Finite Model Theory

Basic results concerning the behaviour of binary relations are well-known today. However, there is a question what properties to study for arities greater than two (see ). In this paper we establish relations as sets of mappings and investigate questions motivated by relational database theory. First we study homomorphisms and the process of update of a relation. Further we deal with the interaction of hornomorphisrn and functional dependencies. The last part is devoted to the investigation of relationships between hornomorphisrn and operations.

In this paper, optimum decomposition of belief networks is discussed. Some methods of decom position are examined and a new method -the method of Minimum Total Number of States (MTNS)-is proposed. The problem of optimum belief network decomposition under our frame work, as under all the other frameworks, is shown to be NP-hard. According to the computational complexity analysis, an algorithm of belief net work decomposition is proposed in (Wen, 1989b) based on simulated annealing. • How can we decompose the space into sub-

We study two extensions of FO 2 [<], first-order logic interpreted in finite words, in which formulas are restricted to use only two variables. We adjoin to this language two-variable atomic formulas that say, 'the letter a appears between positions x and y' and 'the factor u appears between positions x and y'. These are, in a sense, the simplest properties that are not expressible using only two variables. We present several logics, both first-order and temporal, that have the same expressive power, and find matching lower and upper bounds for the complexity of satisfiability for each of these formulations. We give effective conditions, in terms of the syntactic monoid of a regular language, for a property to be expressible in these logics. This algebraic analysis allows us to prove, among other things, that our new logics have strictly less expressive power than full first-order logic FO[<]. Our proofs required the development of novel techniques concerning factorizations of words.

Given two graphs G and H and a function f ⊂ V (G) × V (H), Hedetniemi [9] defined the function graph Gf H by Whenever G ∼ = H, the function graph was called a functigraph by Chen, Ferrero, Gera and Yi . A function graph is a generalization of the α-permutation graph introduced by Chartrand and Harary . The independence number of a graph is the size of a largest set of mutually non-adjacent vertices. In this paper, we study independence number in function graphs. In particular, we give a lower bound in terms of the order and the chromatic number, which improves on some elementary results and has a number of interesting corollaries.

We consider relationship between binary relations in approximation spaces and topologies defined by them. In any approximation space (X, R), a reflexive closure Rω determines an Alexandrov topology T (Rω ) and, for any Alexandrov topology T on X, there exists a reflexive relation RT such that T = TR. From the result, we also obtain that any Alexandrov topology satisfying (clop), A is open if and only if A is closed, can be characterized by reflexive and symmetric relation. Moreover, we provide a negative answer to the problem left open in [1].

Acyclic conjunctive queries form a polyno-mially evaluable fragment of definite non-recursive first-order Horn clauses. Labeled graphs, a special class of relational struc-tures, provide a natural way for represent-ing chemical compounds. We propose an al-gorithm specific to ...

The paper aims at contributing to the problem of translating natural (ethnic) language into the framework of formal logic in a structure-preserving way. There are several problems with encoding knowledge in logic-derived formalisms. Among the most difficult are those connected with natural language quantifiers. Even for relatively simple natural language sentences there are readings hard to encode in the classical formal logic. The solution we propose is to expand the language of the classical first-order logic with a number of new quantifier symbols (corresponding to the natural language quantifiers possibly co-occurring in a sentence) with a nonclassical semantics/interpretation similar in a way to the Mostowski's generalized quantifier, i.e. as a family of subsets of the Cartesian product of variable ranges. We claim that the logic-based formalism involving such nonclassical quantifiers may be considered a good knowledge representation tool closer to the natural language then the classical logic.

In our joint paper (KO) with H. Kihara, we discuss comprehensively inter- polation properties and Beth definability properties of substructural logics, and their algebraic characterizations in comparison with various forms of amalgamation properties of varieties of residuated lattices. This is a brief exposition of some of main results of the paper (KO), We start from an observation on two results on classical logic. The first one says that classical logic has both Craig&#x27;s interpolation property (CIP) and Beth definability property (BDP). The second says that the CIP of classical logic is derived from the amalgamation property (AP) of the class of Boolean algebras, and vice versa. Connections of these three properties can be shown in a more general setting. For instance, the CIP and the BDP are shown to be interderivable for each classical regular logic (see e.g. (GM)). Also, as shown by Maksi- mova (see also (GM)), for each superintuitionistic logic the CIP implies the AP (in ...

