Table Of Content.D Pearce .H Wansing (Eds.)
Nonclassical Logics dna
noitamrofnI Processing
International Workshop
Berlin, Germany, November 9-10, 0991
Proceedings
galreV-regnirpS
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J6rg Siekmann
University of Saarland
German Research Center for Artificial Intelligence (DFKI)
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Volume Editors
David Pearce
Heinrich Wansing
Free University Berlin, Department of Philosophy
Group of Logic, Epistemics and Information
Habelschwerdter Allee 30, W-1000 Berlin 33, FRG
CR Subject Classification (1991): 1.2.3, F.4.1
ISBN 3-540-55745-8 Springer-Verlag Berlin Heidelberg New York
ISBN 0-387-55745-8 Springer-Verlag New York Berlin Heidelberg
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Preface
This volume comprises the proceedings of the First All-Berlin Workshop no lacissalcnoN.
Logics and Information Processing which was held at the Freie Universits on the first
anniversary of the opening of the Berlin Wall, 9-10 November, 1990.
For the field of logic and computer science, the meeting provided one of the first
opportunities for researchers from all parts of the formerly divided city to come together
and exchange ideas. In addition, a grant fl'om the Berlin Senat fiir Wissenschaft dnl~
gnuhcsroF made it possible to invite several colleagues from outside the city. We gratefully
acknowledge this financial support, as well as further assistance from the Freie UniversitSt.
The scope of the workshop was broad, covering various different subfields of logic
(particularly nonclassical logic) and its applications in artificial intelligence. A number
of papers presented are appearing elsewhere and could not be included in this volume.
Nevertheless, the remaining papers appearing here provide a representative selection and
cover all the main themes discussed in the workshop. They have been grouped according
to the four major topics that emerged at the meeting: modal systems, logic programming,
nonmonotonic logics, and proof theory. The classification is only intended to provide a
rough and ready guide, however, since these areas, particularly the last three, overlap
considerably. Thus, for instance, the contributions of Heinrich Herre and Peter Schroeder-
Heister are also of special relevance for logic programming.
We would like to thank the speakers and participants of the workshop, for their stim-
ulating talks and discussions, the authors, who in many cases submitted substantially
revised and extended versions of their papers, and the referees, for their prompt re-
spouses to our requests for comments and evaluations. Lastly, a special word of thanks
to JSrg Siekmann for supporting the publication of these proceedings in LNAI, as well as
to Springer-Verlag for their patience in awaiting the final manuscript.
David aeP ecr dna Heinrich Wansing
Berlin dna Amsterdam, MaTvh 1992
Contents
Modal Systems
Algebraic Aspects of the Relational Knowledge Representation:
Modal Relation Algebras
Ewa Orlowska ...................................................................... 1
A Logic for Memory
32
Christian Schumacher . ............................................................
Actions with Preconditions and Postconditions
64
Michael Morreau ..................................................................
Logic Programming
Testclasses and Closed World Assumptions for Non-Horn Theories
65
J'ffrgen Gehne .....................................................................
Reasoning with Negative Information, II: Hard Negation, Strong Negation
and Logic Programs
36
David Pearce .....................................................................
Lindenbaum-Algebraic Semantics of Logic Programs
08
Gerd Wagner .....................................................................
Nonmonotonic Logics
Conditional Logics and Cumulative Logics
29
Andrd Fuhrmann ..................................................................
Semantics of Nonmonotonic Reasoning in Logic Programming
401
Heinrich Herre ..................................................................
Proof Theory
Formulas-as-types for a Hierarchy of Sublogics of Intuitionistic Propositional Logic
521
Heinrich Wansing ...............................................................
Cut Elimination for Logics with Definitional Reflection
t46
Peter Schroeder-Heister ..........................................................
ALGEBRAIC ASPECTS OF THE RELATIONAL KNOWLEDGE ~EP~ESENTATION:
MODAL RELATION ALGEBRAS
Ewa Orlowska*
Institute of Theorel:ical anti Applied Computer Science
Polish Academy of Sciences
Abstract
Semantics based on relation algebras is introduced for
various applied logics, in particular for modal logics
for ~nowledge representa-tion, epistemic logic, dynamic
logic and temporal logics. Several classes of modal re-
lation algebras are introduced and investigated.
