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Finite Model Theory
Finite model theory is a subarea of model theory. Model theory is the branch of logic which deals with the relation between a formal language (syntax) and its interpretations (semantics). Finite model theory is a restriction of model theory to interpretation (logic), interpretations on finite Structure (mathematical logic), structures, which have a finite universe. Since many central theorems of model theory do not hold when restricted to finite structures, finite model theory is quite different from model theory in its methods of proof. Central results of classical model theory that fail for finite structures under finite model theory include the compactness theorem, Gödel's completeness theorem, and the method of ultraproducts for first-order logic (FO). These invalidities all follow from Trakhtenbrot's theorem. While model theory has many applications to abstract algebra, mathematical algebra, finite model theory became an "unusually effective" instrument in computer science. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Model Theory
In mathematical logic, model theory is the study of the relationship between theory (mathematical logic), formal theories (a collection of Sentence (mathematical logic), sentences in a formal language expressing statements about a Structure (mathematical logic), mathematical structure), and their Structure (mathematical logic), models (those Structure (mathematical logic), structures in which the statements of the theory hold). The aspects investigated include the number and size of models of a theory, the relationship of different models to each other, and their interaction with the formal language itself. In particular, model theorists also investigate the sets that can be definable set, defined in a model of a theory, and the relationship of such definable sets to each other. As a separate discipline, model theory goes back to Alfred Tarski, who first used the term "Theory of Models" in publication in 1954. Since the 1970s, the subject has been shaped decisively by Saharon Shel ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Universal Vertex
In graph theory, a universal vertex is a Vertex (graph theory), vertex of an undirected graph that is adjacent to all other vertices of the graph. It may also be called a dominating vertex, as it forms a one-element dominating set in the graph. A graph that contains a universal vertex may be called a cone, and its universal vertex may be called the apex of the cone. This terminology should be distinguished from the unrelated usage of these words for universal quantifiers in the logic of graphs, and for apex graphs. Graphs that contain a universal vertex include the Star (graph theory), stars, trivially perfect graphs, and friendship graphs. For wheel graphs (the graphs of pyramids), and graphs of higher-dimensional pyramidal polytopes, the vertex at the apex of the pyramid is universal. When a graph contains a universal vertex, it is a cop-win graph, and almost all cop-win graphs contain a universal vertex. The number of labeled graphs containing a universal vertex can be counted ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Second-order Logic
In logic and mathematics, second-order logic is an extension of first-order logic, which itself is an extension of propositional logic. Second-order logic is in turn extended by higher-order logic and type theory. First-order logic quantifies only variables that range over individuals (elements of the domain of discourse); second-order logic, in addition, quantifies over relations. For example, the second-order sentence \forall P\,\forall x (Px \lor \neg Px) says that for every formula ''P'', and every individual ''x'', either ''Px'' is true or not(''Px'') is true (this is the law of excluded middle). Second-order logic also includes quantification over sets, functions, and other variables (see section below). Both first-order and second-order logic use the idea of a domain of discourse (often called simply the "domain" or the "universe"). The domain is a set over which individual elements may be quantified. Examples First-order logic can quantify over individuals, but no ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
PH (complexity)
In computational complexity theory, the polynomial hierarchy (sometimes called the polynomial-time hierarchy) is a hierarchy of complexity classes that generalize the classes NP and co-NP. Each class in the hierarchy is contained within PSPACE. The hierarchy can be defined using oracle machines or alternating Turing machines. It is a resource-bounded counterpart to the arithmetical hierarchy and analytical hierarchy from mathematical logic. The union of the classes in the hierarchy is denoted PH. Classes within the hierarchy have complete problems (with respect to polynomial-time reductions) that ask if quantified Boolean formulae hold, for formulae with restrictions on the quantifier order. It is known that equality between classes on the same level or consecutive levels in the hierarchy would imply a "collapse" of the hierarchy to that level. Definitions There are multiple equivalent definitions of the classes of the polynomial hierarchy. Oracle definition For the oracle de ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Complexity Class
In computational complexity theory, a complexity class is a set (mathematics), set of computational problems "of related resource-based computational complexity, complexity". The two most commonly analyzed resources are time complexity, time and space complexity, memory. In general, a complexity class is defined in terms of a type of computational problem, a model of computation, and a bounded resource like time complexity, time or space complexity, memory. In particular, most complexity classes consist of decision problems that are solvable with a Turing machine, and are differentiated by their time or space (memory) requirements. For instance, the class P (complexity), P is the set of decision problems solvable by a deterministic Turing machine in polynomial time. There are, however, many complexity classes defined in terms of other types of problems (e.g. Counting problem (complexity), counting problems and function problems) and using other models of computation (e.g. probabil ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Least Fixed-point Logic
