mualambda

Mualambda is a fictional formal system described in theoretical computer science discussions as an extension of the lambda calculus that incorporates fixed-point operations inspired by the mu-calculus. The name combines mu, the symbol used for fixed points, and lambda, the anonymous function construct, to indicate its aim: to enable compact representations of recursive, higher-order computations within a single formal framework.

Overview: In mualambda, terms are built from variables, lambda abstractions λx. t, applications t u, and a mu construct μx. t representing the least fixed point of t with respect to x. A companion operator νx. t may be used in versions that emphasize corecursion. Typing systems in mualambda, when present, range from simply typed to dependent types, balancing expressiveness with normalization.

Semantics: Operational semantics extend standard beta-reduction with unfold rules for mu, which replace μx. t with t[μx. t / x] and similarly for ν. Denotational semantics interpret mu as a least fixed point in a complete partial order, aligning with domain-theoretic models.

Relation to existing calculi: Mualambda sits conceptually between lambda calculus and mu-calculus, sharing lambda's function abstraction and mu's fixed-point idea. It is used in thought experiments about recursion, fixed points, and higher-order computation.

Usage and limitations: As a primarily theoretical construct, mualambda has no standardized syntax or implementation. It is discussed in pedagogical contexts to illustrate how recursion and higher-order functions interact, and to explore the boundaries between inductive and coinductive definitions.