1. Introduction
Interpretability of various propositional modal logics and
intuitionistic logic in a logic of binary ,-elations (Orlowska
1988) provides a method of establishing a correspondence between
logical systems and relation algebras. The correspondence shows
thai. applications of relation algebras can be expanded to provide
an algebraization of many nonclassical logics, in particular of
various AI logics. From a logical point of view the corresponden-
ce can be treated as the definition of a new semantics for these
logics and, moreover, it is the starting point-for developing
relational proof systems for those logics. Within the framework
of the theory of relation algebras the correspondence leads to
the concept of nonclassical relation algebras, in particular
modal relation algebras.
In the present paper we define classes of modal algebras of
relations associated with the basic modal logics K,T,B,S4,S5
(Segerberg 1971), logic G of provability (Boolos 1979), modal
logics with operators for 'inaccessible' possible worlds intro-
duced in Humberstone (1.983), temporal logics (Prior 1967, Burgess
1.979, Von Wright ~9G5), dynamic logic (Pratt 1.976, ttarel 1979)
and its extension introduced in Orlowska (1991a), the logic of
vague concepts and epistemic ogi. c (Orlowska 1983, 1989). Many
of those classes are generalized subreducts of the class of
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
*The address for correspondence: Azaiowa 29, 04-539 Warsaw, Po-
land
algebras of binary relations. Most of them are varieties, whoso
corresponding equations, axioms are direct counterparts of the
axioms of the respective logical systems.
In the more general setting developed in section 8, we consi-
der classes of modal relation algebras and classes of Boolean
algebras with operators determined by relation algebras.
The method developed in the paper, a reconstruction of noncla-
ssical logics as relation algebras and a reformu.ation of the
modal axioms as relat:ional equations, can be applied to fini-
tizability investigations. If a .ogical system is finitely a(cid:141)
matizable, then usually the corresponding class of nonclassical
relation algebras is finitely axiomatizable too. On the other
hand, since those algebras are subreducts of the class of algeb-
ras of binary relations, we obtain some of its finitely axiomati-
zable subclasses.
The relationship between modal logics and modal relation
algebras presented in the present paper can he extended to some
other families of nonclassical logics, such as Post's many-
valued logics (Orlowska i99~.b), and relevant logics (Orowska
1990).
2. Preliminaries
In the present section we recall the notion of an algebra of
binary relations (Tarski 1941, Henkin et al 1985, Nemeti 1990),
and the notions of various modal Kripke-style frames which pro-
vide standard semantics for modal propositional logics.
By a full 8Kebr~ of binary relations over a set W we mean an
algebra:
fulIREL(W)=(P(WxW),u~n,-,1,;,-l,I)
where (p(WxW),u,n,-,1) is the Boolean algebra of all the subsets
of WxW, ; and -i are the relational composition and converse,
respectively, and I is the identity relation.
The class of full algebras of binary relations and the class
of algebras of binary relations are defined as follows:
fulIREL={fulIREL(W): W is a set}
REL=S(fulINEL)
where S is the operation of taking isomorphic copies of subalgeb-
Fas.
The class REL has been investiK~ited in Tarski (1955), Monk
(1969), Jonsson (1982, 1991) and many others.
By a modsl Krip#e frame we mean a relational system:
K=(W,R)
where W is a nonempty set (of states), and R is a binary relation
in W (accessibility relation).
h frame K is a T-modal frame if the relation R is reflexive;
it is a B-modal frame if R is reflexive and symmetric; it is an
S4-modal frame if R is reflexive and transitive; and g is an SS-
modal frame if R is an equivalence relation. eW shall also consi-
der modal frames for logic G of provabiity (Boolos 1979). A
modal frame K is a G-frame if the relation R is transitive and if
R -I is well founded.
Formulas of the modal propositional language are built with
propositional variables taken from an infinite, denumerahle set
VARPROP, with classical propositional operations 7,V,A,~, and
with modal operations R,<R>, where R is a relational constant
interpreted as an accessibility relation. The set of modal
formulas is the smallest set including VARPROP and closed with
respect to all the propositional operations listed above.
