S.R. Buss, "The Boolean Formula Value Problem is in ALOGTIME," Proc. 19th Ann. ACM Symp. on Theory of Comput., (1987), pp. 123--131.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....P ) that contains two distinguished terminal symbols ( and ) such that all productions of G are of the form A (s) where A 2 N and s 2 (N [ nf( g) A language that is generated by a parenthesis grammar is called a parenthesis language. Parenthesis languages where rst studied in [22] In [5] it was shown that every parenthesis language is in uNC . Lemma 3. Every recognizable tree language is FOM reducible to a parenthesis language. Furthermore the uniform membership problem for TDTAs is log space reducible to the uniform membership problem for parenthesis grammars. Proof. Let A ....

....T be a xed recognizable tree language. Then the membership problem for T is in uNC . Furthermore there exists a xed deterministic TDTA A such that the membership problem for T (A) is uNC complete under DLOGTIME reductions. Proof. The rst statement follows from Lemma 3 and the results of [5]. For the hardness part let L be a xed regular word language, whose membership problem is uNC complete under DLOGTIME reductions. By [2, Proposition 6.4] such a language exists. If we de ne arity(a) 1 for all a 2 and let # 62 be a constant then we can identify a word a 1 a 2 ....

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S. R. Buss. The Boolean formula value problem is in ALOGTIME. In Proceedings of the 19th Annual Symposium on Theory of Computing (STOC 87), pages 123{

....showing the hardness of the problem under AC many one reducibility. Hence tree isomorphism is NC complete. Trees thus provide the rst class of graphs for which the isomorphism problem captures a natural complexity class. Moreover, so far, the problem of evaluating a Boolean formula [13] and the problem of multiplying permutations on ve points [14] and some of their variations) were the only two NC problems known. Tree isomorphism is a third such problem. As noted by Buss, choosing a graph representation is critical when working at the level of NC . Buss uses Miller and ....

....encode its color. For many tree problems in NC and L, completeness results seem to depend on the representation used. For example, the reachability problem on forests, which is L complete in the pointer representation [21] can be solved in NC (and even TC ) in the string representation [13]. Analogously, the Boolean formula value problem, which is complete for NC in the string representation [8] becomes L complete when described using trees given in the pointer representation [22] In fact, changing from pointer to string representation is FL complete [22] Another important ....

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S. R. Buss, The boolean formula value problem is in ALOGTIME, in: Proc. 19th ACM Symposium on Theory of Computing, 1987, pp. 123{ 131.

....is Alogtime complete with regard to deterministic log time reductions. Proof. i) To decide whether A #(a) holds, we need to know the truth value of each atom appearing in #. Then, all what remains is to evaluate a boolean formula which can be done in Dtime and Atime (see [10]) The value of an 15 atom Rx can be calculated by simulating the corresponding automaton on those components of a which belong to the variables appearing in x. The nave algorithm to do so uses time #(a) d log d ) log d log #(a) For the time complexity bound we perform this ....

....0 V L(M ) Thus, V, E, s, t) DetReach i# A P0 V where A = B, P ) is the structure with the presentation ( 0 # , L(M ) A closer inspection reveals that the above transformation can be defined in first order logic. iii) Evaluation of boolean formulae is Alogtime complete (see [10]) For most questions we can restrict attention to relational vocabularies and replace functions by their graphs at the expense of introducing additional quantifiers. When studying quantifierfree formulae we will not want do to this and hence need to consider the case of quantifier free formulae ....

S. Buss, The boolean formula value problem is in Alogtime, in Proc. 19 ACM Symp. on the Theory of Computing, 1987, pp. 123--131.

.... Some small progress has been made towards proving this [1, 8, 21, 23] parity is in [2] but not AC , ACC [p] and ACC [q] are incomparable if p and q are distinct primes, and majority is in TC but not ACC [2] Thus the first two inclusions in this series are proper, but ACC [6] and P (or even NP) could be identical for all anyone has been able to prove. A reduction from a problem A to a problem B is a mapping of instances of A to instances of B. If the mapping is computationally easy compared to B, then any fast algorithm for B becomes a fast algorithm for A; thus B is ....

