NDLI: Recursive Programs as Definitions in First-Order Logic

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Recursive Programs as Definitions in First-Order Logic

Content Provider	Society for Industrial and Applied Mathematics (SIAM)
Author	Cartwright, Robert
Copyright Year	1984
Abstract	Despite the reputed limitations of first order logic, it is easy to state and prove almost all interesting properties of recursive programs within a simple first order theory, by using an approach we call first order programming logic. Unlike higher order logics based on fixed-point induction, first order programming logic is founded on deductive principles that are familiar to most programmers. Informal structural induction arguments (such as termination proofs for LISP append, McCarthys 91-function, and Ackermanns function) have direct formalizations within the system.The essential elements of first order programming logic are: (1) The data domain D must be a finitely generated set that explicitly includes the undefined object $ \bot $ (representing nontermination) as well as ordinary data objects. (2) Recursive programs over D are treated as logical definitions augmenting a first order theory ofthe data domain. (3) The interpretation of a recursive program is the least fixed-point of the functional corresponding to the program.Since the data domain D is a finitely generated set, the first order axiomatization of D includes a structural induction axiom scheme. This axiom scheme serves as the fundamental proof rule of first order program-ming logic.The major limitation of first order programming logic is that every fixed-point of the functional corresponding to a recursive program is an acceptable interpretation for the program. The logic fails to capture the notion of least fixed-point. To overcome this limitation, we present a simple, effective procedure for transforming an arbitrary recursive program into an equivalent recursive program that has a unique fixed-point, yet retains the logical structure of the original. Given this transformation technique, it is our experience that first order programming logic is sufficiently powerful to prove almost any property of practical interest about the functions computed by recursive programs.
Starting Page	374
Ending Page	408
Page Count	35
File Format	PDF
ISSN	00975397
DOI	10.1137/0213026
e-ISSN	10957111
Journal	SIAM Journal on Computing (SMJCAT)
Issue Number	2
Volume Number	13
Language	English
Publisher	Society for Industrial and Applied Mathematics
Publisher Date	2006-07-13
Access Restriction	Subscribed
Subject Keyword	recursive programs rewrite rules semantics program transformations recursive definitions verification programming logic
Content Type	Text
Resource Type	Article
Subject	Mathematics Computer Science

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