exprimes

Exprimes is a term used in some mathematical and theoretical computer science literature to denote irreducible expressions under a specified reduction or simplification framework. The term is a portmanteau of expression and prime, reflecting a notion of irreducibility analogous to prime numbers in arithmetic.

Definition: Let L be a formal language equipped with a one-step reduction relation ->. An expression e in L is called an exprime (prime expression) with respect to -> if there is no expression e' such that e -> e' and e' is distinct from e. In other words, e is in normal form for the rewriting system. In a terminating system, every expression has a unique normal form, and exprimes coincide with unique normal forms.

Variants: Some authors restrict exprimes to a subset of constants or to expressions that cannot be factorized or decomposed using a chosen decomposition rule. In polynomial rewriting, an exprime might be a polynomial that cannot be factored under the ring's units.

Examples: In a simple arithmetic language with the rule a+b -> c for some constant c, the expressions a and b could be exprimes if neither matches a reduction rule; while 1+2 would not be an exprime if there is a rule to replace 1+2 with 3.

Applications: Normal-form analysis, simplification and compilation, automated theorem proving, symbolic computation.

See also: Normal form, irreducible element, term rewriting system, prime element.

References: This is a general concept used in theoretical discussions; specific definitions vary by rewriting system.