Topics in the theory of computation: selected papers of the International Conference on ’’Foundations of Computation Theory’’, FCT ’83, Borgholm, Sweden, August 21-27, 1983’

188 Pages·1985·4.147 MB·English

by  Marek Karplnski and Jan van Leeuwen (Eds.)

NORTH-HOLLAND MATHEMATICS STUDIES 102
Annals of Discrete Mathematics( 24)
General Editor: Peter L. HAMMER
Rutgers University,N ew Brunswick, NJ, U.S.A.
Advisory Editors
C. BERG6 Universited e Paris, France
M. A. HARRISON, University of California, Berkeley, CA, U.S.A.
V. KLEE, Universityo f Washington, Seattle, WA, U.S.A.
J-H. VAN LINT; CaliforniaI nstitute of Technology,P asadena, CA, U.S.A.
G.-C.R OTA, Massachusetts Institute of Technology, Cambridge, MA, U.S .A.
NORTH-HOLLAND- AMSTERDAM 0 NEW YORK OXFORD
TOPICS IN
THE THEORY OF COMPUTATION
Selected Papers of the International Conference on
'Foundations of Computation Theory',F CT '83,
Borgholm, Sweden, August21-2z 1983
edited by
Marek KARPlNSKl
University of Pittsburgh
and
JanVAN LEEUWEN
University of Utrecht
1985
NORTH-HOLLAND -AMSTERDAM NEW YORK 0 OXFORD
0E lsevier Science Publishers B. V., 1985
All rights reserved, No part of this publication may be reproduced, stored in a retrieval system,
or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording or
otherwise, without the prior permission of the copyright owner.
ISBN: 0 444 87647 2
Publishers:
ELSEVIER SCIENCE PUBLISHERS B.V.
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THE NETHERLANDS
Sole distributors for the U.S.A. and Canada:
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Library of Congress Cataloglng In Pnbllcation Data
International Conference on ”Foundations of Computation
Theory” (1983 : Bxgholm, Sweden)
Topics In the theory of computation.
(Annals of discrete mathematics ; 24) (Ncrth-
Holland mathematics studies ; 102)
1. Computational complexity-Congresses. 2. Machine
theory--Congresses. 3. Electronic data processing--
Mathematics--Congresses. I. Karpifiski, Marek, 1948-
11. Leewen. 3. van (Jan van) 111. Title. IV. Series.
V. Series: krth-Hol&d mathematics studies ; 102.
QA267.156 1983 511 ah-21089
ISBN 0-444-87647-2 (U.S.)
PRINTED IN THE NETHERLANDS
V
PREFACE
This Volume of the Annals of Discrete Mathematics contains nine selected
papers presented at the International Conference on “Foundation of Computation
Theory” held at Borgholm, Sweden, in August 21-27, 1983. The full Conference’s
Proceedings have appeared in: M. Karpinski, ed., Lecture Notes in Computer
Science, Vol. 158, Springer-Verlag, Berlin-New York, 1983, pp. 1-517. The papers
of this Volume are complete and extended versions of the papers presented before
this Conference which were selected, and refereed according to the usual high
standards of the research journal publications.
The topics of nine papers selected by the Editors, should reflect the nature
of this Borgholm Conference. They were chosen on the basis of their immediate
relevance to the most fundamental aspects of the theory of computation, and the
newest developments in this area. The selection foci reflect thematically the idea
of the series of ‘Foundational‘ FCT-Conferences initiated in 1977 (LNCS 466,
Springer-Verlag).
The main areas of Foundations of Computation Theory covered by the Conference,
and in part represented in this Volume, fall into eight categories:
*
Constructive Mathematics in Models of Computation and Programming
(Constable);
*
Abstract Calculi and Denotational Semantics
(Langmaack, Parikh);
*
Theory of Machines, Computations, and Languages
(F reivalds, Trum-W ot schke) ;
*
Nondeterminism, Concurrency, and Distributed Computing
(Shmir-Upfal);
*
Abstract Algebras, Logics, and Combinatorics in Computation Theory
(Harel, Macintyre);
*
General Computability and Decidability
(Harel, Macintyre);
*
Computational and Arithmetic Complexity
(Freivalds);
*
Analysis of Algorithms and Feasible Computing
(v. Braunmiihl-Verbeek, Freivalds).
vi
We hope that the Volume will further help to promote, as well as advertise
the highest quality research in the area of Foundations of Computation Theory.
