Occursdiffers

Occursdiffers is a theoretical relation used in formal language theory and logic to describe a property of occurrences within a term or expression. An occurrence is a specific position in the syntactic structure of a term, such as a node in its abstract syntax tree labeled by a symbol.

The occursdiffers relation holds between two occurrences when they refer to distinct positions in the term’s structure. In other words, two occurrences satisfy occursdiffers if they are not the same node, even if they carry the same symbol. This makes the relation purely positional and structural rather than semantic or binding-based. If the two references point to the same exact position, occursdiffers is false.

In practice, occursdiffers is used in analyses of sharing, duplication, and binding within terms, and can appear in the specification of unification, term rewriting, or constraint-solving systems. It helps formalize questions such as whether two occurrences of a symbol can be treated as independent or must be identified as the same occurrence. The concept is related to, but distinct from, notions like occurs (the presence of a variable within a term) and occurs-check procedures in unification, which focus on variable binding rather than positional identity.

Examples illustrate the idea: in the term f(a, b, a), the two occurrences of a are at different positions, so occursdiffers(a1, a2) holds; in a term with a single occurrence, such as f(a), a1 and a1 refer to the same node, so occursdiffers(a1, a1) does not hold.

See also: occurrences, occurs, unification, term rewriting, abstract syntax tree.