A similarity criterion for sequential programs using truth-preserving partial functions

TL;DR

This paper introduces a criterion based on truth-preserving partial functions to analyze and relate sequential programs, enabling comparison of their computational behavior beyond structural similarities.

Contribution

It proposes a novel approach using truth-preserving partial functions to establish isomorphism between different sequential programs based on their computational properties.

Findings

Defines a class of truth-preserving partial functions for program analysis

Establishes conditions for program isomorphism based on these functions

Allows comparison of programs with different structures performing similar computations

Abstract

The execution of sequential programs allows them to be represented using mathematical functions formed by the composition of statements following one after the other. Each such statement is in itself a partial function, which allows only inputs satisfying a particular Boolean condition to carry forward the execution and hence, the composition of such functions (as a result of sequential execution of the statements) strengthens the valid set of input state variables for the program to complete its execution and halt succesfully. With this thought in mind, this paper tries to study a particular class of partial functions, which tend to preserve the truth of two given Boolean conditions whenever the state variables satisfying one are mapped through such functions into a domain of state variables satisfying the other. The existence of such maps allows us to study isomorphism between different programs, based not only on their structural characteristics (e.g. the kind of programming constructs used and the overall input-output transformation), but also the nature of computation performed on seemingly different inputs. Consequently, we can now relate programs which perform a given type of computation, like a loop counting down indefinitely, without caring about the input sets they work on individually or the set of statements each program contains.