Technical reports

# Programming metalogics with a fixpoint type

**Roy Luis Crole**

February 1992, 164 pages

This technical report is based on a dissertation submitted January 1992 by the author for the degree of Doctor of Philosophy to the University of Cambridge, Churchill College.

**DOI:** 10.48456/tr-247

## Abstract

A programming metalogic is a formal system into which programming languages can be translated and given meaning. The translation should reflect both the structure of the language and make it easy to prove properties of programs. This thesis develops certain metalogics using techniques of category theory and treats recursion in a new way.

The notion of a category with a fixpoint logic is defined. Corresponding to this categorical structure there are type theoretic equational rules which will be present in all of the metalogics considered. These rules define the fixpoint type which will allow the interpretation of recursive declarations. With these core notions FIX categories are defined. These are the categorical equivalent of an equational logic which can be viewed as a very basic programming metalogic. Recursion is treated both syntactically and categorically.

The expressive power of the equational logic is increased by embedding it in an intuitionistic predicate calculus, giving rise to the FIX logic. This contains propositions about the evaluation of computations to values and an induction principle which is derived from the definition of a fixpoint object as an initial algebra. The categorical structure which accompanies the FIX logic is defined, called a FIX hyperdoctrine, and certain existence and disjunction properties of FIX are stated. A particular FIX hyperdoctrine is constructed and used in the proof of the above properties.

PCF-style languages are translated into the FIX logic and computational adequacy results are proved. Two languages are studied: both are similar to PCF except one has call by value recursive function declarations and the other higher order conditionals.

A dependently typed equational logic containing a fixpoint type and a universal type is given together with its related categorical structure, namely a FIX category with attributes. A representation theorem for Scott predomains is proved, which gives rise to a concrete example of such a FIX category with attributes. Recursive domain equations give rise to endofunctions on the universal type; using the fixpoint type we may solve for fixpoints of such endofunctions and thus obtain a solution the original domain as the type coded by the fixpoint.

## Full text

Only available on paper (could be scanned on request).

## BibTeX record

@TechReport{UCAM-CL-TR-247, author = {Crole, Roy Luis}, title = {{Programming metalogics with a fixpoint type}}, year = 1992, month = feb, institution = {University of Cambridge, Computer Laboratory}, address = {15 JJ Thomson Avenue, Cambridge CB3 0FD, United Kingdom, phone +44 1223 763500}, doi = {10.48456/tr-247}, number = {UCAM-CL-TR-247} }