Table Of ContentLecture Notes ni
Computer Science
Edited by .G Goos and .J Hartmanis
521
I I II
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Workshop, HTE ZLirich, May-July 1979
Edited by Erwin Engeler
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nilreB, Heidelberg NewYork 1891
Editorial Board
W. Brauer P. Brinch Hansen D. Gries C. Mo~er G. Seegm~Jtler
.J Stoer N. Wirth
Editor
Erwin Engeter
Mathematik, ETH-Zentrum
CH-8092 Z~rich
AMS Subject Classifications (1980): 68C01, 68 B10, 681315, 03 B45, 03 D45
CR Subjeot Classifications (1981): 5.21, 5.24
ISBN 3-54041160-3 Springer-Veflag Berlin Heidelberg New York
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PREFACE
Since the early beginnings of research on the logic of programs there
have developed a small number of groups who have contributed to
various aspects of the subject. It was my privilege to have contacts
with some of these and, as an early contributor, I felt that the sub-
ject would benefit considerably from bringing people together on an
informal basis to exchange views and results.
With the generous support of the ETH Forschungsinstitut f~r Mathema-
tik, a workshop was held in the summer of 1979. It consisted of in-
formal seminars and discussion periods at which the following visitors
made presentations: D. Harel, J. Makowsky, A. Meyer, Go Mirkowska,
R. Parikh, V. Pratt, A. Salwicki, J. Tiuryn.
In this volume we now present a collection of papers that grew out
of the contacts and tastes formed at the workshop. They constitute
a cross-section of current approaches to the subject as they happened
to be discussed at the workshop. I hope that it proves to be repre-
sentative. In any case, we have tried to include adequate surveys
with historical notes. The papers were refereed, which may help to
excuse the delay. I wish to thank everybody involved in this effort.
September 1981 Erwin Engeler
TABLE OF CONTENTS
D. Kozen (IBM Yorktown Heights, NY, USA)
On the Duality of Dynamic Algebras and Kripke Models ........ I
A. Meyer, R.S. Streett and G. Mirkowska (MIT, Cambridge, Mass.,
USA; University of Warsaw, Poland)
21
The Deducibility Problem in Propositional Dynamic Logic .....
G. Mirkowska (University of Warsaw, Poland)
PAL - Propositional Algorithmic Logic ....................... 32
R. Parikh (BOston University and MIT, Cambridge, Mass., USA)
Propositional Dynamic Logicsof Programs: a Survey ........... 201
A. Salwicki (Polish Acad. Sci., Warsaw, Poland)
On the Algorithmic Theory of Dictionaries ................... 541
A. Salwicki and T. MHldner (Inst. of Math. Machines, Warsaw,
Poland)
On the Algorithmic Properties of Concurrent Programs ........ 961
J. Tiuryn (Warsaw University and MIT)
A. Survey of the Logic of Effective Definitions ............. 89'1
ON THE DUALITY OF DYNAMIC ALGEBRAIS AND KRIPKE MODELS
Dexter Kozen
IBM Thomas J. Watson Research Center
Yorktown Heights, New York 10598
ABSTRACT
We introduce a natural topology on Kripke models of propositional dynamic logic (PDL) and
exhibit a useful duality between certain topological Kripke models and separable dynamic
algebras analogous to the duality between Boolean algebras and their Stone spaces. We prove
several results which clarify the role of the * operator of PDL and characterize the discrepancy
between the standard and nonstandard models in terms of well-understood topological
concepts.
Introduction
Dynamic a~gebras [K,Pr3,Pr4] are two-sorted algebras (K,B,<>) where K is a Kleene
algebra (or relation algebra), B is a Boolean algebra, and <> is a scalar multiplication
K × B--~ B. They provide an algebraic interpretation of propositional dynamic logic (PDL)
which generalizes the standard Kripke model interpretation. In [K] we gave a representation
theorem for dynamic algebras: any separable dynamic algebra is isomorphic to the characteris-
tic algebra of a (possibly nonstandard) Kripke model. This raised the question: is every
dynamic algebra is isomorphic to the characteristic algebra of a standard Kripke model? In
another paper we shall give a negative answer to this question. The construction of the
counterexample uses topological concepts, and motivates the definition of topological Kripke
models. In this paper we develop some properties of these structures. The main result is a
duality between separable dynamic algebras and certain topological Kripke models. This
duality is completely analogous to the duality between Boolean algebras arid their Stone
spaces. It is useful because it gives an alternative topological characterization of models of
PDL in addition to the algebraic one of [K,Pr3,Pr4]. In the last section we give several results
which clarify the discrepancy between the standard and nonstandard Kripke models of PDL in
terms of well-understood topological concepts. We hope that the introduction of concepts and
techniques from general topology will be as fruitful in the study of dynamic logic as it has been
in the study of classical first order logic.
Dynamic algebras
Recall from [K] that a Kleene algebra (or relation algebra) K is a structure
K = (K,u,0,;,X,*)
such that (K, ~ , 0) is an upper semilattice with identity 0, (K, ;, )X is a monoid, and * is a
unary operation satisfying the axioms
a;(/3 u v) = a;/3 u a;v
aC u/3);v = a;y u/3;r
~;0 = 0;a = 0
(I) a;t~*;y = sup n Y';n81;a
where in (I), 0 a = ,X ~n+l _ a;an, and the supremum is with respect to the semilattice order
< inK.
