F. Bedard, F. Lemieux, and P. McKenzie. Extensions to Barrington's M-program model. Theoretical Computer Science, 107:31--61, 1993.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....of S. Groupoid word problems relate to context free languages in the same way as monoid word problems relate to regular languages: Every such word problem is context free, and every contextfree language is a homomorphic pre image of a groupoid word problem (this result is credited to Valiant in [BLM93]) The following de nition is due to B edard, Lemieux, and McKenzie [BLM93] De nition 7. A groupoidal quanti er is a Lindstr om quanti er QL where L is a word problem of some nite groupoid. Usage of groupoidal quanti ers in our logical language is signalled by Q Grp , used in the same way ....

.... way as monoid word problems relate to regular languages: Every such word problem is context free, and every contextfree language is a homomorphic pre image of a groupoid word problem (this result is credited to Valiant in [BLM93] The following de nition is due to B edard, Lemieux, and McKenzie [BLM93]: De nition 7. A groupoidal quanti er is a Lindstr om quanti er QL where L is a word problem of some nite groupoid. Usage of groupoidal quanti ers in our logical language is signalled by Q Grp , used in the same way as described for QMon above. Second order Lindstr om quanti ers on strings ....

F. Bedard, F. Lemieux, and P. McKenzie. Extensions to Barrington's Mprogram model. Theoretical Computer Science, 107:31-61, 1993.

....of algebras, of languages, and of congruences were proved to be isomorphic. Binary algebras, also called groupoids, also nd applications in the study of word languages: indeed the nite groupoids recognize exactly the class of the context free languages (a result found independently in [10] and [3]) under the following convention. A language L A is recognized by groupoid G i there exist a monoid homomorphism : A and a subset F G, such that w 2 L if, and only if w can evaluate to an element in F . When the operation is nonassociative, the outcome of the evaluation of w ....

....[11] Nevertheless, links can be established between classes of groupoids and classes of languages, and research has been done in this direction. In a rst approach, ad hoc examples of groupoids whose recognition power coincides with several signi cant subclasses of CFL s were presented in [3, 14]. Another approach was initiated and partially explored in [8, 2] it consists in classifying groupoids in terms of the properties of their multiplication monoid (denoted M(G) for de nition see Subsection 2.3) and to look at the languages recognized by the groupoids in a given class. An obvious ....

F. B edard, F. Lemieux and P. McKenzie, Extensions to Barrington's M-program model, Theor. Comp. Sc., 107, 1993, pp. 31-61.

.... we showed that valence languages over the monoid of integers correspond to homomorphic images of vertex languages of edge grammars as introduced in [3] Taking abelian monoids as valence regulators, we found algebraic characterizations of the corresponding language families in the spirit of [1, 2]. 2 Preliminaries Throughout the paper we assume the reader to be familiar with the theory of context free languages, see e.g. 19] Let V = fa 1 ; ang, n 1, be an alphabet. The set of all words over V is denoted by V , the empty word by , and V = V n fg. For w 2 V , the ....

F. B'edard, F. L'emieux, and P. McKenzie. Extensions to Barrington's M-program model. Theoretical Computer Science, 107:31--61, 1993.

.... QUESTION : Can this sequence be evaluted to give an element of F , that is, is G(i 1 ; i 2 ; i n ) F non empty It is known that there exists a xed nite groupoid (G; and a xed subset F of G such that the evaluation problem for (G; F ) is LOG(CFL) complete under NC 1 reductions ([BLM93] Corollary 3.4) For the following considerations let (G; be such a groupoid, G = f1; mg, and let F be a corresponding set of target elements. Let : fa 1 ; a 2 ; am ; #; g, uG : a 1 am , u i : a 1 a i , and u i : a i 1 am for i = 1; m. ....

F. Bedard, F. Lemieux, and P. McKenzie. Extensions to Barrington's M-program model. Theoretical Computer Science, 107:31-61, 1993.