....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 ....

S.R. Buss, "The Boolean formula value problem is in ALOGTIME." In Proc. 18th ACM Symp. on the Theory of Computing (1987) 123--131.

....for Boolean Formul Evaluation nd for Tree Contraction 115 [7] N. A. LYNCH, Log space recognition and translation of parenthesis languages J. Assoc. Cornput. Mach. 24 (1977) pp. 583 590. 8] G. L. MILLER AND J. H. REIF, Parallel tree contraction and its appli cation, in Proceedings of the 26th IEEE Symposium on Foundations of Computer Science, IEEE Computer Society, 1985, pp. 478 489. Department of Mathematics University of California, San Diego La ....

....recognition and translation of parenthesis languages J. Assoc. Cornput. Mach. 24 (1977) pp. 583 590. 8] G. L. MILLER AND J. H. REIF, Parallel tree contraction and its appli cation, in Proceedings of the 26th IEEE Symposium on Foundations of Computer Science, IEEE Computer Society, 1985, pp. 478 489. Department of Mathematics University of California, San Diego La Jolla, California 92093 emaih sbuss ucsd.edu 114 the class P of polynomial time predicates from ALOGTIME; since, after all, both loglog n space and deterministic log time are proper subclasses of P. However, it is not ....

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S. R. Buss, The Boolean formula value problem is in ALOGTIME, in Proceedings of the 19-th Annual ACM Symposium on Theory of Computing, May 1987, pp. 123 131.

....technique from [Lyn77] Recall that a parenthesis grammar is a context free grammar with two distinguished terminals: and ) such that each production is of the from A (x) with x parenthesis free. Such a grammar generates a parenthesis language. In our proof we make use of the the fact from [Bus87] that all parenthesis languages are recognizable in ALOGTIME. B) problem is in ALOGTIME for every database B every k and every n. Proof sketch: Note, that we consider here complexity for which it is not known, whether it contains PTIME. Thus we can not use Lemma 6 here. In the proof ....

S.R. Buss. The boolean formula value problem is in ALOGTIME. In Proceedings of the 19th Annual ACM Symposium on Theory of Computing (New York City, May 25--27,

....over in an interesting way to the low level setting. Cook [Coo85] has defined a notion of uniform NC 1 reducibility as a useful notion in studying completeness for complexity classes such as NL; L, etc. This is the analogue of Turing reducibility in low level complexity. Using Buss theorem [Bus87, BCG 92] that the boolean formula value problem is in uniform NC 1 , it can be shown that a language has a (nonuniform) circuit family of polynomial size and logarithmic depth, that is, the language is in nonuniform NC 1 , if and only if it is reducible to a sparse set under uniform NC ....

S. Buss. The boolean formula value problem is in ALOGTIME. In Proc. 19th Annual ACM Symposium on the Theory of Computing, pages 123--131, 1987.

....the hierarchy of complexity classes. Among their results we list in Table 1 those which are most significant to our work. This is one more way of characterizing complexity classes in algebraic terms, which comes after the problem of evaluating formulas and circuits over the Boolean semiring (see [11, 7, 8]) and the computational models of programs over monoids (see [3, 4] and leaf languages (see [6] among others. Using tensor calculus in this context is especially appealing, if only because of the many applications matrix algebra finds in various areas, such as the specification of parallel ....

S. R. Buss. The Boolean formula value problem is in ALOGTIME. In Proceedings 19th Symposium on Theory of Computing, pages 123--131. ACM Press, 1987.

....is to construct a universal formula evaluator algorithm. Such an algorithm takes as input a description of a formula, with all of its inputs speci ed, and produces as output the value of the formula. Parallel algorithms for this problem have been proposed by Gupta [6] Miller and Reif [9] Buss [2]; Buss, Cook, Gupta, and Ramachandran [3] and Kosaraju and Delcher [8] These yield NC algorithms for the problem that also produce, for any given formula of size S, a circuit of depth O(log S) When these circuits are expressed as formulas, the sizes are S ) for various 2. In the case ....

