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The Regular Languages of First-Order Logic with One Alternation
... Our proof is much more elementary, and uses only the most basic facts about finite automata. 1 We use 'string' and 'word' interchangeably. ...
... This paper concerns the use of formulas of first-order logic to define properties of strings over a finite alphabet A. 1 The description of how these formulas work will be rather informal in this introductory section, but we will be more precise later on. Variables in our formulas are interpreted as positions in a string: these should be thought of as positive integers, where the leftmost position in a string is 1. ...
... Recently, Barloy, et. al. [1], proved the central conjecture for Σ 2 , with an intricate proof drawing on circuit complexity, extremal combinatorics, and algebra. ...
We give a simple new proof that regular languages defined by first-order sentences with no quantifier alteration can be defined by such sentences in which only regular atomic formulas appear. Earlier proofs of this fact relied on arguments from circuit complexity or algebra. Our proof is much more elementary, and uses only the most basic facts about finite automata.
... The prefix contains only two blocks of alternating quantifiers, beginning with an existential quantifier: thus the language is in Σ 2 [<]. We note that this complexity measure is conjectured to be closely related to the minimal depth of an equivalent Boolean circuit and that depth is tied to the speed at which the circuit can be evaluated [32] -this conjecture is known to hold up to Σ 2 [<] [4]. It is thus of crucial importance to find what is the minimal number of alternations required to define a given language. ...
We show that for any , it is decidable, given a regular language, whether it is expressible in the fragment of first-order logic FO[<]. This settles a question open since 1971. Our main technical result relies on the notion of polynomial closure of a class of languages , that is, finite unions of languages of the form where each is a letter and each a language of . We show that if a class of regular languages with some closure properties (namely, a positive variety) has a decidable separation problem, then so does its polynomial closure Pol(). The resulting algorithm for Pol() has time complexity that is exponential in the time complexity for and we propose a natural conjecture that would lead to a polynomial time blowup instead. Corollaries include the decidability of half levels of the dot-depth hierarchy and the group-based concatenation hierarchy.
The program-over-monoid model of computation originates with Barrington's
proof that the model captures the complexity class . Here we
make progress in understanding the subtleties of the model. First, we identify
a new tameness condition on a class of monoids that entails a natural
characterization of the regular languages recognizable by programs over monoids
from the class. Second, we prove that the class known as
satisfies tameness and hence that the regular languages recognized by programs
over monoids in are precisely those recognizable in the classical
sense by morphisms from . Third, we show by contrast that the
well studied class of monoids called is not tame. Finally, we
exhibit a program-length-based hierarchy within the class of languages
recognized by programs over monoids from .
We investigate a famous decision problem in automata theory: separation.
Given a class of language C, the separation problem for C takes as input two
regular languages and asks whether there exists a third one which belongs to C,
includes the first one and is disjoint from the second. Typically, obtaining an
algorithm for separation yields a deep understanding of the investigated class
C. This explains why a lot of effort has been devoted to finding algorithms for
the most prominent classes.
Here, we are interested in classes within concatenation hierarchies. Such
hierarchies are built using a generic construction process: one starts from an
initial class called the basis and builds new levels by applying generic
operations. The most famous one, the dot-depth hierarchy of Brzozowski and
Cohen, classifies the languages definable in first-order logic. Moreover, it
was shown by Thomas that it corresponds to the quantifier alternation hierarchy
of first-order logic: each level in the dot-depth corresponds to the languages
that can be defined with a prescribed number of quantifier blocks. Finding
separation algorithms for all levels in this hierarchy is among the most famous
open problems in automata theory.
Our main theorem is generic: we show that separation is decidable for the
level 3/2 of any concatenation hierarchy whose basis is finite. Furthermore, in
the special case of the dot-depth, we push this result to the level 5/2. In
logical terms, this solves separation for : first-order sentences
having at most three quantifier blocks starting with an existential one.