....if the evaluation tree is such that the right child of any inner node is a leaf. An evaluation tree is called linear is for any inner node, at most one child is not a leaf. A linear program (deterministic or not) is one where parenthesizations are restricted to be linear. Theorem 2. 4 a) [1] A language is in SAC 1 if and only if it is recognized by uniform programs over a constant order (polynomial order) groupoid. b) 8] A language is in NP if and only if it is recognized by uniform programs over exponential order groupoids. Without loss of generality, the programs can be linear ....

F. B'edard, F. Lemieux and P.McKenzie, Extensions to Barrington's M-program model, TCS 107 (1993), pp. 31-61.

.... we showed that valence languages over the monoid of integers correspond to homomorphic images of vertex languages of edge grammars as introduced in [3] Taking Abelian monoids as valence regulators, we found algebraic characterizations of the corresponding language families in the spirit of [1, 2]. 2 H. Fernau, R. Stiebe 2 Preliminaries Throughout the paper we assume the reader to be familiar with the theory of context free languages, see e.g. 21] Let V = fa 1 ; a n g, n 1, be an alphabet. The set of all words over V is denoted by V , the empty word by , and V = V nfg. ....

F. B'edard, F. L'emieux, and P. McKenzie. Extensions to Barrington's Mprogram model. Theoretical Computer Science, 107:31--61, 1993.

....to groupoids: Every word problem of a groupoid is context free, and conversely every contextfree language reduces via a homomorphism to such a word problem. Similar to the above we thus see that every AC 0 [F ] for groupoidal F coincides with a class AC 0 [C] for a class C CFL. Theorem 3. 3 [BLM93] AC 0 [CFL] SAC 1 . Subclasses of SAC 1 have been studied in [LMSV99] 4 Relating Polynomial Time to Constant Depth In Sect. 2.2.3 we already mentioned that, e.g. BLeaf P (AC 0 ) PH and BLeaf P (NC 1 ) PSPACE. Recalling Theorems 2.2 and 3.2, we see that the related ....

F. B'edard, F. Lemieux, and P. McKenzie. Extensions to Barrington's M-program model. Theoretical Computer Science, 107:31--61, 1993.

....matrices over the integers f Gamma1; 0; 1g, compute a specific entry in the matrix product. b) Given a sequence of constant dimension matrices over integers expressed in binary notation, compute a specific entry in the matrix product. 2 DLOGTIME uniformity of programs in the sense of [6, 7, 9] differs from our definition of program uniformity in that the former requires DLOGTIME recognition of a description language for programs akin to the direct connection language for circuits. However, the two definitions are equivalent in all interesting situations, because, for example, although ....

....differs from our definition of program uniformity in that the former requires DLOGTIME recognition of a description language for programs akin to the direct connection language for circuits. However, the two definitions are equivalent in all interesting situations, because, for example, although [6, 7, 9] do not formally require the ability to compute the length of a program in log time, any program of interest can be padded to length 2 l , where l is the length of the field of bits allocated to instruction numbers in the description language, and 2 l is computable in log time. Proof. To ....

F. B'edard, F. Lemieux and P. McKenzie, Extensions to Barrington's M-program model, Theoret. Comp. Sci. A 107:1:31-61, 1993.

.... we showed that valence languages over the monoid of integers correspond to homomorphic images of vertex languages of edge grammars as introduced in [3] Taking abelian monoids as valence regulators, we found algebraic characterizations of the corresponding language families in the spirit of [1, 2]. 2 Preliminaries Throughout the paper we assume the reader to be familiar with the theory of context free languages, see e.g. 19] Let V = fa 1 ; ang, n 1, be an alphabet. The set of all words over V is denoted by V , the empty word by , and V = V n fg. For w 2 V , the ....

F. B'edard, F. L'emieux, and P. McKenzie. Extensions to Barrington's M-program model. Theoretical Computer Science, 107:31--61, 1993.