S. R. Buss, The Boolean formula value problem is in ALOGTIME, Proc. 19th Ann. ACM Symp. on Theory of Comput., (1987), pp. 123-131.

....class LOG(DCFL) As a lower bound we will establish L hardness for these problems. In order to prove this result we need less powerful reductions. An NC 1 many one reduction from a language A to a language B is given through a function : of polynomial growth [Bus87] such that, for each w 2 , w 2 A i (w) 2 B holds, and the predicate P (c; i; w) c is the i th symbol of (w) is in NC 1 . Analogously, AC 0 many one reductions are de ned. For establishing the lower bound we proceed as follows. In [MNO88] DCFL CRL is proved by showing how to ....

....general membership problems for the class con sp McNL are NC 1 hard with respect to AC 0 many one reductions, and they belong to LOG(DCFL) Proof. The containment in LOG(DCFL) is obvious due to the inclusion con sp McNL con mon McNL. The lower bound is obtained as follows. According to [Bus87] the Boolean formula evaluation problem is NC 1 complete under AC 0 many one reductions. This problem is de ned as follows: INSTANCE : A completely bracketed Boolean formula 2 f0; 1; g . TASK : Determine the truth value of . Let : f0; 1; g, f#; Y ....

S. Buss. The Boolean formula value problem is in ALOGTIME. In Proc. of 19th STOC, pages 123-131. ACM Press, New York, 1987.

.... of SL under log space bounded Turing reductions, i.e. SL = L SL [19] and the fact that problems in SL can be solved in deterministic space O(log(n) 4 3 ) 3] A collection of SL complete problems can be found in [2] For the de nition of DLOGTIME uniformity and DLOGTIME reductions see e.g. [10, 5]. DLOGTIME uniform NC 1 , brie y uNC 1 , is the class of all languages that can be recognized by a DLOGTIME uniform family of polynomial size, logarithmic depth, fan in two Boolean circuits. It is well known that uNC 1 corresponds to the class ALOGTIME [24] An important subclass of uNC 1 ....

S. R. Buss. The Boolean formula value problem is in ALOGTIME. In Proceedings of the 19th Annual Symposium on Theory of Computing (STOC 87), pages 123{

....circuits. They prefer to have a machine model that they can program in. Thus it is very desirable that uniform NC 1 correspond to logarithmic time on an alternating Turing machine [29] and uniform AC 0 correspond to logarithmic time on an alternating Turing machine making O(1) alternations [10]. Similarly, uniform TC 0 corresponds to logarithmic time and O(1) alternations on a threshold machine [26, 1] Further support for this uniformity condition comes from a series of striking connections to finite model theory. A language is in uniform AC 0 if and only if it can be viewed as ....

S. Buss, The Boolean formula value problem is in ALOGTIME, in Proc. 19th ACM Symposium on Theory of Computing (STOC), pp. 123--131 (1987).

....extended Frege proofs are the maximal proof system among those whose correctness can be proved in PV . It should be possible, although tedious, to establish a similarly close connection between ALV from [10] and Frege proofs, using the fact that evaluation of boolean formulas can be done in NC 1 [6] (cf. also [16] for an effort in this direction) To establish such a tight connection between TV , A2V and AV and their corresponding proof systems, we have to overcome the obstacle that evaluation of boolean formulas is complete for NC 1 , hence it is not possible in AC 0 and ACC(2) and in ....

Samuel R. Buss, The Boolean formula value problem is in ALOGT IME, Proceedings of the 19th Sympos. Theory of Computing, ACM, 1987, pp. 123--131.

.... progress has been made towards proving this [1, 8, 20, 22] parity is in ACC 0 [2] but not AC 0 , ACC 0 [p] and ACC 0 [q] are incomparable if p and q are distinct primes, and majority is in TC 0 but not ACC 0 [2] Thus the first two inclusions in this series are proper, but ACC 0 [6] and P (or even NP) could be identical for all anyone has been able to prove. A reduction from a problem A to a problem B is a mapping of instances of A to instances of B. If the mapping is computationally easy compared to B, then any fast algorithm for B becomes a fast algorithm for A; thus B is ....

