James S. Royer

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.

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.

This book presents developments of a language independent theory of program structure. The theory features a simple, natural notion of control structure which is much broader than in other theories of programming languages such as denotational semantics and program schemes. This notion permits treatment of control structures which involve not only the denotation of programs (i.e., their input/output behavior), but also their structure, size, run times, etc. The theory also treats the relation of control structure and complexity properties of programming languages. The book focuses on expressive interdependencies of control structures (which control structures can be expressed by which others). A general method of proving control structures expressively independent is developed. The book also considers characterizations of the expressive power of general purpose programming languages in terms of control structures. Several new characterizations are presented and two compactness results for such characterizations are shown.