....Similar models were introduced, e.g. by Lange (1986) to describe the logarithmic space alternation hierarchy and by Ibarra and Ravikumar (1988) to model nonuniform sublogarithmic space bounded computations. Our model is equivalent to programs over growing sequences of monoids as introduced by B edard et al. 1993). Because of the involved terminology we do not discuss this issue further. A nonuniform deterministic finite automaton (NUDFA) is a machine with a two way read only input tape, a control unit with k states, and a one way read only program tape. On the program tape, there are instructions of two ....

B' edard, F., F. Lemieux and P. McKenzie (1993). Extensions to Barrington 's M-Program Model. Theoretical Computer Science 107, 31--61.

....113, India. meena imsc.ernet.in regular languages [Pin86] Another example concerns the class LogCFL (the logspace closure of context free languages) which is an important subclass of P and has been studied intensively (both from the perspective of parallel computation and circuit complexity) [Coo85, Lan93, Ven91, BLM93]. Viewing languages as formal power series is an important unifying paradigm in formal language theory [SS78, Gin75, KS85, Sal90] It has led to an arithmetization of the theory and to the unification of disparate looking proofs in the area. Furthermore, the general approach has also yielded ....

F. Bedard, F. Lemieux, and P. McKenzie. Extensions to Barrington's M-program model. Theoretical Computer Science, 107:31--61, 1993.

No context found.

F. Bedard, F. Lemieux, and P. McKenzie. Extensions to Barrington's M-program model. Theoretical Computer Science, 107:31-61, 1993.

.... any w 2 A we have that w 2 L if and only if G( w) F 6= When G is associative, this de nition corresponds to the standard recognition by monoid (e.g. see [9] Many recent papers have investigated the applications of nite groupoids in complexity theory and in formal language theory (e.g. [3, 5, 1, 7, 2]) A fundamental result (implicit in [11] is that a language is context free if and only if it is recognized by a nite groupoid (see also [3] For all a in a groupoid G, we de ne two functions LG (a) G G and RG (a) G G such that xLG (a) ax and xRG (a) xa. Whenever there is no ....

.... see [9] Many recent papers have investigated the applications of nite groupoids in complexity theory and in formal language theory (e.g. 3, 5, 1, 7, 2] A fundamental result (implicit in [11] is that a language is context free if and only if it is recognized by a nite groupoid (see also [3]) For all a in a groupoid G, we de ne two functions LG (a) G G and RG (a) G G such that xLG (a) ax and xRG (a) xa. Whenever there is no ambiguity on which groupoid is used, we simply write L(a) and R(a) The multiplication monoid M(G) is the monoid generated by the set fL(a) R(a) ....

[Article contains additional citation context not shown here]

F. Bedard, F. Lemieux and P.McKenzie, Extensions to Barrington's Mprogram model, TCS 107 (1993), pp. 31-61.

.... Complexity of Computing over Quasigroups Herv e Caussinus y and Francois Lemieux z Abstract In [7] the notions of recognition by semigroups and by programs over semigroups were extended to groupoids. This led to a new characterization of the context free languages and the class SAC 1 . In this paper, we investigate the classes of languages obtained when the groupoids are restricted to be ....

....These examples illustrate how subclasses of semigroups can be related to subclasses of languages, giving new insight for understanding the inner structure of NC 1 . In order to capture larger classes of languages, we need to extend further these notions of recognition. A solution, introduced in [7], resides in the addition of nondeterminism by means of the non associative analogues of semigroups. A groupoid G is a set with a binary operation. In this paper, we consider only finite groupoids. The order of G is its number of elements. Given a word w 2 G , we denote by G(w) the subset of G ....

[Article contains additional citation context not shown here]

F. B'edard, F. Lemieux and P.McKenzie, Extensions to Barrington's M-program model, TCS 107 (1993), pp. 31-61.

....give new characterizations of complexity classes, they establish a close relationship between the algebraic theory of semigroups and complexity theory. At the heart of this research there was a well known result due to Kleene (e.g. see [56] relating finite semigroups to regular languages. In [12] (see also [48] we investigated the effect of replacing semigroups by their nonassociative analogues, called groupoids. We proved a generalization of Kleene s theorem giving a natural correspondence between finite groupoids and contextfree languages. We also showed that using programs over ....

