Subrecursive Programming Systems

This text develops the theory of subrecursive programming systems and applies it to the more general theory of structural complexity theory. Its first goal is to establish relative program succinctness between systems; its second is to illustrate the applicability of these tools.

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1.1. What This Book is About This book is a study of * subrecursive programming systems, * efficiency/program-size trade-offs between such systems, and * how these systems can serve as tools in complexity theory. Section 1.1 states our basic themes, and Sections 1.2 and 1.3 give a general outline of the book. Our first task is to explain what subrecursive programming systems are and why they are of interest. 1.1.1. Subrecursive Programming Systems A subrecursive programming system is, roughly, a programming language for which the result of running any given program on any given input can be completely determined algorithmically. Typical examples are: 1. the Meyer-Ritchie LOOP language [MR67,DW83], a restricted assem- bly language with bounded loops as the only allowed deviation from straight-line programming; 2. multi-tape 'lUring Machines each explicitly clocked to halt within a time bound given by some polynomial in the length ofthe input (see [BH79,HB79]); 3. the set of seemingly unrestricted programs for which one can prove 1 termination on all inputs (see [Kre51,Kre58,Ros84]); and 4. finite state and pushdown automata from formal language theory (see [HU79]). lOr, more precisely, the collection of programs, p, ofsome particular general-purpose programming language (e.
g., Lisp or Modula-2) for which there is a proof in some par- ticular formal system (e.g., Peano Arithmetic) that p halts on all inputs.