....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 AC 0 and NC 1 reductions [6, 9]. 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 ....

S.R. Buss, "The Boolean formula value problem is in ALOGTIME." In Proc. 18th ACM Symp. on the Theory of Computing (1987) 123--131.

....of order 10 over which polynomial length programs characterize NC 1 . This in contrast with the conjecture [BaTh88] that polynomial length M programs require a monoid M of order 60 in order to capture NC 1 . The above results remain valid when the very strict DLOGTIME uniformity criterion [Bu87, BaImSt88] is imposed on the relevant G programs. As corollaries, we exhibit specific groupoids whose word problems (loosely defined as languages of strings of groupoid elements which can be bracketed in such a way as to evaluate to a prescribed element) are complete for LOGCFL, for NL and for NC 1 ....

....for NC 1 respectively, under DLOGTIME reductions. Our groupoid in the case of NC 1 has order 10, again in contrast with Barrington s NC 1 complete word problem over the smallest non solvable group [Ba86] this traces the somewhat obscure difference between the NC 1 complete Formula Value [Bu87, BuCoGuRa89] and Width 5 Graph Accessibility [Ba86, BaImSt88] problems to the structural difference between groupoids and monoids. Our second extension, from monoid M to monoid family fM n g, yields much more unusual complexity classes. In the present paper we restrict our attention to the case of ....

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S.R. Buss, The boolean formula value problem is in ALOGTIME, Proc. of the 19th ACM Symp. on the Theory of Computing (1987), pp. 123-131.

....1 under AC 0 reductions. Proof. The problem of evaluating an expression over any fixed finite groupoid G is in NC 1 . This is because the set of expressions over any finite groupoid, that evaluate to some given element, forms a parenthesis language, and therefore, belongs to NC 1 (see [10]) It remains to show that any function in NC 1 can be reduced to this problem when G contains a nonsolvable loop. If L is a nonsolvable loop of G, then there exists a morphism : L S, where S is a nonabelian simple loop. By Corollary 7.5, any function f : f0; 1g f0; 1g in NC 1 is ....

S.R. Buss, The Boolean Formula Value Problem is in ALOGTIME, Proc. of the 19th ACM Symp. on the Theory of Computing (1987), pp. 123-131.

....AND OR gate of unbounded fan in can be expanded into a NC 1 circuit, we have AC 0 NC 1 . We give for NC 1 two other characterizations. First, it corresponds to those languages recognized by a family of polynomial size Boolean formulae a formula is a circuit that is also a tree (see [70, 17]) Moreover, NC 1 has been proved to be equal to the class of languages recognized by a family of polynomial size constant width branching programs [5] Observe that nondeterministic and deterministic constant width branching programs have the same power. We introduce two other classes between ....

....program over a finite groupoid. 50 Proof. To prove that any language recognized by a uniform family of parenthesized programs is in NC 1 , it suffices to show that any well parenthesized expression over a finite groupoid can be evaluated in NC 1 . This follows from Buss result [17] that any parenthesis context free language belongs to NC 1 . Recall that a parenthesis context free language is a language generated by a grammar whose productions have the form A (ff) where A is a variable and ff contains no parenthesis. Given a groupoid G, we can define a grammar DG whose ....

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S.R. Buss, The boolean formula value problem is in ALOGTIME, Proc. of the 19th ACM Symp. on the Theory of Computing (1987), pp. 123-131.

....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 ....

S.R. Buss, \The Boolean formula value problem is in ALOGTIME." In Proc. 18th ACM Symp. on the Theory of Computing (1987) 123-131.

....log time reductions. Proof. i) To decide whether A j= j( a) holds, we need to know the truth value of each atom appearing in j. Then, all what remains is to evaluate a boolean formula which can be done in DTIME O jjj and ATIME O log jjj DSPACE O log jjj (see [5]) The value of an atom R x can be calculated by simulating the corresponding automaton on those components of a which belong to the variables appearing in x. The nave algorithm to do so uses time O l d ( a) jdj log jdj) and space O log jdj logl d ( a) For the time complexity ....