We investigate properties of finite relational structures over the reals expressed by first-order sentences whose predicates are the relations of the structure plus arbitrary polynomial inequalities, and whose quantifiers can range over the whole set of reals. In constraint programming terminology, this corresponds to Boolean real polynomial constraint queries on finite structures. The fact that quantifiers range over all reals seems crucial; however, we observe that each sentence in the first-order theory of the reals can be evaluated by letting each quantifier range over only a finite set of real numbers without changing its truth value. Inspired by this observation, we then show that when all polynomials used are linear, each query can be expressed uniformly on all finite structures by a sentence of which the quantifiers range only over the finite domain of the structure. In other words, linear constraint programming on finite structures can be reduced to ordinary query evaluation as usual in finite model theory and databases. Moreover, if only "generic" queries are taken into consideration, we show that this can be reduced even further by proving that such queries can be expressed by sentences using as polynomial inequalities only those of the simple form x < y.

This paper is a survery on the technique to prove logics non-finitely generated originated in [H] and later used in [HL] and [HK]. The basic idea is that many (n + 1)-ary quantifiers Here, quantifier Q is n-ary, if ar(Q ) ≤ n where the arity ar(Q ) = sup{n R : R ∈ τ } is the supremum of the numbers of variables in formulas bounded by the quantifier Q . We say that the strict (strong) dimension of the quantifier Q above is n + 1 (similarly for model-classes). In practice, a back-and-forth characterization and a specific construction are needed to prove the non-redundancy. Back-and-forth systems are developed for logics L ∞ω (Q n ) and L ∞ω (Q G ) where Q n is a collection of n-ary quantifiers and Q G a subcollection of Q n satisfying certain symmetricity conditions (G is a permutation group). Non-trivial pairs of L ∞ω (Q n )-equivalent models are then constructed by means of a game.

The Liar Paradox, traditionally phrased as 'This sentence is false', has been a longstanding challenge in logic and semantics. In this paper, I reinterpret the paradox through a functional perspective, treating the statement as a self-referential composition of logical evaluation functions. This approach avoids binary truth value fixation by recognizing the paradox as a system oscillating between outputs, similar to quantum superposition prior to observation.

Classes of graphs with bounded expansion are a generalization of both proper minor closed classes and degree bounded classes. Such classes are based on a new invariant, the greatest reduced average density (grad) of G with rank r, ∇r(G). These classes are also characterized by the existence of several partition results such as the existence of low tree-width and low tree-depth colorings [18][17]. These results lead to several new linear time algorithms, such as an algorithm for counting all the isomorphs of a fixed graph in an input graph or an algorithm for checking whether there exists a subset of vertices of a priori bounded size such that the subgraph induced by this subset satisfies some arbirtrary but fixed first order sentence. We also show that for fixed p, computing the distances between two vertices up to distance p may be performed in constant time per query after a linear time preprocessing. We also show, extending several earlier results, that a class of graphs has sublinear separators if it has sub-exponential expansion. This result result is best possible in general.

We study restricted homomorphism dualities in the context of classes with bounded expansion. This presents a generalization of restricted dualities obtained earlier for bounded degree graphs and also for proper minor closed classes. This is related to distance coloring of graphs and to the "approximative version" of Hadwiger conjecture.

We propose the following computational assumption: in general if we try to compress the depth of a circuit family (parallel time) more than a constant factor we will suffer super-quasipolynomial blowup in the size (number of processors). This assumption is only slightly stronger than the popular assumption about the robustness of NC, and we observe that it has surprising consequences. Note also that the choice of super-quasi-polynomial blowup is the smallest bound that avoids the circuit class collapse of . In this proposal we put our assumption in perspective, discuss its relation to the existing literature, and show that it has two important consequences. The first consequence is NC = SC, where NC is the class characterized by uniform circuits of poly-logarithmic depth and polynomial size, and SC is characterized by algorithms that run in poly-logarithmic space and polynomial time. For the second consequence we use an additional but mild complexity assumption to obtain a strong separation between the graph isomorphism GraphIso and the group isomorphism GroupIso problem. In particular, we show that GraphIso is not reducible to GroupIso using circuits of O(log n) depth.

