The efficiency of primitive recursive functions: A programmer's view

Abstract

Using denotational semantics tools, Colson and others studied primitive recursive (p.r.) algorithms, proving the “ultimate obstinacy” property, which has as a consequence that many computations cannot be efficiently implemented in a language that faithfully expresses the classical definition of p.r. functions.As shown by Ritchie and Meyer, the class of p.r. functions can also be characterised by the programming language Loop. Our purpose is to show that the informal, but precise, operational description of Loop (and its variants) is sufficient to prove non-trivial time lower bounds of p.r. algorithms. In particular, we present a simple proof of a property which is similar to ultimate obstinacy, namely that every p.r. program that implements a non-trivial function must execute xi+c times the body of some loop instruction, where xi is the initial contents of some fixed input register, and c does not depend on the input values. This and other lower bounds were obtained without using the lambda calculus or systems with functional parameters (such as Gödel system T).If the conditional break and the decrement instructions are included in the Loop language, the ultimate obstinacy property does not hold. In this case, we use another approach for obtaining lower bounds. The unconditional break instruction does not avoid the obstinacy property; it is equivalent (in function and efficiency) to the conditional instruction if-then-else. The efficient implementation of other functions like step(x) (step(0)=0; step(x)=1 for x≥1) and min⁡(x,y) is also studied.