....0 jV j 2 L(M) Thus, V;E;s; t) 2 DETREACH iff A j= P0 jV j where A = B;P) is the structure with the presentation (f0g ; L(M) A closer inspection reveals that the above transformation can be defined in first order logic. iii) Evaluation of boolean formulae is ALOGTIMEcomplete (see [5]) For most questions we can restrict attention to relational vocabularies and replace functions by their graphs at the expense of introducing additional quantifiers. When studying quantifier free formulae we will not want do to this and hence need to consider the case of quantifier free formulae ....

S. Buss. The boolean formula value problem is in ALOG- TIME. In Proc. 19th ACM Symp. on the Theory of Computing, pages 123--131, 1987.

....circuit C m;b k 1 defining the corresponding function f m;b k 1 except that in the kth phase, M 2 reads directly a consecutive block of b input bits that are the inputs to the corresponding bottom level gate. Note that the b consecutive input bits can be read in deterministic O(b log n) time [4], and that b is logarithmic in the input length n of the function F k 1 . This proves that the language S 2 is in the class Pi U k . Suppose that S 2 is also in the class Pi S k . By Theorem 6.1(2) the language S 2 is accepted by a Pi poly;c k family of circuits for some constant c. That ....

S. R. Buss, The Boolean formula value problem is in ALOGTIME, Proc. 19th Annual ACM Symposium on Theory of Computing, (1987), pp. 123-131.

....emptiness of intersection problems with nondeterministic sublinear time classes. To work with sublinear time makes it necessary to augment the usual model of a Turing machine with an Index Tape, which enables the machine to access its input tape more quickly. We refer the reader to [16] and [4]. 4 We already pointed out that ; 6= T g Tally DFA 2 DSPACE(g(n) log n) holds (see last section) and therefore also ; 6= T g Tally DFA 2 ATIME( g(n) 2 log 2 n) As an alternative to Theorem 7 we show an improvement of this bound 5 : Theorem 8 ; 6= g Tally DFA log NTIME( g(n) ....

S. R. Buss. The boolean formula value problem is in ALOGTIME. In Proc. 19th Ann. ACM Symp. on Theory of Computing, pages 123--131, 1987.

....class once BIT is admitted. This class FO BIT is a subclass of the very limited combinatorial complexity class AC 0 , which we will define below in terms of boolean circuits. Researchers in circuit complexity were at the same time searching for a natural uniform subclass of AC 0 (e.g. [Bu87]) and Immerman proposed FO BIT as a candidate. 4 Barrington, Immerman and Straubing [BIS88] then showed that a wide variety of possible definitions of uniform AC 0 , including FO BIT , coincide. They extended the first order framework to describe a variety of other circuit complexity ....

....we define a new class, which has been given the name TC 0 [CSV84, PS88] This is a subclass of NC 1 which contains most of the languages known to be in NC 1 , with two principal exceptions. These are the word problem for a non solvable group [Ba89] and the boolean sentence value problem [Bu87], each of which is known to be outside of TC 0 unless TC 0 = NC 1 . NC 1 itself can also be characterized as a constant depth polynomial size unbounded fan in circuit class, where the gates perform multiplication in a fixed non solvable group [Ba89] It is a major open question whether ....

S. R. Buss, "The Boolean formula value problem is in ALOGTIME," 19th ACM Symp. on Theory of Computing (1987), 123-131.

....T over S, and an accepting subset of S (given explicitly by enumeration or implicitly by a condition) decide whether T evaluates to an element in the accepting set. Many special cases of this problem have been studied. For example, intricate NC 1 upper bounds are known when S is the Boolean [13] or a more general semiring [14] the case of groups was crucial Partially supported by Deutsche Forschungsgemeinschaft grant Me 1077 14 1. Part of the work was done while the author was at Universitat Trier, Germany. y Supported by the Qu ebec FCAR and by the NSERC of Canada. to ....

S. R. Buss. The Boolean formula value problem is in ALOGTIME. In Proceedings of the 19th Symposium on Theory of Computing, pages 123-- 131. ACM Press, 1987.