We propose the following computational assumption: in general if we try to compress the depth of a circuit family (parallel time) more than a constant factor we will suffer super-quasipolynomial blowup in the size (number of processors). This assumption is only slightly stronger than the popular assumption about the robustness of NC, and we observe that it has surprising consequences. Note also that the choice of super-quasi-polynomial blowup is the smallest bound that avoids the circuit class collapse of [Vol98]. In this proposal we put our assumption in perspective, discuss its relation to the existing literature, and show that it has two important consequences. The first consequence is NC 6= SC, where NC is the class characterized by uniform circuits of poly-logarithmic depth and polynomial size, and SC is characterized by algorithms that run in poly-logarithmic space and polynomial time. For the second consequence we use an additional but mild complexity assumption to obtain a s...

This paper identifies an industrially relevant class of linear hybrid automata (LHA) called reasonable LHA for which parametric verification of convex safety properties with exhaustive entry states can be verified in polynomial time and time-bounded reachability can be decided in nondeterministic polynomial time for non-parametric verification and in exponential time for parametric verification. Properties with exhaustive entry states are restricted to runs originating in a (specified) inner envelope of some mode-invariant. Deciding whether an LHA is reasonable is shown to be decidable in polynomial time.

The Liar Paradox arises from an attempt to talk about truth and combineit with negation. Mathematical logic provides many methods for analyzingsuch a mixing of syntax (negation) and semantics (truth), but as the Liarindicates there is rather a mismatch in this case, between what we want tosay and what we can say. We may be able to say “This sentence is not true”but whichever way we say it, one of the words in the sentence does not havethe meaning we intend it to have. It is the intention of this paper to make this observation as explicit and general as possible.

Classes of graphs with bounded expansion are a generalization of both proper minor closed classes and degree bounded classes. Such classes are based on a new invariant, the greatest reduced average density (grad) of G with rank r, ∇ r (G). These classes are also characterized by the existence of several partition results such as the existence of low tree-width and low tree-depth colorings [19][18]. These results lead to several new linear time algorithms, such as an algorithm for counting all the isomorphs of a fixed graph in an input graph or an algorithm for checking whether there exists a subset of vertices of a priori bounded size such that the subgraph induced by this subset satisfies some arbirtrary but fixed first order sentence. We also show that for fixed p, computing the distances between two vertices up to distance p may be performed in constant time per query after a linear time preprocessing. We also show, extending several earlier results, that a class of graphs has sublinear separators if it has sub-exponential expansion. This result result is best possible in general.

We study restricted homomorphism dualities in the context of classes with bounded expansion. This presents a generalization of restricted dualities obtained earlier for bounded degree graphs and also for proper minor closed classes. This is related to distance coloring of graphs and to the "approximative version" of Hadwiger conjecture.

The arithmetical hierarchy (AH) is similar to the polynomial hierarchy (PH). Unlike the PH, the AH does not collapse relative to any oracle. A language in the (k + 1)-st level of the AH is computable enumerable (c.e.) relative to the kth level. So, given an oracle in the kth level of the AH, we could use a black-box search to decide whether the input word is in the language. With very large padding arguments, i.e. the paddings grow faster than any relative to the level k of the AH computable function, we would construct a language contained in the k + 1 level of PH, if we use only a finite set of input words. From the oracle in AH, we would construct an analogue oracle at the kth level of PH. For the input words of the finite set, a word is in the language of AH, if and only if it is in the language of PH. And the input word is in the oracle set of AH, if and only if it is in the oracle of PH. As in the language of AH, we must apply a black-box search in the language of PH. So, we would also have exponentially many oracle queries in the language of PH. The PH does not collapse.

We prove that the two-variable fragment of first-order logic has the weak Beth definability property. This makes the two-variable fragment a natural logic separating the weak and the strong Beth properties since it does not have the strong Beth definability property. The proof relies on the special property of 2-variable logic that each model is 2-equivalent to some rather homogeneous model that we dub 2-transitive.

