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Algorithmic Completeness of Imperative Programming Languages


According to the Church-Turing Thesis, effectively calculable functions are functions computable by a Turing machine. Models that compute these functions are called Turing-complete. For example, we know that common imperative languages (such as C, Ada or Python) 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 1) and 2)) 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 3)) and a version of Jones' While programs. No special knowledge is assumed, because all relevant material will be explained from scratch.

Keywords: Algorithm, ASM, Completeness, Computability, Imperative, Simulation.


The short version of the paper had been submitted to Fundamenta Informaticae.

The long version is an older version, with very poor english and presentation, but contains proofs omitted in the short version.

1) Loïc Colson: “About primitive recursive algorithms”, Theoretical Computer Science 83 (1991) 57–69
2) Yiannis N. Moschovakis: “On Primitive Recursive Algorithms and the Greatest Common Divisor Function”, Theoretical Computer Science (2003)
3) Yuri Gurevich: “Sequential Abstract State Machines Capture Sequential Algorithms”, ACM Transactions on Computational Logic (2000)