Table Of ContentLecture Notes ni
Computer Science
Edited by .G Goos and .J Hartmanis
159
CAAP'83
Trees ni Algebra and Programming
8th Colloquium
I'Aquila, March 1983 9-11,
Proceedings
Edited by .G Ausiello and .M Protasi
IIIIII II
galreV-regnirpS
nilreB Heidelberg New York oykoT 1983
lairotidE Board
D. Barstow W. Brauer R Brinch Hansen D. Gries D. Luckham
C. Moler A. Pnueli G. re110mgeeS .J Stoer N. Wirth
srotidE
Giorgio Ausieito
University of Rome
.V Eudossiana ,81 1-00184 Rome
Marco Protasi
University of CAquita
.V Roma 33, 1-67100 I'Aquiia
CR Subject Classifications (1982): ,1D D3, El, E2, F2, F3, ,4F G2, G3
ISBN 3-540-12727-5 Berlin Springer-Verlag Heidelberg New York oykoT
ISBN 0-387-t2727-5 Springer-Verlag New Heidelberg York Berlin oykoT
Library of Congress Cataloging in Publication Data. CAAP'83 (8th: Italy) 1983: l~Aquita,
Trees in algebra and programming. (Lecture notes in computer science; 159) English and
French. |. Trees (Graph theory)-Data processing-Congresses. 2. Data structures
(Computer science)-Congresses. 3. Computational complexity-Congresses. .I Ausietlo,
G. (Giorgio), 194t-. .11 Protasi, M. (Marco), 1950-. Title. IlL .VI Series. QA166.2.C3 1983
511'.5 83-16909
ISBN 0-38?-12727-5 (U.S.)
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8th COLLOQUIUM ON TREES IN ALGEBRA AND PROGRAMMING
Foreword
The preceding colloquia on Trees in Algebra and Programming were
held in Lille (France), under the name of "Colloque de Lille sur les
Arbres en Algebre et en Programmation", with the exception of 1981
when the Colloquium was held in Genoa, Italy.
Traditionally the topics of the Colloquium cover a wider area of
theoretical computer science than it is indicated by the title. In fact
topics include formal aspects and properties of trees and, more gene-
rally, of combinatorial and algebraic structures in all fields of Com-
puter Science: theory of algorithms and computational complexity, for-
mal languages and automata, theory of sequential and parallel programs,
theory of data structures and data bases, algebraic specification of
software, etc.
The 8th Colloquium was held in L'Aquila, Italy, 9-11 March 1983,
organized by G. Ausiello, A. D'Atri, P. Mentrasti and M. Protasi,
under the sponsorship of the University of L'Aquila, the University of
Rome and the European Association for Theoretical Computer Science
(EATCS).
The Program Committee consisted of G. Ausiello (Rome, Chairman),
A. Arnold(Poitiers), E. Astesiano (Genoa), AoBertoni(Milan), C. B~hm
(Rome), M. Dauchet (Lille) H. Ehrig (Berlin), P. Flajolet (Paris),
G.Kahn(Paris) B. Monien (Paderborn), U. Montanari (Pisa), M. Nivat
(Paris), C. Papadimitriou (Athens), M. Protasi (L'Aquila), J. Thatcher
(Yorktown Heights).
Of the total of 50 papers submitted 12 were selected by the Pro-
gram Committee. The members of the Program Committee were assisted by
the following referees:
L. Aiello, V. Ambriola, G. Attardi, P. Bertolazzi, S.L. Bloom, M. Coppo,
G. Costa, B. Courcelle, W. Damm, A. D'Atri, N. De Francesco, P. Degano,
P. Della Vigna, M. Dezani, J.H. Gallier, G. Grandis, I. Guessarian,
A. Kreczmar, G. Levi, G. Longo, A. Maggiolo Schettini, A. Marchetti
Spaccamela, G. Mauri, C. Montangero, M. Ornaghi, P. Padawitz, A. Resti-
vo, B. Robinet, E. Saint-James, D. Sannella, C. Simone, M. Talamo, F.
VI
Turinir Bo Vauquelin~ M. Venturini~ As Walkerf E. Zucca.
Besides five invited lectures were given by M. Dezani, R. Fagin~
Z. Galil; Go Rozenberg and M. Wirsing~
We gratefully acknowledge the suppor t provided by:
- Consiglio Nazionale delle Ricerche; Comitati Nazionali per le Scien-
ze di Ingegneria~ Matematiche e Tecnologiche,
Provincia and Comune de L'Aquila,
-
Scuola Superiore "G. Reiss Romoli".