This is an abstract of the dissertation which solved some problems raised in . The subject is General Algebraic Logic in the sense of Rasiowa , but now for first order logics. Here we discuss the algebraic problems; their connections with (nonclassical and classical) logics were explained in . The variety of cylindric algebras was introduced for the classical first order logic; the present general algebraic (universal algebraic) approach is a generalization of the theory of that variety to make it applicable to other first o. logics as well (cf. Freeman [6]). Throughout, α, β, γ denote infinite ordinals, ω is the set of natural numbers, and Ord is the class of in finite ordinals.

Modal Logic is traditionally concerned with the intensional operators "possibly" and "necessary", whose intuitive correspondence with the standard quantifiers "there exists" and "for all" comes out clearly in the usual Kripke semantics. This observation underlies the well-known translation from propositional modal logic with operators ♦ and , possibly indexed, into the first-order language over possible worlds models . In this way, modal formalisms correspond to fragments of a full first-order (or sometimes higher-order) language over these models, which are both expressively perspicuous and deductively tractable. In this paper, by the "modal fragment" of predicate logic we understand the set of all first-order formulas obtainable as translations of basic (poly-)modal formulas. As the modal fragment is merely a notational variant of the basic modal language, we will often refer to the two interchangeably. Basic modal logic shares several nice properties with full predicate logic, namely, finite axiomatizability, Craig Interpolation and Beth Definability, as well as model-theoretic preservation results such as the Loś-Tarski Theorem characterizing those formulas that are preserved under taking submodels. In addition, basic modal logic has some nice properties not shared with predicate logic as a whole: e.g., its axiomatization does not need side conditions on free or bound variablesand most evidently: basic modal logic is decidable. We shall concentrate on this list in what follows, in the hope that it forms a representative sample. Our aim is to find natural fragments of predicate logic extending the modal one which inherit the above-mentioned nice properties. This quest has two virtues. It forces us to understand why basic modal logic has these nice properties. And it points the way to new insights concerning predicate logic itself. Note that this study takes place over the universe of all models, without special restrictions on accessibility relations. This is the domain of the "minimal modal logic", which

The main result gives a sufficient condition for a class K of finite dimensional cylindric algebras to have the property that not every epimorphism in K is surjective. In particular, not all epimorphisms are surjective in the classes CA n of n-dimensional cylindric algebras and the class of representable algebras in CA n for finite n > 1, solving Problem 10 of [28] for finite n. By a result of Németi, this shows that the Beth-definability property fails for the finite-variable fragments of first order logic as long as the number n of variables available is > 1 and we allow models of size ≥ n + 2, but holds if we allow only models of size ≤ n + 1. We also use our results in the present paper to prove that several results in the literature concerning injective algebras and definability of polyadic operations in CA n are best possible. We raise several open problems.

We introduce a restricted second-order logic $\textrm{SO}^{\textit{plog}}$ for finite structures where second-order quantification ranges over relations of size at most poly-logarithmic in the size of the structure. We demonstrate the relevance of this logic and complexity class by several problems in database theory. We then prove a Fagin’s style theorem showing that the Boolean queries which can be expressed in the existential fragment of $\textrm{SO}^{\textit{plog}}$ correspond exactly to the class of decision problems that can be computed by a non-deterministic Turing machine with random access to the input in time $O((\log n)^k)$ for some $k \ge 0$, i.e. to the class of problems computable in non-deterministic poly-logarithmic time. It should be noted that unlike Fagin’s theorem which proves that the existential fragment of second-order logic captures NP over arbitrary finite structures, our result only holds over ordered finite structures, since $\textrm{SO}^{\textit{plog}}$ i...

In this thesis, we study the descriptive complexity of counting classes based on Boolean circuits. In descriptive complexity, the complexity of problems is studied in terms of logics required to describe them. The focus of research in this area is on identifying logics that can express exactly the problems in specific complexity classes. For example, problems are definable in ESO, existential second-order logic, if and only if they are in NP, the class of problems decidable in nondeterministic polynomial time. In the computation model of Boolean circuits, individual circuits have a fixed number of inputs. Circuit families are used to allow for an arbitrary number of input bits. A priori, the circuits in a family are not uniformly described, but one can impose this as an additional condition, e.g., requiring that there is an algorithm constructing them. For any circuit there is a function counting witnesses (or proofs) for the circuit producing the output 1. Consequently, any class o...

