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A Crevice on the Crane Beach: Finite-Degree Predicates
Abstract
First-order logic (FO) over words is shown to be equiexpressive with FO equipped with a restricted set of numerical predicates, namely the order, a binary predicate MSB, and the finite-degree predicates: FO[Arb] = FO[<, MSB, Fin]. The Crane Beach Property (CBP), introduced more than a decade ago, is true of a logic if all the expressible languages admitting a neutral letter are regular. Although it is known that FO[Arb] does not have the CBP, it is shown here that the (strong form of the) CBP holds for both FO[<, Fin] and FO[<, MSB]. Thus FO[<, Fin] exhibits a form of locality and the CBP, and can still express a wide variety of languages, while being one simple predicate away from the expressive power of FO[Arb]. The counting ability of FO[<, Fin] is studied as an application.
... While it is known that the FOMOD[<, +, ×] does not have the CBP [1], there is still no complete understanding of which vocabularies do and which don't have the CBP. Thus CBP continues to be an area of active research area (see [4], [5], [9]). ...
We show that first order logic (FO) and first order logic extended with modulo counting quantifiers (FOMOD) over purely functional vocabularies which extend addition, satisfy the Crane beach property (CBP) if the logic satisfies a normal form (called positional normal form). This not only shows why logics over the addition vocabulary have the CBP but also gives new CBP results, for example for the vocabulary which extends addition with the exponentiation function. The above results can also be viewed from the perspective of circuit complexity. Showing the existence of regular languages not definable in FOMOD[<, +, *] is equivalent to the separation of the circuit complexity classes ACC0 and NC1 . Our theorem shows that a weaker logic , namely, FOMOD[<,+,2^x] cannot define all regular languages.
We investigate in a method for proving separation results for abstract
classes of languages. A well established method to characterize varieties of
regular languages are identities. We use a recently established generalization
of these identities to non-regular languages by Gehrke, Grigorieff, and Pin: so
called equations, which are capable of describing arbitrary Boolean algebras of
languages.
While the main concern of their result is the existence of these equations,
we investigate in a general method that could allow to find equations for
language classes in an inductive manner.
Thereto we extend an important tool -- the block product or substitution
principle -- known from logic and algebra, to non-regular language classes.
Furthermore, we abstract this concept by defining it directly as an operation
on (non-regular) language classes. We show that this principle can be used to
obtain equations for certain circuit classes, given equations for the gate
types.
Concretely, we demonstrate the applicability of this method by obtaining a
description via equations for all languages recognized by circuit families that
contain a constant number of (inner) gates, given a description of the gate
types via equations.
This paper introduces and begins the study of a well-behaved class of linearly ordered structures, the O-minimal structures. The definition of this class and the corresponding class of theories, the strongly O-minimal theories, is made in analogy with the notions from stability theory of minimal structures and strongly minimal theories. Theorems 2.1 and 2.3, respectively, provide characterizations of O-minimal ordered groups and rings. Several other simple results are collected in . The primary tool in the analysis of O-minimal structures is a strong analogue of "forking symmetry," given by Theorem 4.2. This result states that any (parametrically) definable unary function in an O-minimal structure is piecewise either constant or an order-preserving or reversing bijection of intervals. The results that follow include the existence and uniqueness of prime models over sets (Theorem 5.1) and a characterization of all ℵ0-categorical O-minimal structures (Theorem 6.1).
We consider first-order logic with monoidal quantifiers over words. We show
that all languages with a neutral letter, definable using the addition
numerical predicate are also definable with the order predicate as the only
numerical predicate. Let S be a subset of monoids.
Let LS be the logic closed under quantification over the monoids in S and N
be the class of neutral letter languages. Then we show that: LS[<,+] cap N =
LS[<] Our result can be interpreted as the Crane Beach conjecture to hold for
the logic LS[<,+]. As a corollary of our result we get the result of Roy and
Straubing that FO+MOD[<,+] collapses to FO+MOD[<].
For cyclic groups, we answer an open question of Roy and Straubing, proving
that MOD[<,+] collapses to MOD[<]. Our result also shows that multiplication is
necessary for Barrington's theorem to hold.