....reductions. Proof. i) To decide whether A = #(a) holds, we need to know the truth value of each atom appearing in #. Then, all what remains is to evaluate a boolean formula which can be done in DTIME # O # # ## and ATIME # O # log # ## # DSPACE # O # log # ## (see [5]) The value of an atom Rx can be calculated by simulating the corresponding automaton on those components of a which belong to the variables appearing in x. The nave algorithm to do so uses time O # # d (a) d log d ) # and space O # log d log # d (a) # . For the time ....

.... V # L(M) Thus, V, E, s, t) # DETREACH i# A = P0 V where A = B, P ) is the structure with the presentation ( 0 # , L(M) A closer inspection reveals that the above transformation can be defined in first order logic. iii) Evaluation of boolean formulae is ALOGTIMEcomplete (see [5]) For most questions we can restrict attention to relational vocabularies and replace functions by their graphs at the expense of introducing additional quantifiers. When studying quantifier free formulae we will not want do to this and hence need to consider the case of quantifier free formulae ....

S. Buss. The boolean formula value problem is in ALOG- TIME. In Proc. 19th ACM Symp. on the Theory of Computing, pages 123--131, 1987.

....the size of the circuit C n . Define the set S A = f1 n 01 i 0 e(n) Gammai j bit i of E(C n ) is 1g. Clearly S A is sparse since each E(C n ) is a string of length n O(1) The language BFVP = fhE;xi j E is the encoding of a formula C and C(x) accepts g was shown to be in NC 1 by Buss [Bus87, BCG 92] The circuit that reduces from A to S A , on input x, proceeds by first making all the queries 1 n 01 i 0 e(n) Gammai ; 1 i e(n) in parallel to S A , and then solves the BFVP problem on the instance hE;xi, where E is the sequence of bits produced as answers from the oracle S ....

S. Buss. The boolean formula value problem is in ALOGTIME. In Proc. 19th Annual ACM Symposium on the Theory of Computing, pages 123--131, 1987.

....to their input; that is, the machines have an address tape, and if i is written in binary on the address tape, then the machine can in unit time move its read write head to bit position i of the input. Among other things, such machines can, in logarithmic time, compute the length of their input ([Bu]) In order to avoid uninteresting technicalities regarding encoding of pairing functions, we adopt the convention that a machine computing a k ary function is provided with k input tapes (with an address tape for each input tape) The running time of such a machine must be polylogarithmic in m, ....

S. R. Buss, The Boolean formula value problem is in ALOGTIME, Proc. 19th ACM Symposium on Theory of Computing, 1987, pp. 123--131.

....standard complete sets are easily seen to be isomorphic. Of course, logspace reductions are too powerful to investigate the many interesting subclasses of DLOG (such as ACC 0 , TC 0 , and NC 1 ) For these classes, AC 0 reducibility is the most natural notion of reducibility to use (see [Ba89, Bu87], for example) and once again the standard complete sets are all seen to be isomorphic [ABI93] 3 Q: So, I guess that, with so much written about the isomorphism conjectures, most complexity theoreticians believe that the conjectures are true A: Far from it. In fact, probably most ....

S. Buss, The Boolean Formula Value Problem is in ALOGTIME, in Proc. 19th ACM Symposium on Theory of Computing, pp. 123-131, 1987.

....of the representation. The situation is different if we consider the complexity of evaluating the truth of a propositional formula. This is known to complete for polynomial time if we take the dag representation [Lad75] and complete for alternating logarithmic time under the tree representation [Bus87]. If Phi is finite, it is easy to show that deciding almostsure structure validity for formulas in Phi is in polynomial time. Of course, the constants are exponential in j Phij. If Phi is infinite, then it is easy to show that the problem of deciding almost sure structure validity is in ....

S. R. Buss. The Boolean formula value problem is in ALOGTIME. In Proc. 19th ACM Symp. on Theory of Computing, pages 123--131, 1987.