In this work, we estimate the number of hyperedges in a hypergraph ${\cal H}(U({\cal H}), {\cal F}({\cal H}))$, where $U({\cal H})$ denotes the set of vertices and ${\cal F}({\cal H}))$ denotes the set of hyperedges. We assume a query oracle access to the hypergraph ${\cal H}$. Estimating the number of edges, triangles or small subgraphs in a graph is a well studied problem. Beame \etal~and Bhattacharya \etal~gave algorithms to estimate the number of edges and triangles in a graph using queries to the {\sc Bipartite Independent Set} ({\sc BIS}) and the {\sc Tripartite Independent Set} ({\sc TIS}) oracles, respectively. We generalize the earlier works by estimating the number of hyperedges using a query oracle, known as the {\bf Generalized $d$-partite independent set oracle ({\sc GPIS})}, that takes $d$ (non-empty) pairwise disjoint subsets of vertices $A_1,\ldots,A_d \subseteq U({\cal H})$ as input, and answers whether there exists a hyperedge in ${\cal H}$ having (exactly) one ver...

We study an extension of F O 2 [<], first-order logic interpreted in finite words, in which formulas are restricted to use only two variables. We adjoin to this language two-variable atomic formulas that say, 'the letter a appears between positions x and y'. This is, in a sense, the simplest property that is not expressible using only two variables. We present several logics, both first-order and temporal, that have the same expressive power, and find matching lower and upper bounds for the complexity of satisfiability for each of these formulations. We also give an effective necessary condition, in terms of the syntactic monoid of a regular language, for a property to be expressible in this logic. We show that this condition is also sufficient for words over a two-letter alphabet. This algebraic analysis allows us us to prove, among other things, that our new logic has strictly less expressive power than full first-order logic F O[<].

The main result gives a sufficient condition for a class K of finite dimensional cylindric algebras to have the property that not every epimorphism in K is surjective. In particular, not all epimorphisms are surjective in the classes CA n of n-dimensional cylindric algebras and the class of representable algebras in CA n for finite n > 1, solving Problem 10 of [28] for finite n. By a result of Németi, this shows that the Beth-definability property fails for the finite-variable fragments of first order logic as long as the number n of variables available is > 1 and we allow models of size ≥ n + 2, but holds if we allow only models of size ≤ n + 1. We also use our results in the present paper to prove that several results in the literature concerning injective algebras and definability of polyadic operations in CA n are best possible. We raise several open problems.

Classes of graphs with bounded expansion are a generalization of both proper minor closed classes and degree bounded classes. Such classes are based on a new invariant, the greatest reduced average density (grad) of G with rank r, ∇r(G). These classes are also characterized by the existence of several partition results such as the existence of low tree-width and low tree-depth colorings [18][17]. These results lead to several new linear time algorithms, such as an algorithm for counting all the isomorphs of a fixed graph in an input graph or an algorithm for checking whether there exists a subset of vertices of a priori bounded size such that the subgraph induced by this subset satisfies some arbirtrary but fixed first order sentence. We also show that for fixed p, computing the distances between two vertices up to distance p may be performed in constant time per query after a linear time preprocessing. We also show, extending several earlier results, that a class of graphs has sublinear separators if it has sub-exponential expansion. This result result is best possible in general.

We study generalized quantifiers on finite structures.With every function $$f$$ : ? ? ?we associate a quantifier Q $$_{\text{f}} $$ by letting Q $$_{\text{f}} $$ x? say “there are at least $$_{\text{f}} $$ (n) elementsx satisfying ?, where n is the sizeof the universe.” This is the general form ofwhat is known as a monotone quantifier of type .We study

Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen. Sofern die Verfasser die Dokumente unter Open-Content-Lizenzen (insbesondere CC-Lizenzen) zur Verfügung gestellt haben sollten, gelten abweichend von diesen Nutzungsbedingungen die in der dort genannten Lizenz gewährten Nutzungsrechte. Terms of use: Documents in EconStor may be saved and copied for your personal and scholarly purposes. You are not to copy documents for public or commercial purposes, to exhibit the documents publicly, to make them publicly available on the internet, or to distribute or otherwise use the documents in public. If the documents have been made available under an Open Content Licence (especially Creative Commons Licences), you may exercise further usage rights as specified in the indicated licence.

