Citizendia
Your Ad Here

In logic, especially mathematical logic, a signature lists and describes the non-logical symbols of a formal language. Logic is the study of the principles of valid demonstration and Inference. Mathematical logic is a subfield of Logic and Mathematics with close connections to Computer science and Philosophical logic. Non-logical symbol is a technical term used in Logic In Logic, the non-logical symbols (sometimes also called non-logical constants) of a A formal language is a set of words, ie finite strings of letters, or symbols. In universal algebra, a signature lists the operations that characterize an algebraic structure. Universal algebra (sometimes called general algebra) is the field of Mathematics that studies Algebraic structures themselves not examples ("models" In Algebra, a branch of Pure mathematics, an algebraic structure consists of one or more sets closed under one or more operations, In model theory, signatures are used for both purposes. In Mathematics, model theory is the study of (classes of mathematical structures such as groups, Fields graphs or even models

Signatures play the same role in mathematics as type signatures in computer programming. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and Type signature is a term that is used in computer programming They are rarely made explicit in more philosophical treatments of logic.

Contents

Definition

Formally, a signature can be defined as a triple σ = (Sfunc, Srel, ar), where Sfunc and Srel are disjoint sets not containing any other basic logical symbols, called respectively

and a function ar: Sfunc \cup Srel → \mathbb N_0 which assigns a non-negative integer called arity to every function or relation symbol. In Logic, Mathematics, and Computer science, the arity (synonyms include type, adicity, and rank) of a function A function or relation symbol is called n-ary if its arity is n. A nullary function symbol is called a constant symbol.

A signature with no function symbols is called a relational signature, and a signature with no relation symbols is called an algebraic signature. A finite signature is a signature such that Sfunc and Srel are finite. More generally, the cardinality of a signature σ = (Sfunc, Srel) is defined as |σ| = |Sfunc| + |Srel|.

Other conventions

In universal algebra the word type or similarity type is often used as a synonym for "signature". In model theory, a signature σ is often called vocabulary, or identified with the (first-order) language L to which it provides the non-logical symbols. First-order logic (FOL is a formal Deductive system used in mathematics philosophy linguistics and computer science Non-logical symbol is a technical term used in Logic In Logic, the non-logical symbols (sometimes also called non-logical constants) of a (However, |L| = |σ| + ω. )

As the formal definition is inconvenient for everyday use, the definition of a specific signature is often abbreviated in an informal way, as in:

"The standard signature for abelian groups is σ = (+,–,0), where – is a unary operator. "

Sometimes an algebraic signature is regarded as just a list of arities, as in:

"The similarity type for abelian groups is σ = (2,1,0). "

Formally this would define the function symbols of the signature as something like f0  (binary), f1 (unary) and f2 (nullary), but in reality the usual names are used even in connection with this convention.

In mathematical logic, very often symbols are not allowed to be nullary, so that constant symbols must be treated separately rather than as nullary function symbols. Mathematical logic is a subfield of Logic and Mathematics with close connections to Computer science and Philosophical logic. They form a set Sconst disjoint from Sfunc, on which the arity function ar is not defined. However, this only serves to complicate matters, especially in proofs by induction over the structure of a formula, where an additional case must be considered. Any nullary relation symbol, which is also not allowed under such a definition, can be emulated by a unary relation symbol together with a sentence expressing that its value is the same for all elements. This translation fails only for empty structures (which are often excluded by convention). If nullary symbols are allowed, then every formula of propositional logic is also a formula of first-order logic. This is a technical mathematical article about the area of mathematical logic variously known as "propositional calculus" or "propositional logic" First-order logic (FOL is a formal Deductive system used in mathematics philosophy linguistics and computer science

Use of signatures in logic and algebra

In the context of first-order logic, the symbols in a signature are also known as the non-logical symbols, because together with the logical symbols they form the underlying alphabet over which two formal languages are inductively defined: The set of terms over the signature and the set of (well-formed) formulas over the signature. First-order logic (FOL is a formal Deductive system used in mathematics philosophy linguistics and computer science Non-logical symbol is a technical term used in Logic In Logic, the non-logical symbols (sometimes also called non-logical constants) of a A formal language is a set of words, ie finite strings of letters, or symbols.

In a structure, an interpretation ties the function and relation symbols to mathematical objects that justify their names: The interpretation of an n-ary function symbol f in a structure A with domain A is a function fA: An → A, and the interpretation of an n-ary relation symbol is a relation RA ⊆ An. In Universal algebra and in Model theory, a structure consists of an underlying set along with a collection of Finitary functions and relations Here An = A × A × . . . × A denotes the n-fold cartesian product of the domain A with itself, and so f is in fact an n-ary function, and R an n-ary relation. Cartesian square redirects here For Cartesian squares in Category theory, see Cartesian square (category theory.

Many-sorted signatures

For many-sorted logic and for many-sorted structures signatures must encode information about the sorts. In Universal algebra and in Model theory, a structure consists of an underlying set along with a collection of Finitary functions and relations The most straightforward way of doing this is via symbol types that play the role of generalized arities. [1]

Symbol types

Let S be a set (of sorts) not containing the symbols × or →.

The symbol types over S are certain words over the alphabet S \cup {×, →}: the relational symbol types s1 × . . . × sn, and the functional symbol types s1 × . . . × sn→s', for non-negative integers n and s1,s2,. . . ,sn,s' \in S. (For n = 0, the expression s1 × . . . × sn denotes the empty word. )

Signature

A (many-sorted) signature is a triple (S, P, type) consisting of

Notes

  1. ^ Many-Sorted Logic, the first chapter in Lecture notes on Decision Procedures, written by Calogero G. Zarba.

See also

References

External links

© 2009 citizendia.org; parts available under the terms of GNU Free Documentation License, from http://en.wikipedia.org
 Dapyx Software network: MP3 Explorer | Ebook Manager | Zenithic