....and to # Pi 2 [ That is, we have to deal with logtime Turing machines that are deterministic, nondeterministic, and universally branching. The computational power is very low in this case. For this reason, we have to extend the usual model of random access Turing machines as used for example in [Bus87] to be able to access relations of higher arity rather than to read strings. It is possible to obtain a characterization of #P vial universally branching logtime Turing machines using the logical characterization of NP as it is proven by [Fag74] or [Pap94] First, we describe an encoding e 1 of ....

S. R. Buss. The Boolean formula value problem is in ALOGTIME. In Proc. 19th. Ann. ACM Symp. on Theory of Computing, pages 123 -- 131, 1987.

....subset such as f ; g or f NAND g) of propositional connectives evaluates to 1. For circuit complexity this is the same as the more familiar problem: given a Boolean formula with variables x 1 ; x n and an assignment a 2 f 0; 1 g n , does the assignment satisfy the formula S. Buss [Bus87] was the first to show that this problem belongs to NC 1 , and that it is complete for NC 1 under both AC 0 and DLOGTIME reductions. R. Kosaraju and A. Delcher [KD90] give NC 1 circuits of size n Delta log O(1) n which use the AKS sorting network. Buss defined a PLOF sentence to be one ....

....occurring before shorter ones. Because we picture finite state machines as moving left to right, we reverse the sentence so that it is in prefix notation with shorter operands first, but still use the name PLOF. The problem remains complete for NC 1 under the restriction to PLOF sentences [Bus87, BCGR92]. Theorem 4.3 The BSVP restricted to PLOF sentences is n to n=2 contractible by an SSM that needs only odd even state information. There exist regular languages that are complete for NC 1 [Bar89] but the finite automata accepting them maintain the whole symmetric (or alternating) permutation ....

S. Buss. The boolean formula value problem is in ALOGTIME. In Proc. 19th Annual ACM Symposium on the Theory of Computing, pages 123--131, 1987.

....to their input; that is, the machines have an address tape, and if i is written in binary on the address tape, then the machine can in unit time move its read write head to bit position i of the input. Among other things, such machines can, in logarithmic time, compute the length of their input ([Bu]) In order to avoid uninteresting technicalities regarding encoding of pairing functions, we adopt the convention that a machine computing a k ary function is provided with k input tapes (with an address tape for each input tape) The running time of such a machine must be polylogarithmic in m, ....

S. R. Buss, The Boolean formula value problem is in ALOGTIME, Proc. 19th ACM Symposium on Theory of Computing, 1987, pp. 123--131.

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S. R. Buss, The Boolean formula value problem is in ALOGTIME, in Proceedings of the 19-th Annual ACM Symposium on Theory of Computing, May 1987, pp. 123--131.

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Samuel R. Buss. The Boolean formula value problem is in ALOGTIME. ACM Symposium on Theory of Computing, pages 123--131, 1987.

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Samuel R. Buss. The Boolean formula value problem is in ALOGTIME. In Proceedings of the 19-th Annual ACM Symposium on Theory of Computing, pages 123--131, May 1987.

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S. R. Buss, The Boolean formula value problem is in ALOGTIME, in Proceedings of the 19-th Annual ACM Symposium on Theory of Computing, May 1987, pp. 123--131. 15

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S. R. Buss, The Boolean formula value problem is in ALOGTIME, in Proceedings of the 19-th Annual ACM Symposium on Theory of Computing, May 1987, pp. 123--131.

....result, predating the above mentioned results for Peano arithmetic and set theory. This paper proves that Frege systems also have polynomial size proofs of partial self consistency. Our proof depends critically on the fact that the Boolean formula value problem is in alternating logarithmic time [1,3], or more precisely, on the fact that there are polynomial size propositional formulas which define the truth value of propositional formulas. In section 3 below, we reprove this fact using a simpler construction than was employed in the prior proofs in [1,3] 1 Our proof method also gives a new ....

....is in alternating logarithmic time [1,3] or more precisely, on the fact that there are polynomial size propositional formulas which define the truth value of propositional formulas. In section 3 below, we reprove this fact using a simpler construction than was employed in the prior proofs in [1,3]. 1 Our proof method also gives a new proof of Reckhow s theorem that any two Frege systems p simulate each other [10] We also show that Frege systems simulate extended Frege systems if and only if there are polynomial size Frege proofs of Co ey( We begin by reviewing Frege and extended Frege ....