To ensure timeliness of the production, it was decided to use North-Holland's
camera-ready rapid reproduction system. At this place we want to thank heartily
Mariele Knepper of the Department of Computer Science, University of Bonn,
as well as Nancy Swiantek, Crystal Hamill of the University of Pittsburgh, and
Lydia Defilippo of Carnegie-Mellon University for the masterful supervision of
all operations during the process of selection, refereeing, and preparation for
printing with North-Holland.
The number of anonymous referees of the papers in this book deserves special
mention, their highest scholarly standards in refereeing were one of the major
factors behind the present final shape of publication. It is our great pleasure
to thank all authors for the smooth cooperation with the Editors, remarkably,
within exceedingly constrained time limits.
We hope this Volume succeeded in covering the new and exciting research
areas.
Marek Karpinski
Jan van Leeuwen
Editors
June I984
vii
C O N F E R E N C E C O M M I T E E S
"Foundations of Computation Theory- 1983"
Borgholm, Sweden, August 21-27, 1983
Program Committee:
K.R. Apt (Paris), G. Ausiello (Rome), A.J. Blikle (Warsaw), E. Borger
(Dortmund), W. Brauer (Hamburg), M. Broy (Munich), L. Budach (Berlin),
R. Burstall (Edinburgh), P. van Emde Boas (Amsterdam), F. Gecseg,
(Szeged), J. Gruska (Bratislava), M.A. Harrison (Berkeley),
J. Hartmanis (Ithaca), K. Indermark (Aachen), M. Karpinski (Bonn/
Pittsburgh), D. Kozen (Yorktown Heights), J. van Leeuwen (Utrecht),
L. Lovasz (Budapest), A. Mazurkiewicz (Warsaw), G.L. Miller
(Cambridge Mass.), P. Mosses (Aarhus), B. Nordstrom (Gothenburg),
M. Paterson (Warwick), A. Salomaa (Turku), C.P. Schnorr (Frankfurt),
J.W. Thatcher (Yorktown Heights).
Program Committee Chairman:
Prof. Marek Karpinski
Computer Science Department
University of Bonn
Wegelerstr. 6
D-5300 Bonn 1
W. Germany
Organizing Committee:
E. Sandewall (Chairman)
A. Lingas
J. Maluszynski
Organizing Secretary:
Lillemor Wallgren
Program Secretary:
Mariele Knepper
viii Conference committees
The International Conference on "Foundations of Computation Theory"
followed thematically the series of the FCT-Conferences founded in
1977 in Poznan-Kornik, Poland. The program of the Conference,
including invited lectures, and the selected contributions fell
into eight categories:
* Constructive Mathematics in Models of Computation and Programming
* Abstract Calculi and Denotational Semantics
* Theory of Machines, Computations, and Languages
* Nondeterminism, Concurrency, and Distributed Computing
* Abstract Algebras, Logics, and Combinatorics in Computation Theory
General Computability and Decidability
I(
* Computational and Arithmetic Complexity
* Analysis of Algorithms and Feasible Computing
Organized by:
The Department of Computer and Information Science, Linkoping
University and Institute of Technology
With the cooperation of:
The Departments of Computer Science and Mathematics, University
of Bonn
Under the auspices of:
The Royal Swedish Academy of Engineering Sciences
The European Association for Theoretical Computer Science
Gesellschaft fur Informatik e.V., Bonn
The Swedish Society for Information Processing
Co-sponsors:
ASEA
Ericsson Information System AB
IBM Svenska AB
The Swedish Philips Group
t
0 Elsevier Science Publishers B.V. (North-Holland) I
Input Driven Languages are Recognized in logn Space
by
Burchard von Braunmuhl
Rutger Verbeek
University of Bonn
-Abs-tra ct
We show that the class of input driven languages is contained in
logn space, an improvement over the previously known bound of
2
log n/loglogn space [Me 801. We also show the power of de-
terministic and nondeterministic input driven automata is the
same and that these automata can recognize e.g. the parenthesis
languages and the leftmost Szilard languages (cf. [Me 751,
~ L Y76 1, [Ig 771).
1. Introduction
Since the famous algorithm of Lewis, Hartmanis and Stearns [LSH 651
many attempts have been made to determine the exact lower and upper
bounds for the recognition of context free languages. It is quite
2
improbable that the log n upper bound can be improved for general
CFLs because every improvement would improve the bound for the de-
terministic simulation of nondeterministic Turing machines [Su 751.
