Expr×

Expr× is a notation and software concept introduced to formalize a cross-expression operator in symbolic computation. The term combines 'Expr', shorthand for expression, with the multiplication sign to indicate a structured product of two expression trees. It originated in the symbolic algebra community in the late 2010s as a means to express pairwise composition of subexpressions while retaining their identities for rewriting and simplification. In implementations, Expr× is typically an infix binary operator with left-to-right associativity and user-definable semantics via a context-specific algebra.

Notation and semantics: In a given algebra, E×F denotes the cross product of the expressions E and F, producing a new expression built by combining each subexpression of E with each subexpression of F. In many algebraic settings the operator distributes over addition, so (a+b)×(c+d) expands to ac+ad+bc+bd; in other contexts the operator is defined to preserve structure rather than expand, yielding structured templates rather than expanded polynomials.

Implementation: Reference libraries and languages have experimented with Expr× as an extension to existing CAS or DSLs (domain-specific languages). Some systems implement it as a rewrite rule set on expression trees, others as a templating operator in expression templates libraries. Documentation emphasizes that the exact rules depend on the chosen algebra and domain.

Use and reception: Expr× is used in research on symbolic programming, automated theorem proving, and advanced algebra education. It is considered niche and experimental; debates focus on readability and interoperability with standard notation.

See also: Symbolic computation, Expression templates, Algebra, Cross product, Formal language.