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Set Constraint
In mathematics and theoretical computer science, a set constraint is an equation or an inequation between sets of terms. Similar to systems of ( in)equations between numbers, methods are studied for solving systems of set constraints. Different approaches admit different operators (like "∪", "∩", "\", and function application)If ''f'' is an ''n''-ary function symbol admitted in a term, then "''f''(''E''1,...,''E''''n'')" is a set expression denoting the set , where ''E''1,...,''E''''n'' are set expressions in turn. on sets and different (in)equation relations (like "=", "⊆", and "⊈") between set expressions. Systems of set constraints are useful to describe (in particular infinite) sets of ground terms.This is similar to describing e.g. a rational number as a solution to an equation ''a''⋅''x'' + ''b'' = 0, with integer coefficients ''a'', ''b''. They arise in program analysis, abstract interpretation, and type inference. Relation to regular tree grammars Each regular ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, andin case of abstraction from naturesome basic properties that are considered true starting points of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Theoretical Computer Science
Theoretical computer science (TCS) is a subset of general computer science and mathematics that focuses on mathematical aspects of computer science such as the theory of computation, lambda calculus, and type theory. It is difficult to circumscribe the theoretical areas precisely. The Association for Computing Machinery, ACM's ACM SIGACT, Special Interest Group on Algorithms and Computation Theory (SIGACT) provides the following description: History While logical inference and mathematical proof had existed previously, in 1931 Kurt Gödel proved with his incompleteness theorem that there are fundamental limitations on what statements could be proved or disproved. Information theory was added to the field with a 1948 mathematical theory of communication by Claude Shannon. In the same decade, Donald Hebb introduced a mathematical model of Hebbian learning, learning in the brain. With mounting biological data supporting this hypothesis with some modification, the fields of n ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Term (logic)
In mathematical logic, a term denotes a mathematical object while a formula denotes a mathematical fact. In particular, terms appear as components of a formula. This is analogous to natural language, where a noun phrase refers to an object and a whole sentence refers to a fact. A first-order term is recursively constructed from constant symbols, variables and function symbols. An expression formed by applying a predicate symbol to an appropriate number of terms is called an atomic formula, which evaluates to true or false in bivalent logics, given an interpretation. For example, is a term built from the constant 1, the variable , and the binary function symbols and ; it is part of the atomic formula which evaluates to true for each real-numbered value of . Besides in logic, terms play important roles in universal algebra, and rewriting systems. Formal definition Given a set ''V'' of variable symbols, a set ''C'' of constant symbols and sets ''F''''n'' of ''n''-ary fu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Inequation
In mathematics, an inequation is a statement that an inequality holds between two values. It is usually written in the form of a pair of expressions denoting the values in question, with a relational sign between them indicating the specific inequality relation. Some examples of inequations are: * a 1 * x \neq 0 In some cases, the term "inequation" can be considered synonymous to the term "inequality", while in other cases, an inequation is reserved only for statements whose inequality relation is "not equal to" (≠). Chains of inequations A shorthand notation is used for the conjunction of several inequations involving common expressions, by chaining them together. For example, the chain :0 \leq a < b \leq 1 is shorthand for : which also implies that and . In rare cases, chains without such implications about distant terms are used. For example |
Equation Solving
In mathematics, to solve an equation is to find its solutions, which are the values (numbers, functions, sets, etc.) that fulfill the condition stated by the equation, consisting generally of two expressions related by an equals sign. When seeking a solution, one or more variables are designated as ''unknowns''. A solution is an assignment of values to the unknown variables that makes the equality in the equation true. In other words, a solution is a value or a collection of values (one for each unknown) such that, when substituted for the unknowns, the equation becomes an equality. A solution of an equation is often called a root of the equation, particularly but not only for polynomial equations. The set of all solutions of an equation is its solution set. An equation may be solved either numerically or symbolically. Solving an equation ''numerically'' means that only numbers are admitted as solutions. Solving an equation ''symbolically'' means that expressions can be use ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ground Term
