|
Function Letter
In mathematical logic, an uninterpreted function or function symbol is one that has no other property than its name and ''Arity, n-ary'' form. Function symbols are used, together with constants and variables, to form term (logic), terms. The theory of uninterpreted functions is also sometimes called the free theory, because it is freely generated, and thus a free object, or the empty theory, being the theory (mathematical logic), theory having an empty set of sentence (mathematical logic), sentences (in analogy to an initial algebra). Theories with a non-empty set of equations are known as equational theory, equational theories. The satisfiability problem for free theories is solved by syntactic unification; algorithms for the latter are used by interpreters for various computer languages, such as Prolog. Syntactic unification is also used in algorithms for the satisfiability problem for certain other equational theories, see Unification (computer science). Example As an example ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
|
|
Mathematical Logic
Mathematical logic is the study of Logic#Formal logic, formal logic within mathematics. Major subareas include model theory, proof theory, set theory, and recursion theory (also known as computability theory). Research in mathematical logic commonly addresses the mathematical properties of formal systems of logic such as their expressive or deductive power. However, it can also include uses of logic to characterize correct mathematical reasoning or to establish foundations of mathematics. Since its inception, mathematical logic has both contributed to and been motivated by the study of foundations of mathematics. This study began in the late 19th century with the development of axiomatic frameworks for geometry, arithmetic, and Mathematical analysis, analysis. In the early 20th century it was shaped by David Hilbert's Hilbert's program, program to prove the consistency of foundational theories. Results of Kurt Gödel, Gerhard Gentzen, and others provided partial resolution to th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
|
|
SMT-LIB
In computer science and mathematical logic, satisfiability modulo theories (SMT) is the problem of determining whether a mathematical formula is satisfiable. It generalizes the Boolean satisfiability problem (SAT) to more complex formulas involving real numbers, integers, and/or various data structures such as lists, arrays, bit vectors, and strings. The name is derived from the fact that these expressions are interpreted within ("modulo") a certain formal theory in first-order logic with equality (often disallowing quantifiers). SMT solvers are tools that aim to solve the SMT problem for a practical subset of inputs. SMT solvers such as Z3 and cvc5 have been used as a building block for a wide range of applications across computer science, including in automated theorem proving, program analysis, program verification, and software testing. Since Boolean satisfiability is already NP-complete, the SMT problem is typically NP-hard, and for many theories it is undecidable. R ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
|
|
Term Algebra
Term may refer to: Language *Terminology, context-specific nouns or compound words **Technical term (or ''term of art''), used by specialists in a field ***Scientific terminology, used by scientists *Term (argumentation), part of an argument in debate theory Law and finance *Contractual term, a provision in a contract **Credit repayment terms **Payment terms, "net ''D''" on a trade invoice **Purchase order#Legal, Purchase order, invoice terms more generally *Term life insurance Lengths of time *Academic term, part of a year at school or university *Term of office, a set period a person serves in an elected office *Term of patent, the period of enforcement of patent rights *Term of a pregnancy *Prison sentence Mathematics and physics *Term (logic), a component of a logical or mathematical expression (not to be confused with term logic, or Aristotelian logic) **Ground term, a term with no variables *Term (arithmetic), or addend, an operand to the addition operator **Term of a summ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
|
|
Initial Algebra
In mathematics, an initial algebra is an initial object in the Category (mathematics), category of F-algebra, -algebras for a given endofunctor . This initiality provides a general framework for mathematical induction, induction and recursion. Examples Functor Consider the endofunctor , i.e. sending to , where is a one-point (Singleton (mathematics), singleton) Set (mathematics), set, a terminal object in the category. An algebra for this endofunctor is a set (called the ''carrier'' of the algebra) together with a Function (mathematics), function . Defining such a function amounts to defining a point and a function . Define : \begin \operatorname \colon 1 &\longrightarrow\mathbf \\ * &\longmapsto 0 \end and : \begin \operatorname\colon \mathbf&\longrightarrow\mathbf \\ n &\longmapsto n + 1. \end Then the set of natural numbers together with the function is an initial -algebra. The initiality (the universal property for this case) is not hard to establish; ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
|
|
Algebraic Data Type
In computer programming, especially functional programming and type theory, an algebraic data type (ADT) is a kind of composite data type, i.e., a data type formed by combining other types. Two common classes of algebraic types are product types (i.e., tuples, and records) and sum types (i.e., tagged or disjoint unions, coproduct types or ''variant types''). The values of a product type typically contain several values, called ''fields''. All values of that type have the same combination of field types. The set of all possible values of a product type is the set-theoretic product, i.e., the Cartesian product, of the sets of all possible values of its field types. The values of a sum type are typically grouped into several classes, called ''variants''. A value of a variant type is usually created with a quasi-functional entity called a ''constructor''. Each variant has its own constructor, which takes a specified number of arguments with specified types. The set of all po ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
|
|
Satisfiability Modulo Theories
In computer science and mathematical logic, satisfiability modulo theories (SMT) is the problem of determining whether a mathematical formula is satisfiable. It generalizes the Boolean satisfiability problem (SAT) to more complex formulas involving real numbers, integers, and/or various data structures such as lists, arrays, bit vectors, and strings. The name is derived from the fact that these expressions are interpreted within ("modulo") a certain formal theory in first-order logic with equality (often disallowing quantifiers). SMT solvers are tools that aim to solve the SMT problem for a practical subset of inputs. SMT solvers such as Z3 and cvc5 have been used as a building block for a wide range of applications across computer science, including in automated theorem proving, program analysis, program verification, and software testing. Since Boolean satisfiability is already NP-complete, the SMT problem is typically NP-hard, and for many theories it is undecidable. Re ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
