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The Power of Programs over Monoids in J
TL;DR: In this article, the computational power of programs over monoids in finite aperiodic monoids was investigated, and a fine hierarchy of languages recognized by program over monoid was given, based on the length of programs and some parametrisation of the monoid.
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 $\mathbf{J}$.
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01 Jan 2020
TL;DR: This work surveys recent developments related to the Minimum Circuit Size Problem and proposes a strategy to address the problem.
Abstract: We survey recent developments related to the Minimum Circuit Size Problem.
15 citations
Journal Article•
TL;DR: In this article, the algebraic complexity of the semigroups required to capture a class should be a good indication of its expressive power, i.e., the complexity of a class of problems that can be or cannot be solved within certain resources.
Abstract: Programs over semigroups are a well-studied model of computation for boolean functions. It has been used successfully to characterize, in algebraic terms, classes of problems that can, or cannot, be solved within certain resources. The interest of the approach is that the algebraic complexity of the semigroups required to capture a class should be a good indication of its expressive (or computing) power. In this paper we derive algebraic characterizations for some small classes of boolean functions, all of which have depth-3 AC° circuits, namely k-term DNF, k-DNF, k-decision lists, decision trees of bounded rank, and DNF. The interest of such classes, and the reason for this investigation, is that they have been intensely studied in computational learning theory.
14 citations
References
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01 Jan 1987
TL;DR: It is proved that depth k circuits with gates NOT, OR and MODp where p is a prime require Exp(&Ogr;(n1/2k)) gates to calculate MODr functions for any r ≠ pm.
Abstract: We use algebraic methods to get lower bounds for complexity of different functions based on constant depth unbounded fan-in circuits with the given set of basic operations. In particular, we prove that depth k circuits with gates NOT, OR and MODp where p is a prime require Exp(O(n1/2k)) gates to calculate MODr functions for any r ≠ pm. This statement contains as special cases Yao's PARITY result [ Ya 85 ] and Razborov's new MAJORITY result [Ra 86] (MODm gate is an oracle which outputs zero, if the number of ones is divisible by m).
926 citations
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 .
886 citations
28 Oct 1981
TL;DR: A super-polynomial lower bound is given for the size of circuits of fixed depth computing the parity function and connections are given to the theory of programmable logic arrays and to the relativization of the polynomial-time hierarchy.
Abstract: A super-polynomial lower bound is given for the size of circuits of fixed depth computing the parity function. Introducing the notion of polynomial-size, constant-depth reduction, similar results are shown for the majority, multiplication, and transitive closure functions. Connections are given to the theory of programmable logic arrays and to the relativization of the polynomial-time hierarchy.
718 citations
01 Nov 1986
TL;DR: Improved lower bounds for the size of small depth circuits computing several functions are given and it is shown that there are functions computable in polynomial size and depth k but requires exponential size when the depth is restricted to k 1.
Abstract: We give improved lower bounds for the size of small depth circuits computing several functions. In particular we prove almost optimal lower bounds for the size of parity circuits. Further we show that there are functions computable in polynomial size and depth k but requires exponential size when the depth is restricted to k 1. Our Main Lemma which is of independent interest states that by using a random restriction we can convert an AND of small ORs to an OR of small ANDs and conversely. Warning: Essentially this paper has been published in Advances for Computing and is hence subject to copyright restrictions. It is for personal use only.
667 citations