-
Finally we express our gratitude to the technical staff .E( Basi-
le, D. Danieli~ G. Lauri, N. Ricci) ~ to all colleagues of the Istituto
di Matematica dell'Universit~ de L'Aquila and in particular to the
Chairman prof. G. Saban, who participated in the organization.
Giorgio Ausiello Marco Protasi
Istituto di Automatica Istituto di Matematica
Universit~ di Roma Universit~ de L'Aquila
March 1983
CONTENTS
Invited lectures
M. BROY, M. WIRSING
Generalized heterogeneous algebras and partial
interpretations ........ ........................................ I
M. COPPO, M.DEZANI, G. LONGO
Applicative information systems. ............................... 35
R. FAGIN
Acyclic database schemes (of various degrees): A painless
introduction ................................................... 65
Z. GALIL
Efficient algorithms for finding maximal matching in graphs .... 90
D. JANSSENS, G. ROZENBERG
A survey of NLC grammars. ...................................... 114
Contributed papers
J. BEAUQUIER
Prefix and perfect languages. .................................. 129
E. BEST, M.W. SHIELDS
Some equivalence results for free choice nets and simple nets
and on the periodicity of live free choice nets ................ 141
M. A. BONUCCELLI, E. LODI, F. LUCCIO, P. MAESTRINI, L.PAGLI
VLSI mesh of trees for database processing ..................... 155
M.F. CLAEREBOUT, E. LILIN
Continuit4 des transducteurs d'4tats finis d'arbres ............ 167
M. COPPO, E. GIOVANNETTI
Completeness results for a polymorphic type system ............. 179
B. COURCELLE, F. LAVANDIER
A class of program schemes based on tree rewriting systems ..... 191
F. FAGES , G. HUET
Complete sets of unifiers and matchers in equational theories.. 205
P. FLAJOLET, N. SAHEB
Digital search trees and the generation of an exponentially
distributed variate ............................................ 221
IV
J. GONCZAROWSKI
Decidable properties of monadic recursive schemas with a
depth parameter ............................................ 236
S. ISTRAIL, C. MASALAGIU
Nivat-processing systems:Decision problems related to
protection and synchronization ............................. 255
J.P. JOUANNAUD
Confluent and coherent equational term rewriting systems.
Application to proofs in abstract data types ............... 269
A. LINGAS
An application of maximum bipartite C-matching to subtree
isomorphism ................................................ 284
B. MAHR, J.A~ MAKOWSKY
Characterizing specification languages which admit initial
semantics .................................................. 300
F.S. MAKEDON, C.H. PAPADIMITRIOU, I.H. SUDBOROUGH
Topological bandwidth ...................................... 317
A. MARCHETTI SPACCAMELA, M. TALAMO
Probabilistic analysis of graph colouring algorithms ....... 332
B. MONIEN, E. SPECKENMEYER
Some further approximation algorithms for the vertex
cover problem .............................................. 341
J. PARADAENS, D. VAN GUCHT
An application of the theory of graphs and hypergraphs
to the decomposition of relational database schemes ........ 350
W. RYTTER
Remarks on the pyramidal structure ......................... 367
D.T. SANNELLA, R.M. BURSTALL
Structured theories in LCF ................................. 377
G. SLUTZKI
Alternating tree automata .................................. 392
M. VENTURINI ZILLI
Co-finality in reduction graphs ............................ 405
Generalized Heterogeneous Algebras and Partial
Interpretations*
Manfred Broy
Martin Wirsing
Fakult~t f~r Informatik
Universit~t Passau
Postfach 2540
D-839 Passau
Abstract
A notion of heterogeneous algebras generalizing the concepts of
total, partial and regular algebras si introduced consisting of a
family of carrier sets, a family of total functions and a family
of definedness predicates. Partial interpretations are families of
partial functions mapping such generalized algebras
homomorphically onto partial heterogeneous algebras. Classes of
generalized algebras can be specified equationally by generalized
abstract types. This notion of abstract type si particularly well
suited for the description of programming languages since the
notion of homomorphism between generalized algebras allows to
obtain fully abstract models ni a uniform way - as weakly terminal
models. Sufficient conditions for the existence of initial and
terminal models and interpretations are given, the model classes
of generalized abstract types si analysed using lattice-theoretic
methods and the relationship to the classical concept of partial
functions si explained. The main advantage of this approach - a
uniform treatment of "strict" and "nonstriet" functions - si
shown by an extended example where lla reoursively enumerable
(finite and infinite) sequences of natural numbers are specified
as a generalized abstract type with nonstrict basic functions.