2
Even for deterministic CFLs no better space bound than log n
is known. Only the simultaneous time-space bound for DCFLs is much
better than in the general case of CFLs: using a black pebbling
strategy for the computation graphs of deterministic pushdown
automata Cook [Co 791 has shown that DCFLs can be recognized using
2
logn pebbles (i.e. in log n space) and polynomial time simulta-
neously. The strategy was improved by the authors [BV 8o],[BCMV 831
and shown to be almost optimal (as pebbling strategy) [Ve 811,
[Ve 831. On the other hand several important subclasses of DCFLs are
known to be recognized in logn space (e.g. Dyck sets [RS 721,
parenthesis languages [Me 75],[Ly 761 and leftmost Szilard languages
[Ig 771). All these examples are shown to be included in, or are
easily (i.e. by a homomorphism) transformable to the class of input
driven languages (defined in [Me 80]), which is already known to
2
require a bit less than log n space: according to Mehlhorn [Ke 801
2 B. u. Braunmuehl and R. Verbeek
there is a black pebbling strategy with (log n/ log log n) pebbles
where every pebble requires only O(log1ogn) space.
In this paper we present a logn space algorithm for the recogni-
tion of input driven languages. Our algorithm uses a black and white
pebbling strategy [CS 761 for the computation graphs, where we make
use of the fact, that the structure of the computation graph of an
input driven PDA can be computed by a counter automaton (and hence
in logn space). The pebbles are distributed on the graph in such
a way that their positions need not to be stored but can be recom-
puted. In this way our algorithm uses O(1ogn) pebbles, where each
pebble requires only bounded space. Our approach gives a simple
proof for the space complexity of a larqe subclass of DCFLs. In
terms of pebbling complexity the strategy is optimal: it pebbles
every n-node computation graph using O(1ogn) pebbles in n steps
and there are computation graphs which require n(1ogn) black and
white pebbles [Ve 831.
Since our aim is to prove a space bound for some DCFLs we shall pre-
sent the algorithm directly as simulation algorithm for input driven
PDAs rather than as black and white pebble game.
Unfortunately, there exist some classes of DCFLs that are contained
in DSPACE(1ogn) but are not obviously transformable to input driven
languages (certainly they all are logn- space reducible to any
other nontrivial language): counter languages (where the space bound
is obvious), the languages recognized by finite minimal stacking
PDAs [Ig 781 (which can be proved to be accepted in logn space via
a simple black pebbling strategy with a bounded number of pebbles)
and the two sided Dyck sets [LZ 771, which is the only known
example of a non input driven DCFL in DSPACE(1ogn) for which the
space bound is not obvious.
In section 2, we show that parentheses languages are input driven,
leftmost Szilard languages are homomorphic transformable to input
driven languages and the (nondeterministic) input driven languages
are all deterministic (i.e. acceptable by deterministic input driven
PDAs) which again proves that parentheses languages are DCFLs.
In section 3, we present the simulation algorithm for input driven
languages which uses log n space and n2 time on a Turing machine.
Inputdriven languages are recognized in log n space 3
2. Properties of input driven languages
We call a (deterministic or nondeterministic) real-time pushdown
automaton (PDA) "input driven", if it reacts to some input symbol
always with a push (the stack grows) or always with a pop (the stack
falls). The moves of the stack are controlled only by the actual in-
put symbol. The sequence of push and pop, (i.e. the graph of the
stack height function h(t) (1 2 t 4 n)) is determined by the in-
put and is readable from it.
Definition
A real-time pushdown automaton (PDA) is denoted by
P = (CrrrQr#rqotQf,6)
where Z,r,Q are finite disjoint sets of input symbols, pushdown
symbols and states, # E r (bottom symbol), Qf c_ Q (accepting
states), q, E Q (initial state), and 6 is a finite subset of
Q x r x z x Q x r*.
P is input driven, (idPDA) if 1 is the disjoint union of
u2 .
CorC1,C2 and 6 C Q x r x Ci x Q x Ti (x E Zo means pop,
i=o
x E I2 means replace and push)
A language L 5; I* is input driven (idCFL), if it is recognized by
some input driven PDA.
A configuration is a pair (q,W) containing the state q and the
content of the pushdown store W. For every w E Z*, q,q' E Q,
X E r, W,W' E r*, we define - I W inductively by
<
(SfWX) (q,WXI,
wx
(q,WX) (q',W') iff there are Wl,W2 E T*, Y E r, q1 E Q,
Such that (qrwx) (ql,WIY), (ql,Y,x,q'rW2) E 6
and W' = WlW2.
If xl...xn is the input word, we define the stack height at time
t (that is just before step t) by
h(t) = \W\ if (q,#) (q',W) for 1 5 t I n+l.
xl".x t- 1
is the empty word).
(Xixi-1