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Samuel R. Buss. The Boolean formula value problem is in ALOGTIME. In Pr'oceedigs of the 19-th Aual ACM S/mposium o Theoq/ of omputig, pages 123 131, May 1987.

....question of separating Frege and extended Frege systems and the question of separating N( from P. Nanlely, the lines in a polynomial size Frege proof consist of polynomial size propositional formulas and it is known that polynomial size 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 ....

S. R. Buss, The Boolean formula value problem is in ALOGTIME, in Proceedings of the 19-th Annual ACM Symposium on Theory of Computing, May 1987, pp. 123 131.

....We will assume that the reader is familiar with these capabilities of Alogtime, and also is familiar with both the circuit characterization and the game characterization of Alogtime. For more information on these aspects of alternating logtime, the reader can consult the introductory portions of [3, 5, 4]. There are logspace algorithms for converting pointer representations of trees into string representations of tree, and vice versa, so for logspace and for more powerful complexity classes, the use of string representations is equivalent to the use of pointer representations. 4 Definition When ....

.... ut 6 Conclusions Our main results show that tree isomorphism, tree comparison and tree canonization are in alternating logarithmic time; improving on the logarithmic space algorithms of [8] There are number of other problems known to be in Alogtime, including the Boolean Formula Value Problem [3, 5, 4], and the word problem for S 5 [2] These problems are also known to be complete for Alogtime under deterministic log time reductions. It is still open whether the tree isomorphism, comparison and canonization problems are also complete for Alogtime. A problem with a similar name to the tree ....

S. R. Buss, The Boolean formula value problem is in ALOGTIME, in Proceedings of the 19-th Annual ACM Symposium on Theory of Computing, May 1987, pp. 123--131. 15

....of separating Frege and extended Frege systems and the question of separating NC 1 from P . Namely, the lines in a polynomial size Frege proof consist of polynomial size propositional formulas and it is known that polynomial size formulas can express precisely properties in (nonuniform) NC 1 [24,3,7,6]. 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 [18] Thus, one can intuitively view polynomial size Frege proofs as ....

S. R. Buss, The Boolean formula value problem is in ALOGTIME, in Proceedings of the 19-th Annual ACM Symposium on Theory of Computing, May 1987, pp. 123--131.

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S.R. Buss, "The Boolean Formula Value Problem is in ALOGTIME," Proc. 19th Ann. ACM Symp. on Theory of Comput., (1987), pp. 123--131.

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S. R. Buss. The boolean formula value problem is in ALOGTIME. In Proc. 19th Ann. ACM Symp. on Theory of Computing, pages 123--131, 1987.

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S. Buss, The boolean formula value problem is in ALOGTIME,inProceedings 19th ACM Symposium on the Theory of Computing, 1987, pp. 123--131.

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S.R. Buss. The boolean formula value problem is in ALOGTIME. In Symposium on the Theory of Computing, number 19, pages 123-131. ACM, 1987.

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S. Buss. The Boolean formula value problem is in ALOGTIME. Proceedings of the 19th Annual ACM Symposium on Theory of Computing (STOC'87), pages 123--131, 1987.

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S. Buss. The Boolean formula value problem is in ALOGTIME. Proceedings of the 19th Annual ACM Symposium on Theory of Computating (STOC'87), pages 123-131, 1987.

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S. R. Buss. The Boolean formula value problem is in ALOGTIME. In Proceedings of the Nineteenth Annual ACM Symposium on Theory of Computing, pages 123-131, New York, NY, May 1987.

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S.R. Buss, \The Boolean formula value problem is in ALOGTIME." Proc. 18th ACM Symp. on the Theory of Computing (1987) 123-131.

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S. R. Buss, "The Boolean formula value problem is in ALOGTIME," 19th ACM STOC Symp. (1987), 123-131.

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S. R. Buss. The Boolean formula value problem is in ALOGTIME. In 19th Annual ACM Symposium on Theory of Computing, 123--131, 1987.

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