)* This research was partially sponsored by the Sonderforschungsberelch ,94
Programmiertechnlk, Munich
I. Introduction
As well-known in the meanwhile, data structures as well as programming languages can
be seen as algebras. More precisely it is convenient to use heterogeneous algebras.
Such algebras have been introduced in mathematics for avoiding partial functions. In
computer science heterogeneous algebras are used additionally for imposing adequate
structure on the universe of objects arising in some formal specification. In contrast
to algebraic structures in mathematics computation structures in information
processing often comprise a large number of distinct sorts and operations, such that
structuring the Universe by introducing sorts is much more important. Moreover, in
computer science partial functions arise naturally from nontermlnating computations.
So it is very convenient to use partial heterogeneous algebras as semantic models (of.
Broy, Wirsing 82, Wirsing et al. 80) in computer science.
Partial functions have been used in mathematical logic as an adequate concept for
dealing with computable functions. They provide a natural way to think about
algorithms that sometimes may not terminate. In computer science there is a tendency
to work with monotonic functions on complete partially ordered sets (called domains).
It is one of the goals of this paper to demonstrate, that partial functions always are
sufficient and how they are related to domain theory. Of course, partial algebras can
trivially be totalized by introducing an artificial element (often denoted by &). Then
one obtains trivially ordered algebras ("flat" or "discrete" domains) where all
functions are strict, i.e. if & occurs as argument of a function application, then the
value of the application is &, too. So at a first glance algebraic theories, specified
by first order equations~ with partial algebras as semantic models, seem very limited
if compared with the theory of domains as introduced by Scott (Scott 70). One way
out just seems to incorporate Scott's theory into algebras using continuous ordered
algebras (ADJ 75) and ordered-theoretically specified algebraic theories (M~ller
82).
However, apart from these possibilities we may also use purely equational algebraic
theories for the specification of general computation structures. This shows in
particular the close connection between partial functions and algebras and semantic
models in computer science.
In the following sections we are going to develop the formal foundations for the
specification of algebraic theories, defined by conditional equations and a definedness
predicate, with semantic interpretations that directly correspond to nonstriet
functions and the so-called infinite objects.
After recalling some basic notions (section 2) for algebraic specifications we
introduce in section 3 generalized algebras consisting of families of total functions,
carrier sets and definedness predicates. The definedness predicates divide the objects
of a generalized algebra into two groups, "defined" and "undefined" objects. We show
how total, partial and regular algebras can be represented in this framework.
Regular algebras are structures where the undefined objects do not contain any
information which can not be expressed already by defined objects. Such algebras have
been introduced by Kleene Kleene 52. Basic data structures (for example,
truth-values with "conditional" or "parallel or", naturally extended natural numbers
Manna 74 or any flat domain with naturally extended basic functions) are regular.
In section 4 homomorphisms between generalized algebras are defined. A generalized
homomorphism p : A -> B (where A and B are generalized algebras) is a family of
functions from the defined objects of A Into the defined objects of B which satisfies
the usual homomorphism property (on defined objects). Thus this notion generalizes
the notion of total homomorphism for partial algebras (cf. [Gr~tzer 68]) and - similar
to partial algebras - this notion of homomorphism distinguishes generalized algebras
from "classical" total heterogeneous algebras with one relation symbol.
In section 5 a partial algebra A' is associated to every generalized algebra A by
extending the usual notion of interpretation of terms to partial functions - the
so-called partial interpretations. Since generalized algebras may contain "nonstrict"
functions there might exist defined terms which contain undefined subterms. Such
"partial terms" correspond to (nullary) function symbols in the partial algebra A'
whereas they are undefined in the kernel KERN(A) of A, that is the partial algebra
derived from A' by considering only the partial interpretations of the function
symbols of A. Thus in Contrast to A', the partial terms are undefined in KERN(A).
KERN(A) represents A' if A itself represents a partial algebra (section 6).