By a model of the modal 18nguage we mean a system:
M=(K,m)
where K is a modal Kripke frame, nnd m:VARPROP~P(W) is a meaning
function which assigns sets of states to propositional variables.
We extend m to a satisfaction relation sat for arbitrary formulas
as follows. We say that a formula F is satisfied by a state w in
model M (M,w sat F) if the following conditions are satisfied:
(ml) M,w sat p iff wem(p) for peVARPROP
(m2) M,w sat nG iff not M,w sat G
(m3) M,w sat GIvG2 iff M,w sat Gl or M,w sat G2
(m4) M,w sat GI^G2 iff M,w sat GI and M,w sat Ga
(mS) M,w sat G~G2 iff M,w sat IGIvG2
(mS) M,w sat RG iff for all w'eW if (w,w')eR then M,w' sat G
(mT) M,w sat <R>G iff M,w sat aR~G.
By an extension of a formula F in a model M we mean the
following set:
extMF={weW: M,w sat F}
A formula F is true in mode M iff M,w sat F for all weW,
which is equivalent to eXtMF=W. A formula F is true in a frame K
~ff it is true in every model (K,m). A formula F is valid iff it
is true in all models. The logic defined above is the well known
modal io.gic K. If we restrict the class of models to the models
based on T,B,Sd,SS,G-modal frames, then we obtain the logics
T,B,Sd,SS,G respectively.
In Humberstone (1983) modal operators R and <<R>> have
been introduced and discussed:
(mS) M,w sat RG iff for all w'eW if (w,w')~R then M,w' sat G
(m9) M,w sat <<R>>G iff M,w sat nRnG.
The operators enable us to express in the modal language
properties of states which do not stand in relation R.
In the following we recall a semantic formulation of various
temporal logics. By a temporal frame we mean a relational system
of the form:
K=(W,R,R -~ )
where W is a nonempty set (of moments of time), R is a binary
relation in W which reflects the earlier-later relationship bet-
ween moments of time. Usually it is assumed that R is at least
reflexive and transitive.
In the language of temporal logics we have the classical
propositional operators, the modal temporal operators R,<R>,
R-I,<R-I> interpreted as 'always in the future', 'sometime in
the future', 'always in the past', 'sometime in the past'. The
semantics of the modal temporal operators is the same as before
(clauses (mS),(m7), and their counterparts for R-t). We obtain
various temporal logics by restricting the class of models of the
language to the models in which some specific properties of
accessibility relation are postulated. In the following we list
some of them.
(tl) Vu~v uRv R is serial
(t2) Vu,v(uRv~w(uRw^7~w'(uRw'Aw'Rw))) R is discrete
(t3) ))vRw^wRu(w~+vRu(v,uV R is weakly dense
(t4) Vu,v,w(uRv^uRw~vRw) R is euclidean
(t S)))w IvvvRwvwRv(~w~u^vRu(w,v,u~ R is weakly connected
(t6) ~u,v,w(uRv^uRw~w'(vRw'AwRw')) R is weakly directed
Temporal frames can also include a function s interpreted as a
successor function or next state function. By a standard temporal
.frame with successor we mean a relational system:
K=(W,R,s)
where R is a reflexive, transitive, and weakly connected relation
in ,W and s is a function in W which satisfies the following
conditions:
(sl) (w,s(w))~R
(s2) (s(w),s(w'))eR iff (w,w')eR
(s3) For any W~U if weU and for all u (w,u)eR and ueU imply
s(u)eU, then for all w' (w,w')eR implies w'eU.
Condition (s3) is an induction principle.
The language of the logic determined by that class of frames
(Yon Wright 1985, Segerberg 1979) includes propositional operator
o (operator 'next') whose semantics is:
(miO) M,w sat oG iff M,s(w) sat .G
In applications of modal logics extensions of the standard
modal language are considered which are obtained by adding modal
propositional operations determined by various accessibility
relations. Sometimes an algebraic structure is assumed in the
family of those accessibility relations. To define relation alge-
bras for multimodal logics, in particular for dynamic logic, we
need some extension of class REL.
