Boolean Algebraic Programs as a Methodology for Symbolically Demonstrating Lower and Upper Bounds of Algorithms and Determinism

The lower and upper bound of any given algorithm is one of the most crucial pieces of information needed when evaluating the computational effectiveness for said algorithm. Here a novel method of Boolean Algebraic Programming for symbolic manipulation of Machines, Functions, and Inputs is presented which allows for direct analysis of time complexities and proof of deterministic methodologies. It is demonstrated through the analysis of a particular problem which is proven and solved through the application of Boolean algebraic programming.

💡 Research Summary

The paper introduces a novel methodology called Boolean Algebraic Programming (BAP) for symbolically deriving both lower and upper bounds of algorithmic time complexity. It begins by highlighting the central role of complexity bounds in evaluating computational efficiency and points out the limitations of traditional techniques, which rely heavily on numerical estimation, recurrence solving, or empirical benchmarking. To overcome these challenges, the authors propose a formal framework that translates the operational semantics of any algorithm into a system of Boolean expressions. In this representation, machines, functions, and inputs are modeled as Boolean variables, while control flow constructs (such as conditionals and loops) become logical conjunctions and disjunctions.

A key insight of the work is the reinterpretation of complexity bounds as satisfiability problems. The lower bound of an algorithm corresponds to the minimal number of primitive operations required for the Boolean formula to be satisfied for all possible inputs; this is equivalent to finding a minimum‑satisfying assignment. Conversely, the upper bound is expressed as the maximum number of operations that can be performed before the Boolean formula inevitably becomes true, which maps to a maximum‑satisfying assignment problem. By casting the analysis in this way, the authors can leverage modern SAT/SMT solvers to automatically compute tight bounds without manual induction or case analysis.

Determinism is also given a formal Boolean characterization. A deterministic algorithm yields a unique execution trace for any given input, which translates to a Boolean formula possessing exactly one satisfying model. Non‑deterministic or probabilistic algorithms, in contrast, admit multiple models, and their complexity analysis can be extended to average‑case or expected‑case bounds within the same Boolean framework.

The practical utility of BAP is demonstrated through two canonical case studies. First, a comparison‑based sorting algorithm is encoded: each comparison between elements i and j becomes a Boolean variable C_{i,j}, and the sorted order condition is expressed as a conjunction over all i < j of “C_{i,j} implies A