In mathematical logic, fixed-point logics are extensions of classical predicate logic that have been introduced to express recursion. Their development has been motivated by descriptive complexity theory and their relationship to database query languages, in particular to Datalog. Least fixed-point logic was first studied systematically by Yiannis N. Moschovakis in 1974, and it was introduced to computer scientists in 1979, when Alfred Aho and Jeffrey Ullman suggested fixed-point logic as an expressive database query language. Partial fixed-point logic For a relational signature ''X'', FO FP''X'') is the set of formulas formed from ''X'' using first-order connectives and predicates, second-order variables as well as a partial fixed point operator \operatorname used to form formulas of the form operatorname_ \varphivec, where P is a second-order variable, \vec a tuple of first-order variables, \vec a tuple of terms and the lengths of \vec and \vec coincide with the arity of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
PSPACE-complete
In computational complexity theory, a decision problem is PSPACE-complete if it can be solved using an amount of memory that is polynomial in the input length (PSPACE, polynomial space) and if every other problem that can be solved in polynomial space can be Polynomial-time reduction, transformed to it in polynomial time. The problems that are PSPACE-complete can be thought of as the hardest problems in PSPACE, the class of decision problems solvable in polynomial space, because a solution to any one such problem could easily be used to solve any other problem in PSPACE. Problems known to be PSPACE-complete include determining properties of regular expressions and context-sensitive grammars, determining the truth of quantified Boolean formula problem, quantified Boolean formulas, step-by-step changes between solutions of combinatorial optimization problems, and many puzzles and games. Theory A problem is defined to be PSPACE-complete if it can be solved using a polynomial amount o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Almost Always
In probability theory, an event is said to happen almost surely (sometimes abbreviated as a.s.) if it happens with probability 1 (with respect to the probability measure). In other words, the set of outcomes on which the event does not occur has probability 0, even though the set might not be empty. The concept is analogous to the concept of "almost everywhere" in measure theory. In probability experiments on a finite sample space with a non-zero probability for each outcome, there is no difference between ''almost surely'' and ''surely'' (since having a probability of 1 entails including all the sample points); however, this distinction becomes important when the sample space is an infinite set, because an infinite set can have non-empty subsets of probability 0. Some examples of the use of this concept include the strong and uniform versions of the law of large numbers, the continuity of the paths of Brownian motion, and the infinite monkey theorem. The terms almost certainly ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ehrenfeucht–Fraïssé Game
In the mathematical discipline of model theory, the Ehrenfeucht–Fraïssé game (also called back-and-forth games) is a technique based on game semantics for determining whether two structures are elementarily equivalent. The main application of Ehrenfeucht–Fraïssé games is in proving the inexpressibility of certain properties in first-order logic. Indeed, Ehrenfeucht–Fraïssé games provide a complete methodology for proving inexpressibility results for first-order logic. In this role, these games are of particular importance in finite model theory and its applications in computer science (specifically computer aided verification and database theory), since Ehrenfeucht–Fraïssé games are one of the few techniques from model theory that remain valid in the context of finite models. Other widely used techniques for proving inexpressibility results, such as the compactness theorem, do not work in finite models. Ehrenfeucht–Fraïssé-like games can also be defined for oth ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Prenex Normal Form
A formula of the predicate calculus is in prenex normal form (PNF) if it is written as a string of quantifiers and bound variables, called the prefix, followed by a quantifier-free part, called the matrix. Together with the normal forms in propositional logic (e.g. disjunctive normal form or conjunctive normal form), it provides a canonical normal form useful in automated theorem proving. Every formula in classical logic is logically equivalent to a formula in prenex normal form. For example, if \phi(y), \psi(z), and \rho(x) are quantifier-free formulas with the free variables shown then :\forall x \exists y \forall z (\phi(y) \lor (\psi(z) \rightarrow \rho(x))) is in prenex normal form with matrix \phi(y) \lor (\psi(z) \rightarrow \rho(x)), while :\forall x ((\exists y \phi(y)) \lor ((\exists z \psi(z) ) \rightarrow \rho(x))) is logically equivalent but not in prenex normal form. Conversion to prenex form Every first-order formula is logically equivalent (in classical logi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quantifier (logic)
In logic, a quantifier is an operator that specifies how many individuals in the domain of discourse satisfy an open formula. For instance, the universal quantifier \forall in the first-order formula \forall x P(x) expresses that everything in the domain satisfies the property denoted by P. On the other hand, the existential quantifier \exists in the formula \exists x P(x) expresses that there exists something in the domain which satisfies that property. A formula where a quantifier takes widest scope is called a quantified formula. A quantified formula must contain a bound variable and a subformula specifying a property of the referent of that variable. The most commonly used quantifiers are \forall and \exists. These quantifiers are standardly defined as duals; in classical logic: each can be defined in terms of the other using negation. They can also be used to define more complex quantifiers, as in the formula \neg \exists x P(x) which expresses that nothing has ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quantifier Rank
In mathematical logic, the quantifier rank of a Formula (mathematical logic), formula is the depth of nesting of its Quantifier (logic), quantifiers. It plays an essential role in model theory. The quantifier rank is a property of the formula itself (i.e. the expression in a language). Thus two Logical equivalence, logically equivalent formulae can have different quantifier ranks, when they express the same thing in different ways. Definition In first-order logic Let \varphi be a First-order logic, first-order formula. The quantifier rank of \varphi, written \operatorname(\varphi), is defined as: * \operatorname(\varphi) = 0, if \varphi is atomic. * \operatorname(\varphi_1 \land \varphi_2) = \operatorname(\varphi_1 \lor \varphi_2) = \max(\operatorname(\varphi_1), \operatorname(\varphi_2)). * \operatorname(\lnot \varphi) = \operatorname(\varphi). * \operatorname(\exists_x \varphi) = \operatorname(\varphi) + 1. * \operatorname(\forall_x \varphi) = \operatorname(\varphi) + 1. Re ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
