Formalizing the notions of non-interactive and interactive algorithms
C. A. Middelburg
TL;DR
The paper extends the formalization of algorithms beyond deterministic, non-interactive models to encompass non-deterministic and interactive computation by introducing non-interactive and interactive proto-algorithms. It constructs precise semantic frameworks using rooted labeled directed graphs, alphabets, and interpretations, and develops step-run notions along with two primary equivalence notions: algorithmic and computational simulations. It then analyzes the relationships among isomorphism, algorithmic equivalence, and computational equivalence in both non-interactive and interactive settings, and discusses the implications for expansion, interaction, and the alignment of behavior across representations. The work provides a semantic foundation for complexity analysis and language descriptions of interactive algorithms, while acknowledging open questions about concurrency and the limits of current equivalence notions. Overall, it contributes a rigorous scaffold for analyzing and comparing interactive computation models and highlights directions for future work in concurrency-aware formalizations.
Abstract
An earlier paper gives an account of a quest for a satisfactory formalization of the classical informal notion of an algorithm. That notion only covers algorithms that are deterministic and non-interactive. In this paper, an attempt is made to generalize the results of that quest first to a notion of an algorithm that covers both deterministic and non-deterministic algorithms that are non-interactive and then further to a notion of an algorithm that covers both deterministic and non-deterministic algorithms that are interactive. The notions of an non-interactive proto-algorithm and an interactive proto-algorithm are introduced. Non-interactive algorithms and interactive algorithms are expected to be equivalence classes of non-interactive proto-algorithms and interactive proto-algorithms, respectively, under an appropriate equivalence relation. On both non-interactive proto-algorithms and interactive proto-algorithms, three equivalence relations are defined. Two of them are deemed to be bounds for an appropriate equivalence relation and the third is likely an appropriate one.