Table Of ContentATLANTIS STUDIES IN COMPUTING
VOLUME 2
SERIES EDITORS: JAN A. BERGSTRA, MICHAEL W. MISLOVE
Atlantis Studies in Computing
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Instruction Sequences for
Computer Science
Jan A. Bergstra and Cornelis A. Middelburg
Institute of Informatics, Faculty of Science, University of Amsterdam
Amsterdam, the Netherlands
AMSTERDAM – PARIS – BEIJING
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Atlantis Studies in Computing
Volume 1: Code Generation with Templates - B.J. Arnoldus, M.G.J. Van den Brand, A. Serebrenik
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Print: 978-94-91216-64-0
E-Book: 978-94-91216-65-7
ISSN: 2212-8565
⃝c 2012 ATLANTIS PRESS
Preface
The concept of an instruction sequence is a key concept in practice, but strangely enough
it has as yet not come prominently into the picture in theoretical circles. In much work
on computer architecture, instruction sequences are under discussion. In spite of this, the
notion of an instruction sequence has never been subjected to systematic and precise anal-
ysis. Moreover, in work on computer architecture, the viewpoint is usually taken that a
program is in essence an instruction sequence. By contrast, in the theory of computation,
different viewpoints on what is a program are usually taken. This state of affairs brought
us to define a general notion of an instruction sequence, to subject it to a systematic and
precise analysis, and to provide evidence for the hypothesis that the notion of an instruc-
tion sequence is relevant to diverse subjects from the theory of computation and the area
of computer architecture. Many results of the work in question are brought together in this
book with the aim to bring instruction sequences as a theme in computer science better into
the picture.
To put it otherwise, this book concerns instruction sequences, the behaviours produced
by instruction sequences under execution, the interaction between these behaviours and
components of the execution environment concerning the processing of instructions, the
expressiveness of instruction sequences, and various issues relating to well-known sub-
jects from computer science where we found that the notion of an instruction sequence
is relevant. Most of the issues in question are of a computation-theoretic or computer-
architectural kind. They relate to subjects such as the halting problem, non-uniform com-
putational complexity, instruction sequence performance and instruction set architecture.
Some of the issues considered are somehow related to process algebra, namely remote in-
struction processing and instruction sequence producible processes. Some variations on
instruction sequences of the usual kind, such as instruction sequences without a directional
bias and probabilistic instruction sequences, are also considered.
v
vi Instruction Sequences for Computer Science
This book is primarily intended for researchers in computer science interested in in-
struction sequences as a theme in computer science. It is also meant to be suitable as
supplementary reading in courses for graduate students and advanced undergraduate stu-
dents in computer science. Chapters 5 and 6 may as much appeal to those who are primarily
interested in the subjects from the theory of computation and the area of computer architec-
ture, respectively, that come up in these chapters. Chapter 7 may as much appeal to those
who are primarily interested in process algebra.
Throughout the book, some familiarity with equational logic, universal algebra, and
elementary set theory is assumed. In Sect. 5.2, some familiarity with non-uniform com-
putational complexity is assumed. In Sect. 5.1, Sect. 6.2 and Chap. 7, some familiarity
with computability, instruction set architectures and process algebra, respectively, would
be helpful. Chapter 2 is a prerequisite for Chap. 3, and both chapters are prerequisites for
all subsequent chapters.
Chapter 2 introduces an algebraic theory SPISA of single-pass instruction sequences
and an algebraic theory BTA of mathematical objects that represent in a direct way the
behaviours produced by instruction sequences under execution. The objects concerned are
called threads. It is made precise in the setting of the latter theory which behaviours are
produced by the instruction sequences considered in the former theory. The instruction se-
quences in question include both finite and infinite ones, but the theory provides a notation
by means of which all of them can be represented finitely. This chapter also introduces
alternative notations ISNR and ISNA by means of which all these instruction sequences
can be represented finitely as well, but which are closer to existing assembly languages.
Chapter 3 introduces so-called services, which represent the behaviours exhibited by
the components of an execution environment that are capable of processing particular in-
structions and doing so independently, and extends BTA with an operator meant for the
composition of families of named services and operators that have a direct bearing on the
processing of instructions by services from such service families. In addition, the con-
cept of a functional unit, which is an abstract model of a machine, is introduced. In the
frequently occurring case that the behaviours represented by services can be viewed as
the behaviours of a machine in its different states, the services concerned are completely
determined by a functional unit. Some extensions of ISNR and ISNA with additional in-
structions are explained with the help of some simple functional units.