A dynamic algebra is a structure
D = (K,B,<>)
where K is a Kleene algebra, B is a Boolean algebra, and <> is a scalar multiplication
K x B ~ B satisfying the axioms
<a u /3>X = <a>X V </3>X
<a>(X V Y) = <a>X V <a>Y
<,~>(</~>X) = <a;/3>X
<a>O = <0>X = 0
<?~>X = X
(2) <a*>X = suPn <an>x
where in (2), the supremum is with respect to the lattice order < in B. A dual multiplication
[a]X is defined by
[a]X = -,<a>~X.
Some authors have considered a reverse operator -. This is axiomatized by
(=;#)- tr;,r
=
(3) (~ u #)- = ~- u #--
It can be shown that a*- -- a-* for any tc ~ K. In dynamic algebras, - interacts with scalar
multiplication according to the axiom
(4) X < [a]<a->X
for all X E B. ]Dynamic algebras abound in the theory of computation; see [K,Pr3,Pr4] for
numerous examples.
ekpirK models
The most common examples of dynamic algebras are given by the Kripke models of PDL
[FL]. Recall that a Kripke model is a structure
A .-.- (S,K,B)
where S is a nonempty set of states, B is a Boolean algebra of subsets of S with the set-
theoretic Boolean algebra operations, and K is a Kleene algebra of binary relations on S in
which u is set union in K, ; is relational composition, X is the identity relation, 0 is the null
set, - is the reverse operator
~- = { (s,t) I (t,s) ~ ~ },
and * is any unary relation satisfying (1) and (2), such that the set
(5) <a>X = {s ~' 3t ~ X (s,t) ~ a}
is in B whenever a ~ K and X e B. If (5) is taken as the definition of scalar multiplication
<>, then the structure (K,B,<>) is a dynamic algebra, called the characteristic algebra of A
and denoted C(A). A Kripke model is called standard if a* = a rtc for any a e K, where
(6) a rtc = n n a U
is the reflexive transitive closure of a; otherwise it is called nonstandard. In any Kripke model,
a* is a reflexive, transitive relation containing a, and is the least element of K satisfying this
property. Thus in a nonstandard model, a* contains a rtc, but they need not be equal.
Separability and the representation o~ dynamic algebras
Definition 7. A dynamic algebra (K,B,<>) is separable if for every a, 3/ ¢ K, a #/3, there
exists an X ¢ B such that <a>X # </3>X. A Kleene algebra K is separable if there exists a
separable dynamic algebra over K. []
Not all Kleene algebras are separable [K]. See Pratt [Pr3,Pr4] for a discussion of the
importance of this property.
In [K] we showed that the condition on K of separability was sufficient for a dynamic
algebra D = (K,B,<>) to be isomorphic to C(A) for some (possibly nonstandard) Kripke
model A. We outline here the slightly weaker result that every separable dynamic algebra is
isomorphic to some C(A). The construction uses a technique similar to the Stone representa-
tion theorem for Boolean algebras. Let U, V denote ultrafilters of B, and let S be the set of
all ultrafilters. Let
x'={u~x~u}, B'={X' t XEB},
d = {(U,V) I YXe V <a>X~ U}, K' --- {a' t ae K},
and let S(D) be the structure
S(D) = (S,K',B').
As shown in [K], S(D) is a Kripke model and the map r is a dynamic algebra isomorphism
D--~ C(S(D)).
The representation theorem of [K] did not account for dynamic algebras with a reverse
operator -, but by adding an extra case to the argument of [K, Theorem 48] the result is easily
extended to include such algebras. Specifically, the extra case to be shown is
(8) VX e V <a->Xe U iffVY • U <a>Ye V.
Since a-- = a, we need only show the implication in one direction. Suppose the left side of
(8) holds. IfY • Uthen[a-]<a>Y c U, by(4). By[K, Lemma45], <a>Y• V. []
Topological Kripke models
In this section we introduce topological Kripke models and develop some of their basic
properties.
Definition .9 A topological Kripke model is a Kripke model A = (S,K,B) with topology
on S generated by B. []
If D is a separable dynamic algebra, we shall call S(D) with this topology the Stone space
of D, by analogy with the Stone duality for Boolean algebras. S(D) is compact and Hausdorff
[BS, Theorem 1.6.5].
A topological Kripke model has a base of closed and open (clopen) sets, namely the
elements of B. If A is compact, then the ctopen sets are exactly the elements of B [BS, Lemma
1.6.4]. The topology on S induces a product topology on S x S; terms such as open, closed,
etc. applied to subsets of S × S refer to this topology, cl(X) and cl(a) denote topological
closure.
In the following, A = (S,K,B) is a topological Kripke model, a, 3/ denote subsets of
S x S, and X, Y denote subsets of S. If P is any property of sets, we say that <a> preserves P
sets if <a>X is P whenever X is P. The definition for [a] is similar.
Note that <a> preserves open (closed) sets iff [a] preserves closed (open) sets, and if
a E K and A is compact, then <a> and [a] both preserve clopen sets.