We present a top-down lower bound method for depth-three 20.618...Ön (or 20.849...Ön 2^{0.618...\sqrt n } (or 2^{0.849...\sqrt n }
, respectively). This is the first simple proof of a strong lower bound by a top-down argument for non-monotone circuits.
We develop a new method to analyze the flow of communication in constant-depth circuits. This point of view allows usto prove new lower bounds on the number of wires required to recognize certain languages. We are able to provide explicit languages that can be recognized by AC0 circuits with O(n) gates but not with O(n) wires, and similarly for ACC0 circuits. We are also able to characterize exactly the regular languages that can be recognized with O(n) wires, both in AC0 and ACC0 framework.
We show a property of strings is expressible in the two-variable fragment of first-order logic if and only if it is expressible by both a Σ 2 and a Π 2 sentence. We thereby establish: UTL=FO 2 =Σ 2 ∩Π 2 =UL, where UTL stands for the string properties expressible in the temporal logic with ‘eventually in the future’ and ‘eventually in the past’ as the only temporal operators and UL stands for the class of unambiguous languages. This enables us to show that the problem of determining whether or not a given temporal string property belongs to UTL is decidable (in exponential space), which settles a hitherto open problem. Our proof of Σ 2 ∩Π 2 =FO 2 involves a new comtinatorial characterization of these two classes and introduces a new method of playing Ehrenfeucht-Fraïssé games to verify identities in semigroups.
We investigate the relationship between circuit bottom fan-in and circuit size when circuit depth is fixed. We show that in order to compute certain functions, a moderate reduction in circuit bottom fan-in will cause significant increase in circuit size. In particular, we prove that there are functions that are computable by circuits of linear size and depth k with bottom fan-in 2 but require exponential size for circuits of depth k with bottom fan-in 1. A general scheme is established to study the trade-off between circuit bottom fan-in and circuit size. Based on this scheme, we are able to prove, for example, that for any integer c, there are functions that are computable by circuits of linear size and depth k with bottom fan-in O(log n) but require exponential size for circuits of depth k with bottom fan-in c, and that for any constant ffl ? 0, there are functions that are computable by circuits of linear size and depth k with bottom fan-in log n but require superpolynomial size for...
We investigate quantifier alternation hierarchies in first-order logic on finite words. Levels in these hierarchies are defined by counting the number of quantifier alternations in formulas. We prove that one can decide membership of a regular language in the levels (finite boolean combinations of formulas having only one alternation) and (formulas having only two alternations and beginning with an existential block). Our proofs work by considering a deeper problem, called separation, which, once solved for lower levels, allows us to solve membership for higher levels.
We give a simple proof of a wreath product decomposition for locally ℛ-trivial finite categories. As immediate consequences we obtain two classic results of Stiffler on the decomposition of ℛ-trivial monoids and locally ℛ-trivial semigroups. A small modification of the argument provides a new proof of a recent theorem of Bojańczyk, Straubing and Walukiewicz on the decomposition of forest algebras.
We prove that a regular language defined by a Boolean combination of generalized Σ 1 -sentences built using modular counting quantifiers can be defined by a Boolean combination of Σ 1 -sentences in which only regular numerical predicates appear. The same statement, with “Σ 1 ” replaced by “first-order,” is equivalent to the conjecture that the nonuniform circuit complexity class ACC is strictly contained in NC 1 . The argument introduces some new techniques, based on a combination of semigroup theory and Ramsey theory, which may shed some light on the general case.
This paper is devoted to the languages accepted by constant-depth, polynomial-size families of circuits in which every circuit element computes the sum of its input bits modulo a fixed period q. It has been conjectured that such a circuit family cannot compute the AND function of n inputs. Here it is shown that such circuit families are equivalent in power to polynomial-length programs over finite solvable groups; in particular, the conjecture implies that Barrington’s result on the computational power of branching programs over nonsolvable groups cannot be extended to solvable groups. It is also shown that polynomial-length programs over dihedral groups cannot compute the AND function. Furthermore, it is shown that the conjecture is equivalent to a characterization, in terms of finite semigroups and formal logic, of the regular languages accepted by such circuit families. There is, moreover, considerable evidence for this characterization. This last result is established using new theorems, of independent interest, concerning the algebraic structure of finite categories.