Chapter 4 gives answers to basic expressiveness issues regarding SPISA. In this case,
expressiveness is basically about which behaviours can be produced by instruction se-
Preface vii
quences under execution, which instructions can be removed without reducing the class of
behaviours that can be produced by instruction sequences under execution, how to enlarge
the class of behaviours that can be produced by instruction sequences under execution, et
cetera. This chapter is also concerned with some issues that arise from the investigation
of expressiveness issues regarding SPISA. For example, it is shown that a finite-state exe-
cution mechanism for a set of instruction sequences that by itself can produce each finite-
state behaviour from an instruction sequence belonging to the set of instruction sequences
in question is unfeasible.
Chapter 5 concerns two subjects from the theory of computation, namely the halting
problem and non-uniform computational complexity. Positioning Turing’s result regarding
the undecidability of the halting problem as a result about programs rather than machines,
and taking single-pass instruction sequences as considered in SPISA as programs, the au-
tosolvability requirement that a program of a certain kind must solve the halting problem
for all programs of that kind is analysed. Thinking in terms of single-pass instruction
sequences as considered in SPISA, counterparts of the classical non-uniform complexity
classes P/poly and NP/poly are defined, a notion of completeness for the counterpart of
NP/poly is introduced, several complexity hypotheses are formulated, and it is shown that
a problem closely related to 3SAT is NP-complete as well as complete for the counterpart
of NP/poly.
Chapter 6 concerns two subjects from the area of computer architecture, namely in-
struction sequence performance and instruction set architectures. We study the effect of
eliminating indirect jump instructions from instruction sequences with direct and indirect
jump instructions on the interactive performance of instruction sequences. A strict version
of the concept of a load/store instruction set architecture is proposed for theoretical work
relevant to the design of instruction set architectures, and it is studied how the transfor-
mations on the states of the main memory of a strict load/store instruction set architecture
that can be achieved by executing instruction sequences on it depend on the parameters
involved.
Chapter 7 concerns two subjects related to process algebra, namely protocols to deal
with remote instruction processing and instruction sequence producible processes. If in-
struction processing takes place remotely, this means that a stream of instructions to be
processed arises at one place and the processing of that stream of instructions is handled
at another place. Process algebra is used to describe two protocols to deal with this phe-
nomenon. Because process algebra is considered relevant to computer science, there must
viii Instruction Sequences for Computer Science
be programmed systems whose behaviours are taken for processes as considered in process
algebra. It is shown that all finite-state processes can be produced by single-pass instruc-
tion sequences as considered in SPISA, provided that the cluster fair abstraction rule known
from the algebraic theory of processes called ACP is valid.
Chapter 8 introduces three variations of instruction sequences as considered in SPISA,
namely polyadic instruction sequences, instruction sequences without a directional bias,
and probabilistic instruction sequences. A polyadic instruction sequence is a possibly pa-
rameterized instruction sequence fragment that can produce a joint behaviour together with
other such fragments because the fragment being executed can switch over execution to
another. Instruction sequences without a directional bias require that for each instruction
whose effect involves that execution proceeds in the forward direction, there is a counter-
part whose effect involves that execution proceeds in the backward direction. Probabilistic
instruction sequences are instruction sequences that contain instructions that are themselves
probabilistic by nature.
There are also four appendices. In Appendix A, five challenges for the point of view
from which the approach to semantics followed in Chaps. 2 and 3 originates are sketched.
In AppendixB, some results about functional units for natural numbers are given, which are
except one computability results that are not directly related to existing results that we know
of. In Appendix C, the usefulness of the dynamically instantiated instructions introduced
in Chap. 3 is illustrated by means of an example. In Appendix D, a model of a hypothetical
execution environment for instruction sequences, designed for the purpose of explaining
how instruction sequences as considered in SPISA may be executed, is discussed.
A glossary of the notations introduced in this book and the general mathematical no-
tations used in this book can be found from page 221 onward. At this point, one further
remark about notation may be useful: bold-faced italic letters, with or without decorations,
will be used as syntactical variables in this book.
Acknowledgements
This book brings together and streamlines work done by a group of people which includes,
in addition to the authors, Inge Bethke, Marijke Loots, Alban Ponse and Mark van der
Zwaag. The work in question was partly carried out in the framework of projects funded
by the Netherlands Organisation for Scientific Research (NWO).
Amsterdam, April 2012 J. A. Bergstra and C. A. Middelburg