In section 7 substantially equivalent generalized algebras, that are generalized
algebras which are isomorphic w.r.t, generalized homomorphisms, are studied. It is
shown that substantially equivalent generalized algebras might be not isomorphic w.r.t.
usual homomorphisms (for total algebras). In fact, for a maximal class C of
substantially equivalent generalized algebras the isomorphism classes of C (w.r.t.
usual homomorphisms) form a complete lattice w.r.t, the ordering ~ induced by the
usual homomorphisms. The least element of this lattice is a class of fully abstract
algebras an explicit uniform construction of which is given in section 8. If a
generalized algebra represents a partial or regular algebra then the corresponding
fully abstract algebra has at most one undefined object in every carrier set and can
be considered as a family of flat domains the least elements of which are exactly the
undefined objects.
Initial and (weakly) terminal elements of a class C of generalized algebras are defined
as usual, but w.r.t, generalized homomorphisms (section 9). All initial algebras of C
are substantially equivalent but they might be non-isomorphic (w.r.t. usual
homomorphisms). The same holds for the weakly terminal algebras of C. Moreover,
similar to partial algebras initial and weakly terminal generalized algebras are
minimally defined (under some conditions), that is in such algebras a term is defined
iff it is so In all algebras of the class C.
Finally, in section 10 , generalized abstract types are introduced and studied by using
the results of the previous sections. A generalized abstract type consists of a
signature and a set of axioms which are first-order formulas built over "=" and a
relation symbol "D" representing the definedness predicate. As semantics we consider
all term-generated generalized algebras which satisfy the axioms. We characterize
initial and weakly terminal models and show that for a generalized abstract type T
with positive conditional axioms initial and (under some conditions) weakly terminal
models always exist. The class M of minimally defined models of T has an interesting
structure:
The classes of substantially equivalent models form a complete lattice w.r.t, the
ordering s ~ induced by generalized homomorphlsms. The initial models of T are the
greatest and the weakly terminal models of T are the least element of this lattice.
The isomorphism classes of initial models of T as well as those of the weakly
terminal models of T form a complete lattice I (Z reap.) w.r.t, the ordering ~.
The isomorphism classes of M form also a complete lattice L w.r.t. . ~ The
greatest element of I is also the greatest element of L whereas the least element
of Z is also the least element of L.
Applied to programming languages (as specified for example in [Broy, Wirsing 80]) one
may associate the minimally defined models of a type with least fixed point semantics.
The initial models then eharacterise some kind of operational equivalence between
programs and the weakly terminal models characterise some kind of mathematical
equivalence. In particular, the greatest element of L - the "initial ''2 models -
corresponds to the continuous algebra semantics of programming languages as specified
in [ADJ 80] whereas the least element of L - the "weakly termlnal 2" models - gives
the most abstract semantics which is consistent with the axioms.
In section 11 hierarchies of generalized abstract types are introduced which satisfy
the following hierarchy-constralnt: If a hierarchical generalized abstract type T is
based on a generalized abstract type P then the defined part of every model of T
restricted to P is the defined part of some model of P. Sufficient conditions are
developed which ensure that a hierarchical generalized abstract type properly reflects
the hierarchy and in section 12 fully abstract models of hierarchical generalized
abstract types are constructed.
Finally, in section 13 the set of all finite and infinite recurslvely enumerable
sequences of natural numbers is specified as hierarchical generalized abstract type
based on a type of natural numbers and a type of truth-values with "parallel or" .
2. Basic Notions
We assume the reader to be familiar with the basic notions of algebraic specifications
as found for instance in [Broy, Wirsing 82] or [Wirslng et al. 80], that are the
notions of sisnature X : ,S( F), total (and partial resp.) heterogeneous X-al~ebr a A
: < {As}sgS, {fA)fgF ,> where s A denotes the carrier sets of A and fA the total
(and partial reap.) operations of A, term al~ebra W(X, X) = < {W(X, X}s)sGS,
{fW(X)}fgF > over the family X : {Xs} sgS of free variables, and X-homgmorphism
: A -> B. X-terms (of sort s) are elements from W(X, X) s.
Given a X-algebra A a ~.o.itaulav V : X -> A is family of total functions s s : V X
-> . s A For every valuation one may derive an interpretatign .# : W(Z, X) -> A from
the term algebra into A which is inductively defined as follows:
vAx :def Vs(X) if x g Xs, s~S,
fm fod: #),tecaf
..... ..... ctn)# ,
if f I : s x...x n s -> sgF and ti~W(>-,X)si i=I, ... , n.