All these results can be viewed as separation results for very uniform
circuit classes. For example we separate FO[<,+]-uniform CC0 from
FO[<,+]-uniform ACC0.
In this paper we prove that the class of functions expressible by first order formulas with only two variables coincides with the class of functions computable by AC0 circuits with a linear number of gates. We then investigate the feasibility of using Ehrenfeucht-Fraisse games to prove lower bounds for that class of circuits, as well as for general AC0 circuits
We extend bounds on the expressive power of first-order logic over
finite structures and over ordered finite structures, by generalizing to
the situation where the finite structures are embedded in an infinite
structure M, where M satisfies some simple combinatorial properties
studied in model-theoretic stability theory. We first consider
first-order logic over finite structures embedded in a stable structure,
and show that it has the same generic expressive power as first-order
logic on unordered finite structures. It follows from this that having
the additional structure of, for example, an abelian group or an
equivalence relation, does not allow one to define any new generic
queries. We also consider first-order logic over finite structures
living within any model M that lacks the independence property and show
that its expressive power is bounded by first-order logic over finite
ordered structures. This latter result gives an enormous class of
structures in which the expressive power of first-order logic is sharply
limited; it shows that common queries such as parity and connectivity
cannot be defined for finite structures living within structures from
this huge class. It also gives a pure combinatorial property of an
interpreted structure M that is sufficient to extend results on
first-order logic on ordered structures to first-order logic on finite
structures embedded in M
We study languages with bounded communication complexity in the multiparty "input on the forehead model" with worst-case partition. In the two-party case, languages with bounded complexity are exactly those recognized by programs over commutative monoids [19]. This can be used to show that these languages all lie in shallow ACC0. In contrast, we use coding techniques to show that there are languages of arbitrarily large circuit complexity which can be recognized in constant communication by k players for k ≥ 3. However, we show that if a language has a neutral letter and bounded communication complexity in the k-party game for some fixed k then the language is in fact regular. We give an algebraic characterization of regular languages with this property. We also prove that a symmetric language has bounded k-party complexity for some fixed k iff it has bounded two party complexity.
We consider an extension of first-order logic by modular quantifiers of a fixed modulus q. Drawing on collapse results from finite model theory and techniques of finite semigroup theory, we show that if the only
available numerical predicate is addition, then sentences in this logic cannot define the set of bit strings in which the
number of 1’s is divisible by a prime p that does not divide q. More generally, we completely characterize the regular languages definable in this logic. The corresponding statement, with
addition replaced by arbitrary numerical predicates, is equivalent to the conjectured separation of the circuit complexity
class ACC from NC
1. Thus our theorem can be viewed as proving a highly uniform version of the conjecture.
We propose a new approach to the notion of recognition, which departs from the classical definitions by three specific features. First, it does not rely on automata. Secondly, it applies to any Boolean algebra (BA) of subsets rather than to individual subsets. Thirdly, topol- ogy is the key ingredient. We prove the existence of a minimum recognizer in a very general setting which applies in particular to any BA of subsets of a discrete space. Our main results show that this minimum recognizer is a uniform space whose completion is the dual of the original BA in Stone-Priestley duality; in the case of a BA of languages closed under quotients, this completion, called the syntactic space of the BA, is a com- pact monoid if and only if all the languages of the BA are regular. For regular languages, one recovers the notions of a syntactic monoid and of a free profinite monoid. For nonregular languages, the syntactic space is no longer a monoid but is still a compact space. Further, we give an equational characterization of BA of languages closed under quotients, which extends the known results on regular languages to nonregular lan- guages. Finally, we generalize all these results from BAs to lattices, in which case the appropriate structures are partially ordered.
We consider an extension of first-order logic by modular quantifiers of a fixed modulus q. Drawing on collapse results from finite model theory and techniques of finite semigroup theory, we show that if the only available numerical predicate is addition, then sentences in this logic cannot define the set of bit strings in which the number of 1’s is divisible by a prime p that does not divide q. More generally, we completely characterize the regular languages definable in this logic. The corresponding statement, with addition replaced by arbitrary numerical predicates, is equivalent to the conjectured separation of the circuit complexity class ACC from NC 1 . Thus our theorem can be viewed as proving a highly uniform version of the conjecture.
