7 research outputs found
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
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
On the complexity of the correctness problem for non-zeroness test instruction sequences
This paper concerns the question to what extent it can be efficiently
determined whether an arbitrary program correctly solves a given problem. This
question is investigated with programs of a very simple form, namely
instruction sequences, and a very simple problem, namely the non-zeroness test
on natural numbers. The instruction sequences concerned are of a kind by which,
for each , each function from to can be computed.
The established results include the time complexities of the problem of
determining whether an arbitrary instruction sequence correctly implements the
restriction to of the function from to that
models the non-zeroness test function, for , under several restrictions
on the arbitrary instruction sequence.Comment: 32 pages, minor revision with several obscurities made clea
Program algebra for Turing-machine programs
This paper presents an algebraic theory of instruction sequences with
instructions for Turing tapes as basic instructions, the behaviours produced by
the instruction sequences concerned under execution, and the interaction
between such behaviours and Turing tapes provided by an execution environment.
This theory provides a setting for the development of theory in areas such as
computability and computational complexity that distinguishes itself by
offering the possibility of equational reasoning and being more general than
the setting provided by a known version of the Turing-machine model of
computation. The theory is essentially an instantiation of a parameterized
algebraic theory which 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.Comment: 19 pages, Sect. 2--4 are largely shortened versions of Sect. 2--4 of
arXiv:1808.04264, which, in turn, draw from preliminary sections of several
earlier papers; 21 pages, some remarks in Sect.1 and Sect.10 adde
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
Program algebra for random access machine programs
This paper presents an algebraic theory of instruction sequences with
instructions for a random access machine (RAM) as basic instructions, the
behaviours produced by the instruction sequences concerned under execution, and
the interaction between such behaviours and RAM memories. This theory provides
a setting for the development of theory in areas such as computational
complexity and analysis of algorithm that distinguishes itself by offering the
possibility of equational reasoning to establish whether an instruction
sequence computes a given function and being more general than the setting
provided by any known version of the RAM model of computation. In this setting,
a semi-realistic version of the RAM model of computation and a bit-oriented
time complexity measure for this version are introduced.Comment: 25 pages, Sect. 2--4 are largely shortened versions of Sect. 2--4 of
arXiv:1808.04264, which, in turn, draw from preliminary sections of several
other papers. arXiv admin note: substantial text overlap with
arXiv:1901.0884