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Programming Metalogics with a Fixpoint Type
, 1992
"... A programming metalogic is a formal system into which programming languages can be translated and given meaning. The translation should both reflect the structure of the language and make it easy to prove properties of programs. This thesis develops certain metalogics using techniques of category th ..."
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A programming metalogic is a formal system into which programming languages can be translated and given meaning. The translation should both reflect 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 fixpoint object 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 same properties. PCFstyle languages are translated into the FIX logic and computational adequacy reaulta are proved. Two languages are studied: Both are similar to PCF except one has call by value recursive function declararations and the other higher order conditionals. ...
Some Mathematical Aspects of Information Technology: Fixed Points and the Formal Semantics of Programming Languages
, 1996
"... this article. On the other hand, many others are not mentioned at all, and there is indeed an immense literature covering the various topics of which our bibliography is but a tiny fraction. Devising a complete classification of all the areas of mathematics which are of importance in IT would be an ..."
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this article. On the other hand, many others are not mentioned at all, and there is indeed an immense literature covering the various topics of which our bibliography is but a tiny fraction. Devising a complete classification of all the areas of mathematics which are of importance in IT would be an interesting and valuable project in its own right, though time consuming and beyond the abilities of the author, and in any case is not the objective of this article. Instead, we propose to take one concept, that of fixed point , and attempt to relate it to two of the main areas, SE and IKBS, which were identified earlier. The notion of fixed point is, of course, of great importance within mathematics, and it turns out also to be central in the areas we intend to consider in the context of programming language semantics. There may well be applications of ideas concerning fixed points elsewhere within IT, but they will not fall within our scope. Thus, specifically, we consider the use of fixed points in relation to the problem of giving formal, machine independent meaning (a formal semantics) to computer programs. To do such is fundamental to the problem of formal verification of software, or the use of formal methods as it is known in industry, and we take up this issue for procedural programs in x2. In x3