Formalisation of Intensionality as Algorithms (project abstract)

In a sequence of papers, Moschovakis developed a class of languages of recursion as a new approach to the mathematical notion of algorithm and development of computational semantics, e.g., see Moschovakis [7], for FLR, and Moschovakis [8], for Lar. In particular, the language and theory of acyclic recursion Lar is intended for modeling the logical concepts of meaning and synonymy, from the perspective of the theory of computability, by targeting adequateness of computational semantics of NL. Lar is a higher order type theory, which is a proper extension of Gallin’s TY2, Gallin [3], and, thus, of Montague’s Intensional Logic (IL). Lar has a highly expressive language, an effective reduction calculus and strong mathematical properties. It models the notion of algorithm by abstract mathematical objects, which are tuple of functions defined by mutual recursion, called acyclic recursors. The referential intensions of the meaningful Lar terms are acyclic recursors defined by their canonical forms, which are recursion terms. For the construction of recursion terms (where-terms), the language Lar uses a recursion operator, denoted by the constant where that applies over a head term A0 and a set of assignments, called body, {p1 := A1, . . . , pn := An}, where each Ai is a term of the same type as the recursion variable pi (1 ≤ i ≤ n): A0 where{p1 := A1, . . . , pn := An}. The where-terms represent recursive computations by designating functional recursors: intuitively, the denotation of the term A0 depends on the functions denoted by p1, . . . , pn that are computed recursively by the system of assignments {p1 := A1, . . . , pn := An}. In an acyclic system of assignments, the computations close-off. The formal syntax of Lar allows only recursion terms with acyclic systems of assignments, while the FLR allows cyclicity, but is limited with respect to its type system. The languages of recursion (e.g., FLR and Lar) have two semantic layers: denotational semantics and ref-