Mapping programs to equations

Extracting the function of a program from a static analysis of its source code is a valuable capability in software engineering; at a time when there is increasing talk of using AI (Artificial Intelligence) to generate software from natural language specifications, it becomes increasingly important to determine the exact function of software as written, to figure out what AI has understood the natural language specification to mean. For all its criticality, the ability to derive the domain-to-range function of a program has proved to be an elusive goal, due primarily to the difficulty of deriving the function of iterative statements. Several automated tools obviate this difficulty by unrolling the loops; but this is clearly an imperfect solution, especially in light of the fact that loops capture most of the computing power of a program, are the locus of most of its complexity, and the source of most of its faults. This dissertation investigates a three-step process to map a program written in a C-like language into a function from inputs to outputs, or from initial states to final states. The semantics of iterative statements are captured (while loops, repeat loops, for loops), including nested iterative statements, by means of the concept of invariant relation; an invariant relation is a reflexive transitive relation that links program states separated by an arbitrary number of iterations. But the function derived for large and complex programs may be too unwieldy to be useful, not unlike drinking from a fire hose. In order to enable the user to query the program at scale, four functions are proposed. We propose four functions: Assume(), which enables the user to make assumptions about program states or program parts; Capture(), which enables the user to capture the state of the program at some label of the function of some program part; Verify(), which enables the user to verify a unary assertion about the state of the program at some label, or a binary assertion about a program part; and Establish(), which is envisioned to use program repair techniques to modify the program so as to make a Verify() query return true