use the fact that NLOGSPACE is closed under complement [19, 30], and conclude that consistency is in NLOGSPACE. Finally, we show that consistency of simple, monadic schemas is hard using a reduction of the reachability problem for graphs of out degree 2 which is known to be logspace complete in NLOGSPACE [20] Let G = V; E) be a graph where each vertex has ....

R. Szelepcs'enyi. The Method of Forcing for Nondeterministic Automata. Bull. EATCS 33, 96--100, 1987.

....for stcon. All of these algorithms can be implemented on the standard JAG [8, 14] Using the NNJAG s ability to access the names of the nodes in the graph, Poon [13] shows how to implement Immerman s and Szelepcs enyi s nondeterministic O(logn) space algorithm for directed s t nonconnectivity [11, 16] on a nondeterministic NNJAG. It is not clear that this algorithm can be implemented on a standard nondeterministic JAG. Our main results are to prove lower bounds of ST = Omega Gamma n 2 = log n) and S 1=2 T = Omega Gamma mn 1=2 ) for stcon on the JAG model, and of S 1=3 T = ....

R. Szelepcs'enyi. The method of forcing for nondeterministic automata. Acta Informatica, 26:279--284, 1988.

....enumerations of T 2 and T 3 in order to decide the question whether a given word w is contained in I G (G) then it will be enumerated by T 2 ) or not (then it will be enumerated by T 3 ) 2 As an application of this theorem, we can derive Theorem I.9. 1 in [Sal73] from the recently proved fact [Sze88b, Sze88a, Imm88] that the family of context sensitive grammars allows effective complementation. It is not possible to reason the other way round coming from the decidability of the word problem leading to effective complementation, as the family of context free grammars shows. Note that the proof resembles ....

R. Szelepcs'enyi. The method of forcing for nondeterministic automata. EATCS Bulletin, 33:96--100, 1988.

....problems are of equal complexity. For nondeterministic or one sided error probabilistic algorithms, however, the complexities may differ. In particular, if a problem L is solvable nondeterministically in O(log n) space, then the complement of L is, too, by the result of Immerman and Szelepcs enyi [24, 34]. However, their algorithms are rather slow. For example, a logarithmic space nondeterministic RAM can solve st connectivity in time O(n) but to solve the complementary st nonconnectivity problem by the Immerman or Szelepcs enyi algorithms requires time Omega0 n 4 ) Is nonconnectivity ....

....easy to see that a nondeterministic two pebble WAG can recognize nonbipartite graphs (guess and verify an odd cycle) but not so easy to see a direct way to recognize bipartite graphs. In fact this is also possible, by the following corollary to Theorem 7 and Immerman and Szelepcs enyi s Theorem [24, 34]. Corollary 10: Let H be an undirected graph problem, and let S(n) Omega0 log n) If H is solvable using space O(S(n) by a nondeterministic JAG or WAG J , then so is its complement G 0H. Proof: Simulate J by a nondeterministic, S(n) space bounded general machine M . By a straightforward ....

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R. Szelepcs'enyi. The method of forcing for nondeterministic automata. Acta Informatica, 26:279--284, 1988.

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R. Szelepcsenyi. The method of forcing for nondeterministic automata. Bull. European Assoc. Theoret. Comput. Sci., 33:96-100, 1987.

No context found.

R. Szelepcs'enyi, "The method of forcing for nondeterministic automata," Bull. Eur. Ass. Theoretical Comp. Sci 33 (Oct. 1987), 96-99.

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R'obert Szelepcs'enyi, "The Method of Forcing for Nondeterministic Automata," Bull. European Association Theor. Comp. Sci. (Oct. 1987), 96-100.

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