It is well known that monadic second-order logic with linear order captures exactly regular languages. On the other hand, if addition is allowed, then J.F.Lynch has proved that existential monadic secondorder logic captures at least all the languages in NTIME(n), and then expresses some NP-complete languages (e.g. knapsack problem). It seems that most combinatorial NP-complete problems (e.g. traveling salesman, colorability of a graph) do not belong to NTIME(n). But it has been proved that they do belong to NLIN (the similar class for RAM's). In the present paper, we prove that existentia] monadic second-order logic with addition captures the class NLIN, so enlarging considerably the set of natural problems expressible in this logic. Moreover, we also prove that this logic still captures NLIN even if first-order part of the second-order formulas is required to be V*3*, so improving the recent similar result of a.g.Lynch about NTIME(n).

It is well known that monadic second-order logic with linear order captures exactly regular languages. On the other hand, if addition is allowed, then J.F.Lynch has proved that existential monadic secondorder logic captures at least all the languages in NTIME(n), and then expresses some NP-complete languages (e.g. knapsack problem). It seems that most combinatorial NP-complete problems (e.g. traveling salesman, colorability of a graph) do not belong to NTIME(n). But it has been proved that they do belong to NLIN (the similar class for RAM's). In the present paper, we prove that existentia] monadic second-order logic with addition captures the class NLIN, so enlarging considerably the set of natural problems expressible in this logic. Moreover, we also prove that this logic still captures NLIN even if first-order part of the second-order formulas is required to be V*3*, so improving the recent similar result of a.g.Lynch about NTIME(n).

We examine the following version of a classic combinatorial search problem introduced by R\&#39;enyi: Given a finite set $X$ of $n$ elements we want to identify an unknown subset $Y \subset X$ of exactly $d$ elements by testing, by as few as possible subsets $A$ of $X$, whether $A$ contains an element of $Y$ or not. We are primarily concerned with the model where the family of test sets is specified in advance (non-adaptive) and each test set is of size at most a given $k$. Our main results are asymptotically sharp bounds on the minimum number of tests necessary for fixed $d$ and $k$ and for $n$ tending to infinity.

A universal schema for diagonalization was popularized by N. S. Yanofsky (2003) in which the existence of a (diagonolized-out and contradictory) object implies the existence of a fixed-point for a certain function. It was shown that many self-referential paradoxes and diagonally proved theorems can fit in that schema. Here, we fit more theorems in the universal schema of diagonalization, such as Euclid's theorem on the infinitude of the primes and new proofs of G. Boolos (1997) for Cantor's theorem on the non-equinumerosity of a set with its powerset. Then, in Linear Temporal Logic, we show the non-existence of a fixed-point in this logic whose proof resembles the argument of Yablo's paradox. Thus, Yablo's paradox turns for the first time into a genuine mathematico-logical theorem in the framework of Linear Temporal Logic. Again the diagonal schema of the paper is used in this proof; and also it is shown that G. Priest's inclosure schema (1997) can fit in our universal diagonal/fixed-point schema. We also show the existence of dominating (Ackermann-like) functions (which dominate a given countable set of functions-like primitive recursives) using the schema.

Earlier papers [BV22, BV23b, BV23a] introduced the notions of a core and an index of a relation (an index being a special case of a core). A limited form of the axiom of choice was postulated-specifically that all partial equivalence relations (pers) have an index-and the consequences of adding the axiom to axiom systems for point-free reasoning were explored. In this paper, we define a partial ordering on relations, which we call the thins ordering. We show that our axiom of choice is equivalent to the property that core relations are the minimal elements of the thins ordering. We also characterise the relations that are maximal with respect to the thins ordering. Apart from our axiom of choice, the axiom system we employ is paired to a bare minimum and admits many models other than concrete relations-we do not assume, for example, the existence of complements; in the case of concrete relations, the theorem is that the maximal elements of the thins ordering are the empty relation and the equivalence relations. This and other properties of thins provide further evidence that our axiom of choice is a desirable means of strengthening point-free reasoning on relations.