|
|
Congruence Closure
In mathematics, a subset of a given set is closed under an operation on the larger set if performing that operation on members of the subset always produces a member of that subset. For example, the natural numbers are closed under addition, but not under subtraction: is not a natural number, although both 1 and 2 are. Similarly, a subset is said to be closed under a ''collection'' of operations if it is closed under each of the operations individually. The closure of a subset is the result of a closure operator applied to the subset. The ''closure'' of a subset under some operations is the smallest superset that is closed under these operations. It is often called the ''span'' (for example linear span) or the ''generated set''. Definitions Let be a set equipped with one or several methods for producing elements of from other elements of . Operations and (partial) multivariate function are examples of such methods. If is a topological space, the limit of a sequence of eleme ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
|
|
Common Subexpression
In compiler theory, common subexpression elimination (CSE) is a compiler optimization that searches for instances of identical expressions (i.e., they all evaluate to the same value), and analyzes whether it is worthwhile replacing them with a single variable holding the computed value. Example In the following code: a = b * c + g; d = b * c * e; it may be worth transforming the code to: tmp = b * c; a = tmp + g; d = tmp * e; if the cost of storing and retrieving tmp is less than the cost of calculating b * c an extra time. Principle The possibility to perform CSE is based on available expression analysis (a data flow analysis). An expression b*c is available at a point ''p'' in a program if: * every path from the initial node to p evaluates b*c before reaching ''p'', * and there are no assignments to b or c after the evaluation but before ''p''. The cost/benefit analysis performed by an optimizer will calculate whether the cost of the store to tmp is less than the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
|
|
Decision Problem
In computability theory and computational complexity theory, a decision problem is a computational problem that can be posed as a yes–no question on a set of input values. An example of a decision problem is deciding whether a given natural number is prime. Another example is the problem, "given two numbers ''x'' and ''y'', does ''x'' evenly divide ''y''?" A decision procedure for a decision problem is an algorithmic method that answers the yes-no question on all inputs, and a decision problem is called decidable if there is a decision procedure for it. For example, the decision problem "given two numbers ''x'' and ''y'', does ''x'' evenly divide ''y''?" is decidable since there is a decision procedure called long division that gives the steps for determining whether ''x'' evenly divides ''y'' and the correct answer, ''YES'' or ''NO'', accordingly. Some of the most important problems in mathematics are undecidable, e.g. the halting problem. The field of computational ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
|
|
Signature (logic)
In logic, especially mathematical logic, a signature lists and describes the non-logical symbols of a formal language. In universal algebra, a signature lists the operations that characterize an algebraic structure. In model theory, signatures are used for both purposes. They are rarely made explicit in more philosophical treatments of logic. Definition Formally, a (single-sorted) signature can be defined as a 4-tuple \sigma = \left(S_, S_, S_, \operatorname\right), where S_ and S_ are disjoint sets not containing any other basic logical symbols, called respectively * '' function symbols'' (examples: +, \times), * ''s'' or '' predicates'' (examples: \,\leq, \, \in), * '' constant symbols'' (examples: 0, 1), and a function \operatorname : S_ \cup S_ \to \N which assigns a natural number called ''arity'' to every function or relation symbol. A function or relation symbol is called n-ary if its arity is n. Some authors define a nullary (0-ary) function symbol as ''constant symbol'', ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
|
|
Satisfiability Modulo Theories
In computer science and mathematical logic, satisfiability modulo theories (SMT) is the problem of determining whether a mathematical formula is satisfiable. It generalizes the Boolean satisfiability problem (SAT) to more complex formulas involving real numbers, integers, and/or various data structures such as lists, arrays, bit vectors, and strings. The name is derived from the fact that these expressions are interpreted within ("modulo") a certain formal theory in first-order logic with equality (often disallowing quantifiers). SMT solvers are tools that aim to solve the SMT problem for a practical subset of inputs. SMT solvers such as Z3 and cvc5 have been used as a building block for a wide range of applications across computer science, including in automated theorem proving, program analysis, program verification, and software testing. Since Boolean satisfiability is already NP-complete, the SMT problem is typically NP-hard, and for many theories it is undecidable. Re ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
|
 |
Unification (computer Science)
In logic and computer science, specifically automated reasoning, unification is an algorithmic process of solving equations between symbolic expression (mathematics), expressions, each of the form ''Left-hand side = Right-hand side''. For example, using ''x'',''y'',''z'' as variables, and taking ''f'' to be an uninterpreted function, the Singleton (mathematics), singleton equation set is a syntactic first-order unification problem that has the substitution as its only solution. Conventions differ on what values variables may assume and which expressions are considered equivalent. In first-order syntactic unification, variables range over first-order terms and equivalence is syntactic. This version of unification has a unique "best" answer and is used in logic programming and programming language type system implementation, especially in Hindley–Milner based type inference algorithms. In higher-order unification, possibly restricted to higher-order pattern unification, terms may ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |