Xtermen

Xtermen are a formal construct used in logic and computer science to represent terms with shared structure. They generalize traditional tree-like terms by allowing subterms to be shared through a directed acyclic graph, enabling more compact representations and more efficient symbolic manipulation in areas such as term rewriting, symbolic computation, and program analysis.

Definition and structure: An xtermen consists of a finite set of labeled nodes and a designated root node. Each node carries a symbol from a given signature and has a fixed arity that determines its input edges. Edges connect nodes to their arguments, and unlike ordinary terms, a single node may serve as the argument for multiple parents, creating shared substructures. The graph is acyclic, ensuring a well-defined evaluation order, although extensions exist that permit controlled cyclic references for fixed-point computations.

Operations and variants: Substitution in xtermen replaces occurrences of a variable node with another xtermen, while evaluation maps the graph to a concrete value according to an interpretation of the signature. Two xtermen are considered equivalent up to renaming of variable nodes (alpha-equivalence) and sharing structure. Variants include extended forms that allow optional cycles for modeling recursion, as well as rooted versions that emphasize a single focal term.

Applications: Xtermen are used to optimize symbolic computation and compilers through common subexpression sharing, to simplify algebraic expressions in model checking and automated reasoning, and to represent knowledge with shared components in knowledge bases. They align with concepts from term graphs and graph rewriting, providing a bridge between tree-based and graph-based representations.

Etymology: The term xtermen combines “term” with a marker (the x) to indicate cross-linked or shared structure, reflecting its core feature of subterm sharing within a unified graph.