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Wikipedia / Term algebra

In universal algebra and mathematical logic, a term algebra is a freely generated algebraic structure over a given signature. 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.

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.

A similar notion is that of a Herbrand universe in logic, usually used under this name in logic programming, 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. 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 is a set of function symbols, with each having an associated arity. A constant is a function symbol of arity 0.

are disjoint.) Then the set of terms is the set of all well-formed strings is the smallest set such that:

  • .

  • to them represents again a term.

The term algebra is defined as follows:

  • .

  • .

  • and for each n.

A term algebra is called absolutely free:

  • .

  • .

  • .

Example

.

has the natural numbers.

.

is considered. We use to flag its members, which otherwise may be hard to recognize due to their uncommon syntactic form. We have e.g.

  • is a variable symbol;

  • is a constant symbol; hence

  • is a 2-ary function symbol; hence, in turn,

  • is a 2-ary function symbol.

corresponds to a mathematical expression built from the admitted symbols and written in Polish prefix notation in usual infix notation.

To give some counter-examples, we have e.g.

  • is neither an admitted variable symbol nor an admitted constant symbol;

  • , for the same reason,

  • ).

are both "sluggards" in that they just record what computation should be done, rather than doing it.

. For example,

.

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 of a signature σ consists of all ground atoms of σ: formulas of the form R(t1,...,tn), where t1,...,tn 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

The first-order theory of any term algebra can be shown to be decidable using quantifier elimination. The complexity of the decision problem is in NONELEMENTARY because binary constructors are injective and thus pairing functions.

See also

References

Further reading