Signature (logic)
Encyclopedia
In logic
, especially mathematical logic
, a signature lists and describes the non-logical symbol
s 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.
Signatures play the same role in mathematics
as type signature
s in computer programming
. They are rarely made explicit in more philosophical treatments of logic.
and a function ar: Sfunc
Srel → 
which assigns a non-negative integer called arity
to every function or relation symbol. A function or relation symbol is called n-ary if its arity is n. A nullary (0-ary) 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|.
As the formal definition is inconvenient for everyday use, the definition of a specific signature is often abbreviated in an informal way, as in:
Sometimes an algebraic signature is regarded as just a list of arities, as in:
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. 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
.
, 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 language
s are inductively defined: The set of terms over the signature and the set of (well-formed) formulas over the signature.
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. 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.
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
{×, →}: the relational symbol types s1 × ... × sn, and the functional symbol types s1 × ... × sn→s' , for non-negative integers n and s1,s2,...,sn,s' 
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
Logic
In philosophy, Logic is the formal systematic study of the principles of valid inference and correct reasoning. Logic is used in most intellectual activities, but is studied primarily in the disciplines of philosophy, mathematics, semantics, and computer science...
, especially mathematical logic
Mathematical logic
Mathematical logic is a subfield of mathematics with close connections to foundations of mathematics, theoretical computer science and philosophical logic. The field includes both the mathematical study of logic and the applications of formal logic to other areas of mathematics...
, a signature lists and describes the non-logical symbol
Non-logical symbol
In logic, the formal languages used to create expressions consist of symbols which can be broadly divided into constants and variables. The constants of a language can further be divided into logical symbols and non-logical symbols .The non-logical symbols of a language of first-order logic consist...
s of a formal language
Formal language
A formal language is a set of words—that is, finite strings of letters, symbols, or tokens that are defined in the language. The set from which these letters are taken is the alphabet over which the language is defined. A formal language is often defined by means of a formal grammar...
. In universal algebra
Universal algebra
Universal algebra is the field of mathematics that studies algebraic structures themselves, not examples of algebraic structures....
, a signature lists the operations that characterize an algebraic structure
Algebraic structure
In abstract algebra, an algebraic structure consists of one or more sets, called underlying sets or carriers or sorts, closed under one or more operations, satisfying some axioms. Abstract algebra is primarily the study of algebraic structures and their properties...
. In model theory
Model theory
In mathematics, model theory is the study of mathematical structures using tools from mathematical logic....
, signatures are used for both purposes.
Signatures play the same role in mathematics
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
as type signature
Type signature
In computer science, a type signature or type annotation defines the inputs and outputs for a function, subroutine or method. A type signature includes at least the function name and the number of its arguments...
s in computer programming
Computer programming
Computer programming is the process of designing, writing, testing, debugging, and maintaining the source code of computer programs. This source code is written in one or more programming languages. The purpose of programming is to create a program that performs specific operations or exhibits a...
. They are rarely made explicit in more philosophical treatments of logic.
Definition
Formally, a (single-sorted) 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- function symbols (examples: +, ×, 0, 1) and
- relation symbols or predicates (examples: ≤, ∈),
and a function ar: Sfunc
Srel → which assigns a non-negative integer called arityArity
In logic, mathematics, and computer science, the arity of a function or operation is the number of arguments or operands that the function takes. The arity of a relation is the dimension of the domain in the corresponding Cartesian product...
to every function or relation symbol. A function or relation symbol is called n-ary if its arity is n. A nullary (0-ary) 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. However, the cardinality of the language L will always be infinite; if σ is finite then |L| will be ℵ0.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 groupAbelian groupIn abstract algebra, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on their order . Abelian groups generalize the arithmetic of addition of integers...
s 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
Mathematical logic
Mathematical logic is a subfield of mathematics with close connections to foundations of mathematics, theoretical computer science and philosophical logic. The field includes both the mathematical study of logic and the applications of formal logic to other areas of mathematics...
, very often symbols are not allowed to be nullary, so that constant symbols must be treated separately rather than as nullary function symbols. 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
First-order logic
First-order logic is a formal logical system used in mathematics, philosophy, linguistics, and computer science. It goes by many names, including: first-order predicate calculus, the lower predicate calculus, quantification theory, and predicate logic...
.
Use of signatures in logic and algebra
In the context of first-order logicFirst-order logic
First-order logic is a formal logical system used in mathematics, philosophy, linguistics, and computer science. It goes by many names, including: first-order predicate calculus, the lower predicate calculus, quantification theory, and predicate 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 language
Formal language
A formal language is a set of words—that is, finite strings of letters, symbols, or tokens that are defined in the language. The set from which these letters are taken is the alphabet over which the language is defined. A formal language is often defined by means of a formal grammar...
s are inductively defined: The set of terms over the signature and the set of (well-formed) formulas over the signature.
In a structure
Structure (mathematical logic)
In universal algebra and in model theory, a structure consists of a set along with a collection of finitary operations and relations which are defined on it....
, 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. Here An = A × A × ... × A denotes the n-fold cartesian product
Cartesian product
In mathematics, a Cartesian product is a construction to build a new set out of a number of given sets. Each member of the Cartesian product corresponds to the selection of one element each in every one of those sets...
of the domain A with itself, and so f is in fact an n-ary function, and R an n-ary relation.
Many-sorted signatures
For many-sorted logic and for many-sorted structures signatures must encode information about the sorts. The most straightforward way of doing this is via symbol types that play the role of generalized arities.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
{×, →}: the relational symbol types s1 × ... × sn, and the functional symbol types s1 × ... × sn→s 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
- a set S of sorts,
- a set P of symbols, and
- a map type which associates to every symbol in P a symbol type over S.
External links
- Stanford Encyclopedia of Philosophy: "Model theory" -- by Wilfred Hodges.
- PlanetMath: Entry "Signature" describes the concept for the case when no sorts are introduced.
- Baillie, Jean, "An Introduction to the Algebraic Specification of Abstract Data Types."