# On the Computational Power of Programs over \(\mathsf {BA}_2\) Monoid

01 Mar 2021-pp 29-40

Abstract: The PLP conjecture for monoids states that for every monoid M, either M is universal (that is, for every language \(L \subseteq \varSigma ^*\) there is a program over M which accepts the language L) or it has the polynomial length property (that is, every program over the monoid M has an equivalent program of length \({\mathsf {poly}}(n)\)). The conjecture has been confirmed (Tesson-Therien (2001)) for the case of groups and several subclasses of aperiodic monoids such as the variety DA and the monoids divided by the monoid U. However, the case of the set of monoids divided by the monoid \(\mathsf {BA}_2\) is still open, which if resolved, confirms the conjecture for all aperiodic monoids.

Topics: Monoid (72%)

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Abstract: The model of programs over (finite) monoids, introduced by Barrington and Th{e}rien, gives an interesting way to characterise the circuit complexity class $\mathsf{NC^1}$ and its subclasses and showcases deep connections with algebraic automata theory. In this article, we investigate the computational power of programs over monoids in $\mathbf{J}$, a small variety of finite aperiodic monoids. First, we give a fine hierarchy within the class of languages recognised by programs over monoids from $\mathbf{J}$, based on the length of programs but also some parametrisation of $\mathbf{J}$. Second, and most importantly, we make progress in understanding what regular languages can be recognised by programs over monoids in $\mathbf{J}$. We show that those programs actually can recognise all languages from a class of restricted dot-depth one languages, using a non-trivial trick, and conjecture that this class suffices to characterise the regular languages recognised by programs over monoids in $\mathbf{J}$.

2 citations

##### References

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01 Nov 1986

TL;DR: The method of proof is extended to investigate the complexity of the word problem for a fixed permutation group and show that polynomial size circuits of width 4 also recognize exactly nonuniform NC 1.

Abstract: We show that any language recognized by an NC 1 circuit (fan-in 2, depth O (log n )) can be recognized by a width-5 polynomial-size branching program. As any bounded-width polynomial-size branching program can be simulated by an NC 1 circuit, we have that the class of languages recognized by such programs is exactly nonuniform NC 1 . Further, following Ruzzo ( J. Comput. System Sci. 22 (1981), 365–383) and Cook ( Inform. and Control 64 (1985) 2–22) , if the branching programs are restricted to be ATIME(logn)-uniform, they recognize the same languages as do ATIME(log n )-uniform NC 1 circuits, that is, those languages in ATIME(log n ). We also extend the method of proof to investigate the complexity of the word problem for a fixed permutation group and show that polynomial size circuits of width 4 also recognize exactly nonuniform NC 1 .

853 citations

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Abstract: Introduction. In the following all semigroups are of finite order. One semigroup S, is said to divide another semigroup S2, written SlIS2, if S, is a homomorphic image of a subsemigroup of S2. The semidirect product of S2 by Sl, with connecting homomorphism Y, is written S2 X y Sl. See Definition 1.6. A semigroup S is called irreducible if for all finite semigroups S2 and Si and all connecting homomorphisms Y, S I (S2 X Y SJ) implies S I S2 or S I S1. It is shown that S is irreducible if and only if either:

321 citations

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TL;DR: Using Thérien's classification of finite monoids, new characterizations are given of the classes of automata and a new proof that the dot-depth hierarchy of algebraic automata theory is infinite is given.

Abstract: Recently a new connection was discovered between the parallel complexity class NC1 and the theory of finite automata in the work of Barrington on bounded width branching programs. There (nonuniform) NC1 was characterized as those languages recognized by a certain nonuniform version of a DFA. Here we extend this characterization to show that the internal structures of NC1 and the class of automata are closely related.In particular, using Therien's classification of finite monoids, we give new characterizations of the classes AC0, depth-kAC0, and ACC, the last being the AC0 closure of the mod q functions for all constant q. We settle some of the open questions in [3], give a new proof that the dot-depth hierarchy of algebraic automata theory is infinite [8], and offer a new framework for understanding the internal structure of NC1.

160 citations

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TL;DR: The power of NUDFA's over nilpotent groups is characterized and some optimal lower bounds for NUD FA's over certain groups which are solvable but not nilpotsent are proved.

Abstract: A new model, non-uniform deterministic finite automata (NUDFA's) over general finite monoids, has recently been developed as a strong link between the theory of finite automata and low-level parallel complexity. Achievements of this model include the proof that width 5 branching programs recognize exactly the languages in non-uniform NC1, NUDFA characterizations of several important subclasses of NC1, and a new proof of the old result that the dot-dephth hierarchy is infinite, using M. Sipser's (1983, in “Proceedings, 15th ACM Symposium on the Theory of Computing,” Association for Computing Machinery, New York, pp. 61–69) work on constant depth circuits. Here we extend this theory to NUDFA's over solvable groups (NUDFA's over non-solvable groups have the maximum possible computing power). We characterize the power of NUDFA's over nilpotent groups and prove some optimal lower bounds for NUDFA's over certain groups which are solvable but not nilpotent. Most of these results appeared in preliminary form in ( D. A. Barrington and D. Therien, 1987 , in “Automata, Languages, and Programming: 14th International Colloquium,” Springer-Verlag, Berlin, pp. 163–173).

126 citations