....following many directions. We now detail the contributions made in this thesis. We define three variations of the recognition by programs over groupoids. In the first one we allow the programs to use a different groupoid for each input length. These programs over growing groupoids, introduced in [12], are no more powerful than standard programs whenever the growth of the 2 By projection we mean a mapping, where for any i 1 there exist j 1 such that the i th symbol in OE(w) is determined by the j th symbol of w. 12 groupoids does not exceed some polynomial (as a function of the input ....

[Article contains additional citation context not shown here]

F. B'edard, F. Lemieux and P.McKenzie, Extensions to Barrington's Mprogram model, TCS 107 (1993), pp. 31-61.

....a rich theory with many deep results and applications, and it remains an active field that continues to challenge researchers. This makes more striking the observation that no such theory exists for context free languages. Nevertheless, this topic has been the subjet of recent investigations (e.g. [18, 21, 10, 13, 19, 7, 8, 20]) that we briefly describe here. A groupoid G is a set with a binary operation that can be non associative. All groupoids considered in this paper are finite. Groupoids can be used as language recognizers as follows. For any w 2 G , denote with G(w) the set of all elements g 2 G such that w can ....

....w 2 A we have that w 2 L if and only if G(OE(w) F 6= When G is associative, this definition corresponds to the recognition by monoid defined above. Our interest in groupoids comes from the fact that a language is context free if and only if it is recognized by a finite groupoid (e.g. see [18, 10]) In the absence of a general theory of groupoids, a classification of the contextfree languages based on the algebraic properties of the groupoids that recognize them is still a major research project. This approach could also have implications in complexity theory since context free languages ....

F. B'edard, F. Lemieux and P.McKenzie, Extensions to Barrington's M-program model, TCS 107 (1993), pp. 31-61.

.... progress has been made towards proving this [1, 13, 26, 29] 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 rst 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 ....

....we can assume that the number M of symbols in f is a power of 2 (pad it with blanks if necessary) Let f (i) be the expresion obtained from f by replacing each symbol s, that is not a variable, with sB M i 1 1 , where B is the blank symbol. The rest of the idea is similar to that in [6]. We construct an expression for each gate in the circuit so that is the expression corresponding to the output gate. A gate g on level 1 whose inputs are x and y is represented by the expression g = f(x; y) A gate g on level i 1 whose inputs are g 1 and g 2 (which are on level i 1) is ....

F. Bedard, F. Lemieux and P. McKenzie, \Extension to Barrington's M-program Model" Theoretical Comp. Sci. 107 (1993) 31-61.

....using first order logic augmented with a monoidal quantifier for G. Loosely speaking, such a quantifier provides a constrained oracle call to the word problem for G (defined essentially as the problem of computing the product of a sequence of elements of G) B edard, Lemieux and McKenzie [4] later noted that there is a fixed finite groupoid whose word problem is complete for the class LOGCFL of languages reducible in logarithmic space to a context free language [6, 18] A groupoid G is a set with a binary operation on which no constraint such as associativity or commutativity is ....

F. B'edard, F. Lemieux, and P. McKenzie. Extensions to Barrington's M-program model. Theoretical Computer Science, 107:31--61, 1993.

....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 elucidating the power of bounded width branching programs [6] the case of groupoids was shown to capture LOGCFL [7], the case of Boolean and integer matrices relates to NL and to the determinant [3, 10, 16, 24, 36, 41] the case of F matrices for various fields F captures linear algebra and counting classes like PhiL [11, 15] and the case of sets of natural numbers with union and sum as operations captures ....

F. B'edard, F. Lemieux, and P. McKenzie. Extensions to Barrington's Mprogram model. Theoretical Computer Science, 107:31--61, 1993.

