Anti-unification (computer science)


Anti-unification is the process of constructing a generalization common to two given symbolic expressions. As in unification, several frameworks are distinguished depending on which expressions are allowed, and which expressions are considered equal. If variables representing functions are allowed in an expression, the process is called "higher-order anti-unification", otherwise "first-order anti-unification". If the generalization is required to have an instance literally equal to each input expression, the process is called "syntactical anti-unification", otherwise "E-anti-unification", or "anti-unification modulo theory".
An anti-unification algorithm should compute for given expressions a complete, and minimal generalization set, that is, a set covering all generalizations, and containing no redundant members, respectively. Depending on the framework, a complete and minimal generalization set may have one, finitely many, or possibly infinitely many members, or may not exist at all; it cannot be empty, since a trivial generalization exists in any case. For first-order syntactical anti-unification, Gordon Plotkin gave an algorithm that computes a complete and minimal singleton generalization set containing the so-called "least general generalization".
Anti-unification should not be confused with dis-unification. The latter means the process of solving systems of inequations, that is of finding values for the variables such that all given inequations are satisfied. This task is quite different from finding generalizations.

Prerequisites

Formally, an anti-unification approach presupposes
Given a set of variable symbols, a set of constant symbols and sets of -ary function symbols, also called operator symbols, for each natural number, the set of terms is recursively defined to be the smallest set with the following properties:
For example, if x ∈ V is a variable symbol, 1 ∈ C is a constant symbol, and add ∈ F2 is a binary function symbol, then x ∈ T, 1 ∈ T, and add ∈ T by the first, second, and third term building rule, respectively. The latter term is usually written as x+1, using Infix notation and the more common operator symbol + for convenience.

Higher-order term

Substitution

A substitution is a mapping from variables to terms; the notation refers to a substitution mapping each variable to the term, for, and every other variable to itself. Applying that substitution to a term is written in postfix notation as ; it means to replace every occurrence of each variable in the term by. The result of applying a substitution to a term is called an instance of that term.
As a first-order example, applying the substitution to the term

Generalization, specialization

If a term has an instance equivalent to a term, that is, if for some substitution, then is called more general than, and is called more special than, or subsumed by,. For example, is more general than if is commutative, since then.
If is literal identity of terms, a term may be both more general and more special than another one only if both terms differ just in their variable names, not in their syntactic structure; such terms are called variants, or renamings of each other.
For example, is a variant of, since and.
However, is not a variant of, since no substitution can transform the latter term into the former one, although achieves the reverse direction.
The latter term is hence properly more special than the former one.
A substitution is more special than, or subsumed by, a substitution if is more special than for each variable.
For example, is more special than, since and is more special than and, respectively.

Anti-unification problem, generalization set

An anti-unification problem is a pair of terms.
A term is a common generalization, or anti-unifier, of and if and for some substitutions.
For a given anti-unification problem, a set of anti-unifiers is called complete if each generalization subsumes some term ; the set is called minimal if none of its members subsumes another one.

First-order syntactical anti-unification

The framework of first-order syntactical anti-unification is based on being the set of first-order terms and on being syntactic equality.
In this framework, each anti-unification problem has a complete, and obviously minimal, singleton solution set.
Its member is called the least general generalization of the problem, it has an instance syntactically equal to and another one syntactically equal to.
Any common generalization of and subsumes.
The lgg is unique up to variants: if and are both complete and minimal solution sets of the same syntactical anti-unification problem, then and for some terms and, that are renamings of each other.
Plotkin has given an algorithm to compute the lgg of two given terms.
It presupposes an injective mapping, that is, a mapping assigning each pair of terms an own variable, such that no two pairs share the same variable.
The algorithm consists of two rules:
For example, ; this least general generalization reflects the common property of both inputs of being square numbers.
Plotkin used his algorithm to compute the "relative least general generalization " of two clause sets in first-order logic, which was the basis of the Golem approach to inductive logic programming.

First-order anti-unification modulo theory