In mathematical logic, a ground term of a formal system is a term that does not contain any variables. Similarly, a ground formula is a formula that does not contain any variables. In first-order logic with identity, the sentence Q(a) \lor P(b) is a ground formula, with a and b being constant symbols. A ground expression is a ground term or ground formula. Examples Consider the following expressions in first order logic over a signature containing the constant symbols 0 and 1 for the numbers 0 and 1, respectively, a unary function symbol s for the successor function and a binary function symbol + for addition. * s(0), s(s(0)), s(s(s(0))), \ldots are ground terms; * 0 + 1, \; 0 + 1 + 1, \ldots are ground terms; * 0+s(0), \; s(0)+ s(0), \; s(0)+s(s(0))+0 are ground terms; * x + s(1) and s(x) are terms, but not ground terms; * s(0) = 1 and 0 + 0 = 0 are ground formulae. Formal definitions What follows is a formal definition for first-order languages. Let a first-order language b ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rational Number
In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ). The set of all rational numbers, also referred to as "the rationals", the field of rationals or the field of rational numbers is usually denoted by boldface , or blackboard bold \mathbb. A rational number is a real number. The real numbers that are rational are those whose decimal expansion either terminates after a finite number of digits (example: ), or eventually begins to repeat the same finite sequence of digits over and over (example: ). This statement is true not only in base 10, but also in every other integer base, such as the binary and hexadecimal ones (see ). A real number that is not rational is called irrational. Irrational numbers include , , , and . Since the set of rational numbers is countable, and the set of real numbers is uncountable ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Integer
An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign (−1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language of mathematics, the set of integers is often denoted by the boldface or blackboard bold \mathbb. The set of natural numbers \mathbb is a subset of \mathbb, which in turn is a subset of the set of all rational numbers \mathbb, itself a subset of the real numbers \mathbb. Like the natural numbers, \mathbb is countably infinite. An integer may be regarded as a real number that can be written without a fractional component. For example, 21, 4, 0, and −2048 are integers, while 9.75, , and are not. The integers form the smallest group and the smallest ring containing the natural numbers. In algebraic number theory, the integers are sometimes qualified as rational integers to distinguish them from the more general algebraic integers ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Abstract Interpretation
In computer science, abstract interpretation is a theory of sound approximation of the semantics of computer programs, based on monotonic functions over ordered sets, especially lattices. It can be viewed as a partial execution of a computer program which gains information about its semantics (e.g., control-flow, data-flow) without performing all the calculations. Its main concrete application is formal static analysis, the automatic extraction of information about the possible executions of computer programs; such analyses have two main usages: * inside compilers, to analyse programs to decide whether certain optimizations or transformations are applicable; * for debugging or even the certification of programs against classes of bugs. Abstract interpretation was formalized by the French computer scientist working couple Patrick Cousot and Radhia Cousot in the late 1970s. Intuition This section illustrates abstract interpretation by means of real-world, non-computing example ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Type Inference
Type inference refers to the automatic detection of the type of an expression in a formal language. These include programming languages and mathematical type systems, but also natural languages in some branches of computer science and linguistics. Nontechnical explanation Types in a most general view can be associated to a designated use suggesting and restricting the activities possible for an object of that type. Many nouns in language specify such uses. For instance, the word leash indicates a different use than the word line. Calling something a table indicates another designation than calling it firewood, though it might be materially the same thing. While their material properties make things usable for some purposes, they are also subject of particular designations. This is especially the case in abstract fields, namely mathematics and computer science, where the material is finally only bits or formulas. To exclude unwanted, but materially possible uses, the concept of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Regular Tree Grammar
In theoretical computer science and formal language theory, a regular tree grammar is a formal grammar that describes a set of directed trees, or terms. A regular word grammar can be seen as a special kind of regular tree grammar, describing a set of single-path trees. Definition A regular tree grammar ''G'' is defined by the tuple ''G'' = (''N'', Σ, ''Z'', ''P''), where * ''N'' is a finite set of nonterminals, * Σ is a ranked alphabet (i.e., an alphabet whose symbols have an associated arity) disjoint from ''N'', * ''Z'' is the starting nonterminal, with , and * ''P'' is a finite set of productions of the form ''A'' → ''t'', with , and , where ''T''Σ(''N'') is the associated term algebra, i.e. the set of all trees composed from symbols in according to their arities, where nonterminals are considered nullary. Derivation of trees The grammar ''G'' implicitly defines a set of trees: any tree that can be derived from ''Z'' using the rule set ''P'' is said to be describe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