.... algebras, usually called groupoids, are exactly the recognizers needed for context free languages: this relationship has been well known in theory of tree automata (see [12] and can be traced back to the paper of Mezei and Wright [15] It has also been used in complexity theoretic work (e.g. see [26, 5]) In view of this connection, it is natural to try to characterize the languages recognized by various specific subclasses of finite groupoids. One such class, that has been extensively studied in the past Work supported by FCAR (Qu ebec) and CRSNG (Canada) y D epartement de math ematiques ....

....the groupoid G recognizes the language L iff there exist an A generated subgroupoid of G isomorphic to A ( ff and a subset F of this subgroupoid such that x 2 L iff there is some tree t such that [ t; x) ff is in F . One pleasant feature of this notion of language recognition is Lemma 2. 1 [5] L is recognizable by a finite groupoid iff L is context free. The finite group topology on A is the smallest topology such that every morphism from A onto a finite group is continuous. It is equivalent to say that the group languages form a basis for this topology. It was first introduced ....

F. B'edard, F. Lemieux and P.McKenzie, Extensions to Barrington's M-program model, TCS 107 (1993), pp. 31-61.

....using first order logic augmented with a monoidal quantifier for G. Loosely speaking, such a quantifier provides a constrained oracle call to the word problem for G (defined essentially as the problem of computing the product of a sequence of elements of G) B edard, Lemieux and McKenzie [BLM93] later noted that there is a fixed finite groupoid whose word problem is complete for the class LOGCFL of languages reducible in logarithmic space to a context free language [Coo71, Sud78] A groupoid G is a set with a binary operation satisfying no discernible property, and the word problem for ....

....be expanded into SAC 1 sub circuits, since groupoid word problems are context free languages. There results a logspace uniform SAC 1 circuit, proving membership in LOGCFL. LOGCFL QGFO bit is seen by appealing to the fixed G whose word problem is LOGCFL complete under DLOGTIME reducibility [BLM93]. Since DLOGTIME is shown expressible in FO bit by [BIS90] the inclusion follows. 4.2 Capturing LOGCFL without BIT Theorem 4.2. There is a fixed groupoid G such that LOGCFL QGFO. Proof. We first show how to express plus and times and their negations as FO (Q Grp ) formulas (i.e. formulas ....

[Article contains additional citation context not shown here]

F. B'edard, F. Lemieux, and P. McKenzie. Extensions to Barrington's M-program model. Theoretical Computer Science, 107:31--61, 1993.

....two cases when the operation is associative. In the non associative setting, the WordProblem has been taken to mean the following: given an unbracketed sequence of elements of A, does there exist a parenthesization that evaluates to some target element; this question has been extensively studied [17, 7, 5] and will not be pursued here. Our goal is to investigate how the algebraic properties of (A; Delta) such as associativity, commutativity, solvability and so on, affect the computational complexity of the problems Expression Evaluation and Circuit Value. Our work is the direct continuation of ....

F. B'edard, F. Lemieux and P.McKenzie, "Extensions to Barrington's M-program model", Theoretical Computer Science 107,199331--61

No context found.

F. Bedard, F. Lemieux, and P. McKenzie. Extensions to Barrington's M-program model. Theoretical Computer Science, 107:31--61, 1993.

No context found.

F. B'edard, F. Lemieux, and P. McKenzie. Extensions to Barrington's M-program model. Theoretical Computer Science, 107:31--61, 1993.

No context found.

F. B'edard, F. Lemieux and P. McKenzie, Extensions to Barrington's M-program model, Theoret. Comp. Sci. A 107:1:31-61, 1993.

No context found.

B'edard F., Lemieux F., McKenzie P., Extensions to Barrington's M-program model. Structure in Complexity Theory: Fifth Annual Conference (1990), 200-209.

No context found.

F. B'edard, F. Lemieux and P. McKenzie (1990), Extensions to Barrington 's M-program model, 5th IEEE Structure in Complexity Theory Symp., 200-210. Revised version to appear in Theoret. Comp. Sci.

Online articles have much greater impact   More about CiteSeer.IST   Add search form to your site   Submit documents   Feedback  

CiteSeer.IST - Copyright Penn State and NEC