Using purely syntactical arguments, it is shown that every nontrivial pseudovariety of monoids contained in DO whose corresponding variety of languages is closed under unambiguous product, for instance DA, is local in the sense of Tilson.
We survey the current state of knowledge on the circuit complexity of regular languages and we prove that regular languages that are in AC0 and ACC0 are all computable by almost linear size circuits, extending the result of Chandra et al. (J. Comput. Syst. Sci. 30:222–234, 1985). As a consequence we obtain that in order to separate ACC0 from NC1 it suffices to prove for some ε>0 an Ω(n
1+ε
) lower bound on the size of ACC0 circuits computing certain NC1-complete functions.
This article is a contribution to the algebraic theory of automata, but it also contains an application to Büchi’s sequential
calculus. The polynomial closure of a class of languagesC is the set of languages that are finite unions of languages of the formL
0
a
1
L
1 ···a
nLn, where thea
i’s are letters and theL
i’s are elements ofC. Our main result is an algebraic characterization, via the syntactic monoid, of the polynomial closure of a variety of languages.
We show that the algebraic operation corresponding to the polynomial closure is a certain Mal’cev product of varieties. This
result has several consequences. We first study the concatenation hierarchies similar to the dot-depth hierarchy, obtained
by counting the number of alternations between boolean operations and concatenation. For instance, we show that level 3/2
of the Straubing hierarchy is decidable and we give a simplified proof of the partial result of Cowan on level 2. We propose
a general conjecture for these hierarchies. We also show that if a language and its complement are in the polynomial closure
of a variety of languages, then this language can be written as a disjoint union of marked unambiguous products of languages
of the variety. This allows us to extend the results of Thomas on quantifier hierarchies of first-order logic.
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.
We present a new, combinatorial proof of the classical theorem that the analytic sets are not closed under complement. Possible connections with questions in complexity theory are discussed.
The study of categories as generalized monoids is shown to be essential to the understanding of monoid decomposition theory. A new ordering for categories, called division, is introduced which allows useful comparison of monoids and categories. Associated with every morphism ϕ: M → N is a category Dϕ, called the derived category of ϕ, which encodes the essential information about the morphism. The derived category is shown to be a rightful generalization of the kernel of a group morphism.Collections of categories that admit direct products and division are studied. Called varieties, these collections are shown to be completely determined by the path equations their members satisfy. Several important examples are discussed. In a major application, a strong connection is established between category varieties and certain semigroup varieties associated with recognizable languages.
A language L over an alphabet A is said to have a neutral letter if there is a letter e∈A such that inserting or deleting e's from any word in A* does not change its membership or non-membership in L.The presence of a neutral letter affects the definability of a language in first-order logic. It was conjectured that it renders all numerical predicates apart from the order predicate useless, i.e., that if a language L with a neutral letter is not definable in first-order logic with linear order, then it is not definable in first-order logic with any set N of numerical predicates. Named after the location of its first, flawed, proof this conjecture is called the Crane Beach conjecture (CBC, for short). The CBC is closely related to uniformity conditions in circuit complexity theory and to collapse results in database theory.We investigate the CBC in detail, showing that it fails for N={+,×}, or, possibly stronger, for any set N that allows counting up to the m times iterated logarithm, for any constant m. On the positive side, we prove the conjecture for the case of all monadic numerical predicates, for the addition predicate +, for the fragment BC(Σ1) of first-order logic, for regular languages, and for languages over a binary alphabet. We explain the precise relation between the CBC and so-called natural-generic collapse results in database theory. Furthermore, we introduce a framework that gives better understanding of what exactly may cause a failure of the conjecture.
We show that any language recognized by an NC1 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 NC1 circuit, we have that the class of languages recognized by such programs is exactly nonuniform NC1. Further, following Ruzzo (J. Comput. System Sci.22 (1981), 365–383) and Cook (Inform. and Control64 (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 NC1 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 NC1.
