Algorithmic Completeness of Imperative Programming Languages

Abstract

According to the Church-Turing Thesis, effectively calculable functions are functions computable by a Turing machine. Models that compute these functions are called Turingcomplete. For example, we know that common imperative languages (such as C, Ada or P ython) are Turing complete (up to unbounded memory). Algorithmic completeness is a stronger notion than Turing-completeness. It focuses not only on the input-output behavior of the computation but more importantly on the step-by-step behavior. Moreover, the issue is not limited to partial recursive functions, it applies to any set of functions. A model could compute all the desired functions, but some algorithms (ways to compute these functions) could be missing (see [10, 27] for examples related to primitive recursive algorithms). This paper's purpose is to prove that common imperative languages are not only Turing-complete but also algorithmically complete, by using the axiomatic definition of the Gurevich's Thesis and a fair bisimulation between the Abstract State Machines of Gurevich (defined in [16]) and a version of Jones' While programs. No special knowledge is assumed, because all relevant material will be explained from scratch.

Algorithmic Completeness for BSP Languages

“…Moreover, we say that P is without overwrite on X if ∆(P, X) is consistent (see p.5). We proved in [Mar18] that:…”

“…As in [Mar18], we think that a minimal imperative language is more convenient than a transition function to determine classes in time or space, or simply to be compared to more usual programming languages. Therefore, we define at the Section 5 p.9 the operational semantics of our core imperative programming language While BSP , which is our candidate for algorithmic completeness (for BSP algorithms).…”

“…If P does not terminate locally on X, which will be denoted by P ↑ X, then we assume that time(P, X) = ∞. We proved in [Mar18] that:…”

“…The simplest idea to translate a While program P into an ASM program is to add a variable for the current line of the program and follow the flow of execution. We detail in [Mar18] why this naive solution does not work as intended. Instead, we will add a bounded number of boolean variables b P1 , .…”

“…In [Mar18], the first author proved that usual imperative programming languages are not only Turing-complete but also algorithmically complete. This has been done by proving that a minimal imperative language While is algorithmically equivalent to the Abstract State Machines of Gurevich: While ASM, which means that the executions are the same, up to fresh variables and temporal dilation (more details are given at Section 6 p.11).…”