L'accès aux archives de la collection « Astérisque » (http://smf4.emath.fr/ Publications/Asterisque/) implique l'accord avec les conditions générales d'utilisation (http://www.numdam.org/conditions). Toute utilisation commerciale ou impression systématique est constitutive d'une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/

We show that if a complexity classC is closed downward under polynomial- time majority truth-table reductions ( p mtt), then practically every other \polynomial&quot; closure property it enjoys is inherited by the corresponding bounded 2-sided error class BP(C). For instance, the Arthur-Merlin game class AM (Bab85) enjoys practically every closure property of NP. Our main lemma shows that for any

We prove that the Lambek syntactic calculus allowing empty premises is complete with respect to the class of all free monoid models (i. e., the class of all string models, allowing the empty string). 3 3>i W* = (U ti^A)) Π W* = U (^-(A) Π W;). Note that ti ^A) Π W* = j>i J>i iϋ»(A) for any j > i (according to Lemma 3.2). Now \J (wj(A) Π W*) = 3>î A =^A. 4 A simple quasimodel Definition. We define the non-negative count # as the following mapping from types to non-negative integers. #(A B) ^) , if L* h The non-negative count of a sequence of types is defined in the natural way. By definition, #(Λ) ^ 0. Lemma 4.1 For any type A its non-negative satisfies the inequalities 0 < Free monoid completeness of the Lambek calculus allowing empty premises 177 PROOF. Induction on ||A||.

We describe a simple computing technique for the tournament choice problem. It rests upon relational modeling and uses the BDD-based computer system RelView for the evaluation of the relation-algebraic expressions that specify the solutions and for the visualization of the computed results. The Copeland set can immediately be identified using RelView's labeling feature. Relation-algebraic specifications of the Condorcet non-losers, the Schwartz set, the top cycle, the uncovered set, the minimal covering set, the Banks set, and the tournament equilibrium set are delivered. We present an example of a tournament on a small set of alternatives, for which the above choice sets are computed and visualized via RelView. The technique described in this paper is very flexible and especially appropriate for prototyping and experimentation, and as such very instructive for educational purposes. It can easily be applied to other problems of social choice and game theory.

We consider a group of individuals who face a binary collective decision. Each group member holds some private information, and all agree about what decision should be taken in each state of nature. However, the state is unknown, and members can differ in their valuations of the two types of mistakes that might occur, and in their prior beliefs about the true state. For a slightly randomized majority rule, we show that informative voting by all voters is the unique Nash equilibrium, that this equilibrium is strict, and that the Condorcet asymptotic efficiency result holds in this setting.

Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen. Sofern die Verfasser die Dokumente unter Open-Content-Lizenzen (insbesondere CC-Lizenzen) zur Verfügung gestellt haben sollten, gelten abweichend von diesen Nutzungsbedingungen die in der dort genannten Lizenz gewährten Nutzungsrechte. Terms of use: Documents in EconStor may be saved and copied for your personal and scholarly purposes. You are not to copy documents for public or commercial purposes, to exhibit the documents publicly, to make them publicly available on the internet, or to distribute or otherwise use the documents in public. If the documents have been made available under an Open Content Licence (especially Creative Commons Licences), you may exercise further usage rights as specified in the indicated licence.

In this thesis, we study small, yet important, circuit complexity classes within NC 1 , such as ACC 0 and TC 0. We also investigate the power of a closely related problem called Iterated to help me out. To me, Carol is the role model for every Graduate Secretary. I would like to thank Aduri Pavan, my Master thesis advisor, for his guidance and support during the initial stage of my research life. I would like to thank Xiaoyang Gu, one of my best friends, for bringing me into the world of Complexity Theory and for helping me solve many technical and non-technical problems over the years.

The aim of this paper is to use formal power series techniques to study the structure of small arithmetic complexity classes such as GapNC 1 and GapL. More precisely, we apply the Kleene closure of languages and the formal power series operations of inversion and root extraction to these complexity classes. We define a counting version of Kleene closure and show that it is intimately related to inversion and root extraction within GapNC 1 and GapL. We prove that Kleene closure, inversion, and root extraction are all hard operations in the following sense: There is a language in AC 0 for which inversion and root extraction are GapL-complete, and there is a finite set for which inversion and root extraction are GapNC 1-complete, with respect to appropriate reducibilities. The latter result raises the question of classifying finite languages so that their inverses fall within interesting subclasses of GapNC 1 , such as GapAC 0. We initiate work in this direction by classifying the complexity of the Kleene closure of finite languages. We formulate the problem in terms of finite monoids and relate its complexity to the internal structure of the monoid.