Let A be a finite alphabet and A∗ the free monoid generated by A. A language is any subset of A∗. Assume that all the languages of the form {a}, where a is either the empty word or a letter in A, are given. Close this basic family of languages under Boolean operations; let (0) be the resulting Boolean algebra of languages. Next, close (0) under concatenation and then close the resulting family under Boolean operations. Call this new Boolean algebra (1), etc. The sequence (0), (1),…, k,… of Boolean algebras is called the dot-depth hierarchy. The union of all these Boolean algebras is the family of star-free or aperiodic languages which is the same as the family of noncounting regular languages. Over an alphabet of one letter the hierarchy is finite; in fact, (2) = (1). We show in this paper that the hierarchy is infinite for any alphabet with two or more letters.
It is shown that for every k, polynomial-size, depth-k Boolean circuits are more powerful than polynomial-size, depth-(k−1) Boolean circuits. Connections with a problem about Borel sets and other questions are discussed.
Recently a new connection was discovered between the parallel complexity class NC ¹ and the theory of finite automata in the work of Barrington on bounded width branching programs. There (nonuniform) NC ¹ 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 NC ¹ and the class of automata are closely related.
In particular, using Thérien's classification of finite monoids, we give new characterizations of the classes AC ⁰ , depth- k AC ⁰ , and ACC , the last being the AC ⁰ 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 NC ¹ .
The notion of a p-variety arises in the algebraic approach to Boolean circuit complexity. It has great signi cance, since many known and conjectured lower bounds on circuits are equivalent to the assertion that certain classes of semigroups form p-varieties. In this paper, we prove that semigroups of dot-depth one form a pvariety. This example has the following implication: if a Boolean combination of 1 formulas, using arbitrary numerical predicates, de nes a regular language, one can then nd an equivalent 1 formula all of whose numerical predicates are regular.
We give several characterizations, in terms of formal logic, semigroup theory, and operations on languages, of the regular languages in the circuit complexity class AC0, thus answering a question of Chandra, Fortune, and Lipton. As a by-product, we are able to determine effectively whether a given regular language is in AC0 and to solve in part an open problem originally posed by McNaughton. Using recent lower-bound results of Razborov and Smolensky, we obtain similar characterizations of the family of regular languages recognized by constant-depth circuit families that include unbounded fan-in mod p addition gates for a fixed prime p along with unbounded fan-in boolean gates. We also obtain logical characterizations for the class of all languages recognized by nonuniform circuit families in which mod m gates (where m is not necessarily prime) are permitted. Comparison of this characterization with our previous results provides evidence for a conjecture concerning the regular languages in this class. A proof of this conjecture would show that computing the bit sum modulo p, where p is a prime not dividing m, is not AC0-reducible to addition mod m, and thus that MAJORITY is not AC0-reducible to addition mod m. Peer Reviewed http://deepblue.lib.umich.edu/bitstream/2027.42/30017/1/0000385.pdf
: 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 Gamma 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. 1. Introduction Proving lower bounds for the resources needed to compute certain functions is one of the most interesting branches of theoretical computer science. One of the ultimate goals of this branch is of course to show that P 6= NP . However, it seems that we are yet quite far from achieving this goal and that new techniques have to...
Submitted to Acta Informatica, special issue for the 70th birthday of Klaus-Jörn Lange . Michaël Cadilhac and Charles Paperman. 2021. The Regular Languages of Wire Linear AC0
- Michaël Cadilhac
- Charles Paperman
- Cadilhac Michaël
PAC Learning Depth-3 AC0 Circuits of Bounded Top Fanin
- Ning Ding
- Yanli Ren
- Dawu Gu
- Ding Ning
Borel Sets and Circuit Complexity
- Michael Sipser
- David S Johnson
- Ronald Fagin
- Michael L Fredman
- David Harel
- Richard M Karp
- Nancy A Lynch
- Christos H Papadimitriou
- Ronald L Rivest
- Walter L Ruzzo
- Sipser Michael