By a full ex~ended algebra of binary relations over a set W we
mean an algebra
fulIEREL(W)=(P(WxW),u,n,-,I,;,-',I,*,/,\)
where (P(WxW),u,n,-,l,;,-~,I) is the full algebra of binary
relations ever W, * is the operation of taking the reflexive and
transitive closure of a relation, and /,\ are the operations of
right and left residuation, respectively, that is
x/y=-(y-~:-x)
x\y=-(-y;x-1).
The status of the operation * is similar to the status of the
operation of infinite join in generalized Boolean algebras (Ra-
siowa and Sikorski 1963).
The class of full extended algebras of binary relations and
the class of extended algebras of binary relations are defined
as follows:
fulIEREL=(fulIEREL(W): W is a set}
EREL=S(fulIEREL).
By a mul~imodal frame we mean a system K=(W,A) , where A is an
algebra from class EREL or from some of its subreducts, that is h
might be a subalgebra of a reduct of EREL. Multimodal frames are
semantical structures for various applied modal logics.
By a specifica~ion frame we mean a system K=(W,A) such that A
is an algebra from a subreduet of EREL obtaied by restricting
EREL to the operators u,;,/,\. The residuations are interpreted
as program specifications (Hoare and Jifeng 1988) in the
following way. Let relations O~YxZ and R~XxZ be given. The
weakest prespecification Q\R of Q to achieve R is the greatest
relation P~XxY such that P;OcR. Similarly, let relations P~XxY
and RcXxZ be given. The weakest postspecifieation R/P of P to
achieve R is the greatest relation OcYxZ such that P;Q~R.
A ~vnamic frmme is a frame K=(W,A) such that A is an algebra
from a subreduet of EREL obtained by restricting operations in
the algebras from EREL to the operators u,;,$. In the language of
dynamic logic we have a family of modal operators ~ and <R>,
where R is a relational term over that subreduet of EREL. In
dynamic frames accessibility relations are interpreted as prog-
rams, and relational operations are interpreted as program con-
structors. To define semantics of the extension of dynamic logic
introduced in Orlowska (1991a) we need multimodal frames deter-
mined by the whole class EREL of algebras. The residuations are
interpreted as program specifications.
An indiscernibillty frame is a frame K=(W,A), where A is an
algebra from a subreduct of EREL obtained by restricting opera-
tions in algebras from EREL to n and by assuming that the rela-
tions are equivalence relations. The relations may reflect a
degree of precision of an agent who recognizes objects from
universe .W If his perception ability is not perfect, then the
agent might not be able to grasp single objects. He only recog-
nizes classes of objects such that in each class there are ele-
ments which he considers to be the same up to his discrimination
resources. In this way equivalence classes of indiscernihility
relation play the role of atomic, indivisible structures which
carl be recognized by the agent. The discrimination ability of a
group of agents may be reflected by the intersection of the
indiscernib~lity relations corresponding to the members of the
group. Consider the following example. Suppose that the knowledge
of a group {a,b} of agents about the size of objects
wl,w2,w3,w4,w5 is:
a b
wl small small
w2 small very small
w3 large very large
w4 medium medium
w5 large medium
The indiscernibility relations lad(a) and ind(b) are defined
by the clause:
(w,u)eind(a)(ind(b)) iff the size of w equals the size of u
according to agent a(b).
The relations provide the following equivalence classes:
ind(a): {wi,w2} {w3,wb} {w4}
ind(b): {wl} {w2} {w3} {w4,w5}
The knowledge of agent a is not complete, he cannot distin-
guish between wl and w2, and between w3 and w5 with respect to
their siz 9. Similarly, agent b cannot distinguish between w4 and
w5 with respect to their size. The joint knowledge of group {a,h}
of agents is determined by the relation ind(a,b) defined as:
(w,u)eind(a,b) iff the size of w equals the size of u accor-
ding to both a and b.
Thus we have ind(a,b)=ind(a)nind(b). The equivalence classes
