On algorithmic equivalence of instruction sequences for computing bit string functions

Every partial function from bit strings of a given length to bit strings of a possibly different given length can be computed by a finite instruction sequence that contains only instructions to set and get the content of Boolean registers, forward jump instructions, and a termination instruction. We look for an equivalence relation on instruction sequences of this kind that captures to a reasonable degree the intuitive notion that two instruction sequences express the same algorithm.Comment: 27 pages, the preliminaries have textual overlaps with the preliminaries in arXiv:1308.0219 [cs.PL], arXiv:1312.1529 [cs.PL], and arXiv:1312.1812 [cs.PL]; 27 pages, three paragraphs about Milner's algorithmic equivalence hypothesis added to concluding remarks; 26 pages, several minor improvements of the presentation mad

Instruction sequence size complexity of parity

Each Boolean function can be computed by a single-pass instruction sequence that contains only instructions to set and get the content of Boolean registers, forward jump instructions, and a termination instruction. Auxiliary Boolean registers are not necessary for this. In the current paper, we show that, in the case of the parity functions, shorter instruction sequences are possible with the use of an auxiliary Boolean register in the presence of instructions to complement the content of auxiliary Boolean registers. This result supports, in a setting where programs are instruction sequences acting on Boolean registers, a basic intuition behind the storage of auxiliary data, namely the intuition that this makes possible a reduction of the size of a program.Comment: 12 pages, the preliminaries are largely the same as the preliminaries in arXiv:1312.1812 [cs.PL] and some earlier papers; 13 pages, minor errors corrected; 13 pages, presentation improved; 14 pages, remarks about related work added; 14 pages, presentation improve

Very Simple Chaitin Machines for Concrete AIT

In 1975, Chaitin introduced his celebrated Omega number, the halting probability of a universal Chaitin machine, a universal Turing machine with a prefix-free domain. The Omega number's bits are {\em algorithmically random}--there is no reason the bits should be the way they are, if we define ``reason'' to be a computable explanation smaller than the data itself. Since that time, only {\em two} explicit universal Chaitin machines have been proposed, both by Chaitin himself. Concrete algorithmic information theory involves the study of particular universal Turing machines, about which one can state theorems with specific numerical bounds, rather than include terms like O(1). We present several new tiny Chaitin machines (those with a prefix-free domain) suitable for the study of concrete algorithmic information theory. One of the machines, which we call Keraia, is a binary encoding of lambda calculus based on a curried lambda operator. Source code is included in the appendices. We also give an algorithm for restricting the domain of blank-endmarker machines to a prefix-free domain over an alphabet that does not include the endmarker; this allows one to take many universal Turing machines and construct universal Chaitin machines from them

A short introduction to program algebra with instructions for Boolean registers

A parameterized algebraic theory of instruction sequences, objects that represent the behaviours produced by instruction sequences under execution, and objects that represent the behaviours exhibited by the components of the execution environment of instruction sequences is the basis of a line of research in which issues relating to a wide variety of subjects from computer science have been rigorously investigated thinking in terms of instruction sequences. In various papers that belong to this line of research, use is made of an instantiation of this theory in which the basic instructions are instructions to read out and alter the content of Boolean registers and the components of the execution environment are Boolean registers. In this paper, we give a simplified presentation of the most general such instantiated theory.Comment: 21 pages, this paper is to a large extent a compilation of material from several earlier publications; 23 pages, presentation improved and section on uses for the theory added. arXiv admin note: text overlap with arXiv:1702.0351