Term algebra

In universal algebra and mathematical logic, a term algebra is a freely generated algebraic structure over a given signature.^[1]^[2] For example, in a signature consisting of a single binary operation, the term algebra over a set X of variables is exactly the free magma generated by X. Other synonyms for the notion include absolutely free algebra and anarchic algebra.^[3]

From a category theory perspective, a term algebra is the initial object for the category of all X-generated algebras of the same signature, and this object, unique up to isomorphism, is called an initial algebra; it generates by homomorphic projection all algebras in the category.^[4]^[5]

A similar notion is that of a Herbrand universe in logic, usually used under this name in logic programming,^[6] which is (absolutely freely) defined starting from the set of constants and function symbols in a set of clauses. That is, the Herbrand universe consists of all ground terms: terms that have no variables in them.

An atomic formula or atom is commonly defined as a predicate applied to a tuple of terms; a ground atom is then a predicate in which only ground terms appear. The Herbrand base is the set of all ground atoms that can be formed from predicate symbols in the original set of clauses and terms in its Herbrand universe.^[7]^[8] These two concepts are named after Jacques Herbrand.

Term algebras also play a role in the semantics of abstract data types, where an abstract data type declaration provides the signature of a multi-sorted algebraic structure and the term algebra is a concrete model of the abstract declaration.

Universal algebra[]

A type $\tau$ is a set of function symbols along with their associated arities. For any non-negative integer $n$ , let $\tau _{n}$ denote the function symbols in $\tau$ of arity $n$ . A function of arity 0 is treated as a constant.

Let $\tau$ be a type, and let $X$ be a non-empty set of symbols, representing the variable symbols. (For simplicity, assume $X$ and $\tau$ are disjoint.) Then the set of terms $T(X)$ of type $\tau$ over $X$ is the smallest set such that:

$X\cup \tau _{0}\subseteq T(X)$ .
For all $n\geq 1$ and for all function symbols $f\in \tau _{n}$ and terms $t_{1},...,t_{n}\in T(X)$ , we have the string $f(t_{1},...,t_{n})\in T(X)$ .

The term algebra ${\mathcal {T}}(X)$ of type $\tau$ over $X$ is defined as follows:^[9]

The domain of ${\mathcal {T}}(X)$ is $T(X)$ .
For each nullary function $f$ in $\tau _{0}$ , $f^{{\mathcal {T}}(X)}()$ is defined as the string $f$ .
For all $n\geq 1$ and for each n-ary function $f$ in $\tau$ and elements $t_{1},...,t_{n}$ in the domain, $f^{{\mathcal {T}}(X)}(t_{1},...,t_{n})$ is defined as the string $f(t_{1},...,t_{n})$ .

A term algebra is called absolutely free because for any algebra ${\mathcal {A}}$ of type $\tau$ , and for any function $g:X\to {\mathcal {A}}$ , $g$ extends to a unique homomorphism $g^{\ast }:{\mathcal {T}}(X)\to {\mathcal {A}}$ , which simply evaluates each term $t\in {\mathcal {T}}(X)$ to its corresponding value $g^{\ast }(t)\in {\mathcal {A}}$ . Formally, for each $t\in {\mathcal {T}}(X)$ :

If $t\in X$ , then $g^{\ast }(t)=g(t)$ .
If $t=f\in \tau _{0}$ , then $g^{\ast }(t)=f^{\mathcal {A}}()$ .
If $t=f(t_{1},...,t_{n})$ where $f\in \tau _{n}$ and $n\geq 1$ , then $g^{\ast }(t)=f^{\mathcal {A}}(g^{\ast }(t_{1}),...,g^{\ast }(t_{n}))$ .

Herbrand base[]

The signature σ of a language is a triple <O, F, P> consisting of the alphabet of constants O, function symbols F, and predicates P. The Herbrand base^[10] of a signature σ consists of all ground atoms of σ: of all formulas of the form R(t₁, …, t_n), where t₁, …, t_n are terms containing no variables (i.e. elements of the Herbrand universe) and R is an n-ary relation symbol (i.e. predicate). In the case of logic with equality, it also contains all equations of the form t₁ = t₂, where t₁ and t₂ contain no variables.