The class of languages definable by majority quantifiers using the order predicate is investigated. It is shown that the additional use of first order or counting quantifiers does not increase this class. Further on, addition is in this connection a definable numerical predicate, while the converse does not hold. The emptiness problem for this class turns out to be undecidable.
We study Monadic Second-Order Logic (MSO) over finite words, extended with (non-uniform arbitrary) monadic predicates. We show that it defines a class of languages that has algebraic, automata-theoretic, and machine-independent characterizations. We consider the regularity question: Given a language in this class, when is it regular? To answer this, we show a substitution property and the existence of a syntactical predicate.
We give three applications. The first two are to give very simple proofs that the Straubing Conjecture holds for all fragments of MSO with monadic predicates and that the Crane Beach Conjecture holds for MSO with monadic predicates. The third is to show that it is decidable whether a language defined by an MSO formula with morphic predicates is regular.
It is proved that any 0-minimal structure M (in which the underlying order is dense) is strongly 0-minimal (namely, every N elementarily equivalent to M is 0-minimal). It is simultaneously proved that if M is 0- minimal, then every definable set of n-tuples of M has finitely many “definably connected components.
We investigate in a method for proving lower bounds for abstract circuit classes. A well established method to characterize varieties of regular languages are identities. We use a recently established generalization of these identities to non-regular languages by Gehrke, Grigorieff, and Pin: so called equations, which are capable of describing arbitrary Boolean algebras of languages. While the main concern of their result is the existence of these equations, we investigate in a general method that could allow to find equations for circuit classes in an inductive manner. Thereto we extend an important tool – the block product or substitution principle – known from logic and algebra, to non-regular language classes. Furthermore, we abstract this concept by defining it directly as an operation on (non-regular) language classes. We show that this principle can be used to obtain equations for certain circuit classes, given equations for the gate types. Concretely, we demonstrate the applicability of this method by obtaining a description via equations for all languages recognized by circuit families that contain a constant number of (inner) gates, given a description of the gate types via equations.
We consider two-variable first-order logic on finite words with a fixed
number of quantifier alternations. We show that all languages with a neutral
letter definable using the order and finite-degree predicates are also
definable with the order predicate only. From this result we derive the
separation of the alternation hierarchy of two-variable logic on this
signature.
Part 1 Mathematical preliminaries: words and languages automata and regular languages semigroups and homomorphisms. Part 2 Formal languages and formal logic: examples definitions. Part 3 Finite automata: monadic second-order sentences and regular languages regular numerical predicates infinite words and decidable theories. Part 4 Model-theoretic games: the Ehrenfeucht-Fraisse game application to FO [decreasing] application to FO [+1]. Part 5 Finite semigroups: the syntactic monoid calculation of the syntactic monoid application to FO [decreasing] semidirect products categories and path conditions pseudovarieties. Part 6 First-order logic: characterization of FO [decreasing] a hierarchy in FO [decreasing] another characterization of FO [+1] sentences with regular numerical predicates. Part 7 Modular quantifiers: definition and examples languages in (FO + MOD(P))[decreasing] languages in (FO + MOD)[+1] languages in (FO + MOD)[Reg] summary. Part 8 Circuit complexity: examples of circuits circuits and circuit complexity classes lower bounds. Part 9 Regular languages and circuit complexity: regular languages in NC1 formulas with arbitrary numerical predicates regular languages and non-regular numerical predicates special cases of the central conjecture. Appendices: proof of the Krohn-Rhodes theorem proofs of the category theorems.
