Deciding Equations in the Time Warp Algebra

TL;DR

This paper introduces the time warp algebra, a mathematical structure modeling information growth in programs, proves its equational theory is decidable despite lacking the finite model property, and provides an OCaml implementation using Z3.

Contribution

It defines the time warp algebra, analyzes its properties, and develops a decision procedure for its equations, combining algebraic theory with practical implementation.

Findings

Time warp algebra forms a distributive involutive residuated lattice.

The algebra's equational theory is decidable despite no finite model property.

An OCaml implementation using Z3 was developed for decision procedures.

Abstract

Join-preserving maps on the discrete time scale $\omega^+$, referred to as time warps, have been proposed as graded modalities that can be used to quantify the growth of information in the course of program execution. The set of time warps forms a simple distributive involutive residuated lattice -- called the time warp algebra -- that is equipped with residual operations relevant to potential applications. In this paper, we show that although the time warp algebra generates a variety that lacks the finite model property, it nevertheless has a decidable equational theory. We also describe an implementation of a procedure for deciding equations in this algebra, written in the OCaml programming language, that makes use of the Z3 theorem prover.