Decidability[]

Term algebras can be shown decidable using quantifier elimination. The complexity of the decision problem is in NONELEMENTARY.^[11]

References[]

^ Wilfrid Hodges (1997). A Shorter Model Theory. Cambridge University Press. pp. 14. ISBN 0-521-58713-1.
^ Franz Baader; Tobias Nipkow (1998). Term Rewriting and All That. Cambridge University Press. p. 49. ISBN 0-521-77920-0.
^ Klaus Denecke; Shelly L. Wismath (2009). Universal Algebra and Coalgebra. World Scientific. pp. 21–23. ISBN 978-981-283-745-5.
^ T.H. Tse (2010). A Unifying Framework for Structured Analysis and Design Models: An Approach Using Initial Algebra Semantics and Category Theory. Cambridge University Press. pp. 46–47. doi:10.1017/CBO9780511569890. ISBN 978-0-511-56989-0.
^ Jean-Yves Béziau (1999). "The mathematical structure of logical syntax". In Carnielli, Walter Alexandre; D'Ottaviano, Itala M. L. (eds.). Advances in Contemporary Logic and Computer Science: Proceedings of the Eleventh Brazilian Conference on Mathematical Logic, May 6-10, 1996, Salvador, Bahia, Brazil. American Mathematical Society. p. 9. ISBN 978-0-8218-1364-5. Retrieved 18 April 2011.
^ Dirk van Dalen (2004). Logic and Structure. Springer. p. 108. ISBN 978-3-540-20879-2.
^ M. Ben-Ari (2001). Mathematical Logic for Computer Science. Springer. pp. 148–150. ISBN 978-1-85233-319-5.
^ Monroe Newborn (2001). Automated Theorem Proving: Theory and Practice. Springer. p. 43. ISBN 978-0-387-95075-4.
^ Stanley Burris; H. P. Sankappanavar (1981). A Course in Universal Algebra. Springer. pp. 68–69, 71. ISBN 978-1-4613-8132-7.{{cite book}}: CS1 maint: multiple names: authors list (link)
^ Rogelio Davila. Answer Set Programming Overview.
^ Jeanne Ferrante; Charles W. Rackoff (1979). The Computational Complexity of Logical Theories. Springer.

External links[]

Weisstein, Eric W. "Herbrand Universe". MathWorld.

[1] Wilfrid Hodges (1997). A Shorter Model Theory. Cambridge University Press. pp. 14. ISBN 0-521-58713-1.

[2] Franz Baader; Tobias Nipkow (1998). Term Rewriting and All That. Cambridge University Press. p. 49. ISBN 0-521-77920-0.

[DeneckeWismath2009-3] Klaus Denecke; Shelly L. Wismath (2009). Universal Algebra and Coalgebra. World Scientific. pp. 21–23. ISBN 978-981-283-745-5.

[Tse2010-4] T.H. Tse (2010). A Unifying Framework for Structured Analysis and Design Models: An Approach Using Initial Algebra Semantics and Category Theory. Cambridge University Press. pp. 46–47. doi:10.1017/CBO9780511569890. ISBN 978-0-511-56989-0.

[CarnielliD'Ottaviano1999-5] Jean-Yves Béziau (1999). "The mathematical structure of logical syntax". In Carnielli, Walter Alexandre; D'Ottaviano, Itala M. L. (eds.). Advances in Contemporary Logic and Computer Science: Proceedings of the Eleventh Brazilian Conference on Mathematical Logic, May 6-10, 1996, Salvador, Bahia, Brazil. American Mathematical Society. p. 9. ISBN 978-0-8218-1364-5. Retrieved 18 April 2011.

[Dalen2004-6] Dirk van Dalen (2004). Logic and Structure. Springer. p. 108. ISBN 978-3-540-20879-2.

[Ben-Ari2001-7] M. Ben-Ari (2001). Mathematical Logic for Computer Science. Springer. pp. 148–150. ISBN 978-1-85233-319-5.

[Newborn2001-8] Monroe Newborn (2001). Automated Theorem Proving: Theory and Practice. Springer. p. 43. ISBN 978-0-387-95075-4.

[Burris1981-9] Stanley Burris; H. P. Sankappanavar (1981). A Course in Universal Algebra. Springer. pp. 68–69, 71. ISBN 978-1-4613-8132-7.{{cite book}}: CS1 maint: multiple names: authors list (link)

[10] Rogelio Davila. Answer Set Programming Overview.

[11] Jeanne Ferrante; Charles W. Rackoff (1979). The Computational Complexity of Logical Theories. Springer.

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Term algebra

Contents

Universal algebra[]

Herbrand base[]

Decidability[]

See also[]

References[]

Further reading[]

External links[]