On Algorithms, Effective Procedures, and Their Definitions

Abstract

I examine the classical idea of ‘algorithm’ as a sequential, step-by-step, deterministic procedure (i.e., the idea of ‘algorithm’ that was already in use by the 1930s), with respect to three themes, its relation to the notion of an ‘effective procedure’, its different roles and uses in logic, computer science, and mathematics (focused on numerical analysis), and its different formal definitions proposed by practitioners in these areas. I argue that ‘algorithm’ has been conceptualized and used in contrasting ways in the above areas, and discuss challenges and prospects for adopting a final foundational theory of (classical) ‘algorithms’.

The open texture of ‘algorithm’ in legal language

“…Even if a perfect mathematical definition of a term is available, the applicability of this formal definition of an informal situation that originated in a natural language is not straightforward nor univocal. Indeed, in philosophy of computation, there are a lot of discussions over the status of the so-called Church-Turing thesis (e.g., Shapiro 2006bShapiro , 2013Sieg 2009Sieg , 2013Copeland and Shagrir 2019;Quinon 2019;De Benedetto 2021;Papayannopoulos 2023), i.e., the thesis equating our informal notion of effective calculability with (one of) our formal notion(s) of classical computability (i.e., Turing computability, general recursiveness, Post computability, and the like). Within these discussions, the intuitive concept of computation has been argued to exhibit a certain degree of openness in its application and exact definition (e.g., Sieg 2009Sieg , 2013Shapiro 2013;Quinon 2019;De Benedetto 2021) that cannot be found in (one of) its formal equivalent(s).…”