Arithmetical definability has been extensively studied over the natural numbers. In this paper, we take up the study of arithmetical definability over finite structures, motivated by the correspondence between uniform AC0 and FO(PLUS, TIMES). We prove finite analogs of three classic results in arithmetical definability, namely that < and TIMES can first-order define PLUS, that < and DIVIDES can first-order define TIMES, and that < and COPRIME can first-order define TIMES. The first result sharpens the equivalence FO(PLUS, TIMES) =FO(BIT) to FO(<, TIMES) = FO(BIT), answering a question raised by Barrington et al. about the Crane Beach Conjecture. Together with previous results on the Crane Beach Conjecture, our results imply that FO(PLUS) is strictly less expressive than FO(<, TIMES) = FO(<, DIVIDES) = FO(<,COPRIME). In more colorful language, one could say that, for parallel computation, multiplication is harder than addition.
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.
Uniformity notions more restrictive than the usual FO[<, +, *]-uniformity = FO[<, Bit]-uniformity are introduced. It is shown that the general framework exhibited by Barrington et al. still holds if the fan-in of the gates in the corresponding circuits is considered
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 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(&Ogr;(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).
A letter e ∈Σ is said to be neutral for a language L if it can be inserted and deleted at will in a word without affecting membership in L. The Crane Beach Conjecture, which was recently disproved, stated that any language containing a neutral letter and definable in first-order with arbitrary numerical predicates () is in fact FO [<] definable and is thus a regular, star-free language. More generally, we say that a logic or a computational model has the Crane Beach property if the only languages with neutral letter that it can define/compute are regular.
We develop an algebraic point of view on the Crane Beach properties using the program over monoid formalism which has proved of importance in circuit complexity. Using recent communication complexity results we establish a number of Crane Beach results for programs over specific classes of monoids. These can be viewed as Crane Beach theorems for classes of bounded-width branching programs. We also apply this to a standard extension of FO using modular-counting quantifiers and show that the boolean closure of this logic’s Σ1 fragment has the CBP.
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
The counting ability of weak formalisms is of interest as a measure of their expressive power. The question was investigated in the 1980's in several papers in complexity theory and in weak arithmetic. In each case, the considered formalism (AC--circuits, first--order logic, , respectively) was shown to be able to count precisely up to a polylogarithmic number. An essential part of each of the proofs is the construction of a 1--1 mapping from a small subset of into a small initial segment. In each case the expressibility of such a mapping depends on some strong argument (group theoretic device or prime number theorem) or intricate construction. We present a coding device based on a collision-free hashing technique, leading to a completely elementary proof for the polylog counting capability of first--order logic (with built--in arithmetic), --circuits, rudimentary arithmetic, the Linear Hierarchy, and monadic--second order logic with addition. @InProceedings{durand_et_al:DSP:2007:976, author = {Arnaud Durand and Clemens Lautemann and Malika More}, title = {Counting Results in Weak Formalisms}, booktitle = {Circuits, Logic, and Games}, year = {2007}, editor = {Thomas Schwentick and Denis Th{'e}rien and Heribert Vollmer }, number = {06451}, series = {Dagstuhl Seminar Proceedings}, ISSN = {1862-4405}, publisher = {Internationales Begegnungs- und Forschungszentrum f{"u}r Informatik (IBFI), Schloss Dagstuhl, Germany}, address = {Dagstuhl, Germany}, URL = {http://drops.dagstuhl.de/opus/volltexte/2007/976}, annote = {Keywords: Small complexity classes, logic, polylog counting} }
Padding techniques are well-known from Computational Complexity Theory. Here, an analogous concept is considered in the context of existential second-order logics. Informally, a graph H is a padded version of a graph G, if H consists of an isomorphic copy of G and some isolated vertices. A set A of graphs is called weakly expressible by a formula ' in the presence of padding, if ' is able to distinguish between (sufficiently) padded versions of graphs from A and padded versions of graphs that are not in A. From results of Lynch [Lyn82, Lyn92] it can be easily concluded that (essentially) every NP- set of graphs is weakly expressible by an existential monadic second-order (MonSigma 1 1 ) formula with polynomial padding and built-in addition. In particular, NP 6= coNP if and only if there is a coNP-set of graphs that is not weakly expressible by a MonSigma 1 1 -formula in the presence of addition, even if polynomial padding is allowed. In some sense, this implies that MonSigma ...