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Logic meets algebra: the case of regular languages
 LOGICAL METHODS IN COMPUTER SCIENCE
, 2007
"... The study of finite automata and regular languages is a privileged meeting point of algebra and logic. Since the work of Büchi, regular languages have been classified according to their descriptive complexity, i.e. the type of logical formalism required to define them. The algebraic point of view ..."
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The study of finite automata and regular languages is a privileged meeting point of algebra and logic. Since the work of Büchi, regular languages have been classified according to their descriptive complexity, i.e. the type of logical formalism required to define them. The algebraic point of view on automata is an essential complement of this classification: by providing alternative, algebraic characterizations for the classes, it often yields the only opportunity for the design of algorithms that decide expressibility in some logical fragment. We survey the existing results relating the expressibility of regular languages in logical fragments of MSO[S] with algebraic properties of their minimal automata. In particular, we show that many of the best known results in this area share the same underlying mechanics and rely on a very strong relation between logical substitutions and blockproducts of pseudovarieties of monoid. We also explain the impact of these connections on circuit complexity theory.
An EhrenfeuchtFraïssé Game Approach to Collapse Results in Database Theory
, 2008
"... We present a new EhrenfeuchtFraïssé game approach to collapse results in database theory. We show that, in principle, every natural generic collapse result may be proved via a translation of winning strategies for the duplicator in an EhrenfeuchtFraïssé game. Following this approach we can deal wi ..."
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We present a new EhrenfeuchtFraïssé game approach to collapse results in database theory. We show that, in principle, every natural generic collapse result may be proved via a translation of winning strategies for the duplicator in an EhrenfeuchtFraïssé game. Following this approach we can deal with certain infinite databases where previous, highly involved methods fail. We prove the natural generic collapse for Zembeddable databases over any linearly ordered context structure with arbitrary monadic predicates, and for Nembeddable databases over the context structure 〈R, <,+, MonQ, Groups〉, where Groups is the collection of all subgroups of 〈R,+ 〉 that contain the set of integers and MonQ is the collection of all subsets of a particular infinite set Q of natural numbers. This, in particular, implies the collapse for arbitrary databases over 〈N, <,+, MonQ 〉 and for Nembeddable databases over 〈R, <,+, Z, Q〉. I.e., firstorder logic with < can express the same ordergeneric queries as firstorder logic with <, +, etc. Restricting the complexity of the formulas that may be used to formulate queries to Boolean combinations of purely existential firstorder formulas, we even obtain the collapse for N
An algebraic point of view on the CraneBeach conjecture
, 2006
"... 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 CraneBeach Conjecture, which was recently disproved, stated that any language containing a neutral letter and definable in FO is in fact FO[<] definabl ..."
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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 CraneBeach Conjecture, which was recently disproved, stated that any language containing a neutral letter and definable in FO is in fact FO[<] definable and is thus a regular, starfree 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 boundedwidth branching programs. We also apply this to a standard extension of FO using modularcounting quantifiers and show that the boolean closure of this logic’s Σ1 fragment has the CBP.
Linear Circuits, TwoVariable Logic and Weakly Blocked Monoids
"... Abstract. Following recent works connecting twovariable logic to circuits and monoids, we establish, for numerical predicate sets P satisfying a certain closure property, a onetoone correspondence between F O[<, P]uniform linear circuits, twovariable formulas with P predicates, and weak bloc ..."
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Abstract. Following recent works connecting twovariable logic to circuits and monoids, we establish, for numerical predicate sets P satisfying a certain closure property, a onetoone correspondence between F O[<, P]uniform linear circuits, twovariable formulas with P predicates, and weak block products of monoids. In particular, we consider the case of linear TC 0, majority quantifiers, and finitely typed monoids. This correspondence will hold for any numerical predicate set which is F O[<]closed and whose predicates do not depend on the input length. 1
Modular Quantifiers
"... In the late nineteeneighties much of our research concerned the application of semigrouptheoretic methods to automata and regular languages, and the connection between computational complexity and this algebraic theory of automata. It was during this period that we became aware of the work of Wolf ..."
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In the late nineteeneighties much of our research concerned the application of semigrouptheoretic methods to automata and regular languages, and the connection between computational complexity and this algebraic theory of automata. It was during this period that we became aware of the work of Wolfgang Thomas. Thomas had undertaken the study of concatenation hierarchies of starfree regular languages—a subject close to our hearts — by modeltheoretic methods. He showed that the levels of the dotdepth hierarchy corresponded precisely to level of the quantifier alternation hierarchy within firstorder logic[26], and applied EhrenfeuchtFraïssé games to prove that the dotdepth hierarchy was strict [27], a result previously obtained by semigrouptheoretic means [4, 18]. Finite model theory, a subject with which we’d had little prior acquaintance, suddenly appeared as a novel way to think about problems that we had been studying for many years. We were privileged to have been introduced to this field by so distinguished a practitioner as Wolfgang Thomas, and to have then had the opportunity to work together with him. The study of languages defined with modular quantifiers, the subject of the present survey, began with this collaboration.
LOGIC MEETS ALGEBRA: THE CASE OF REGULAR LANGUAGES PASCAL TESSON AND DENIS THÉRIEN
"... Abstract. The study of finite automata and regular languages is a privileged meeting point of algebra and logic. Since the work of Büchi, regular languages have been classified according to their descriptive complexity, i.e. the type of logical formalism required to define them. The algebraic point ..."
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Abstract. The study of finite automata and regular languages is a privileged meeting point of algebra and logic. Since the work of Büchi, regular languages have been classified according to their descriptive complexity, i.e. the type of logical formalism required to define them. The algebraic point of view on automata is an essential complement of this classification: by providing alternative, algebraic characterizations for the classes, it often yields the only opportunity for the design of algorithms that decide expressibility in some logical fragment. We survey the existing results relating the expressibility of regular languages in logical fragments of MSO[S] with algebraic properties of their minimal automata. In particular, we show that many of the best known results in this area share the same underlying mechanics and rely on a very strong relation between logical substitutions and blockproducts of pseudovarieties of monoid. We also explain the impact of these connections to circuit complexity theory. 1.
Nondefinability of languages by generalized firstorder formulas over (N,+)
"... Abstract. We consider firstorder logic with monoidal quantifiers over words. We show that all languages with a neutral letter, definable using the addition 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 ..."
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Abstract. We consider firstorder logic with monoidal quantifiers over words. We show that all languages with a neutral letter, definable using the addition 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 prove that LS [<,+] ∩N = LS [<] ∩N Our result can be interpreted as the Crane Beach conjecture to hold for the logic LS [<,+]. As a consequence 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 as a numerical predicate is necessary for Barrington’s theorem to hold and also to simulate majority quantifiers. All these results can be viewed as separation results for highly uniform circuit classes. For example we separate FO[<,+]uniform CC0 from FO[<,+]uniform ACC0. 1