Representing Incomplete Knowledge in Abductive Logic Programming 1 Introduction

29 Pages·2015·0.25 MB·English

Representing Incomplete Knowledge in Abductive
Logic Programming
Marc Denecker Danny De Schreye
Department of Computer Science, K.U.Leuven,
Celestijnenlaan 200A, B-3001 Heverlee, Belgium.
e-mail : fmarcd, [email protected]
Abstract
Recently,Gelfond andLifschitz presentedaformallanguage forrepresenting
incomplete knowledge on actions and states, and a sound translation from this
language toextended logic programming. Wepresent analternative translation
toabductive logic programming withintegrity constraints andprovethesound-
nessandcompleteness. Inaddition, weshowhowanabductiveprocedurecanbe
used, not only for explanation,but also for deduction and proving satis(cid:12)ability
under uncertainty.
From a more general perspective, this work can be viewed as a -successful-
experiment in the declarative representation of and automated reasoning on
incomplete knowledge using abductive logic programming.
1 Introduction
An outstanding problem in logic programming is the representation of incomplete
knowledge. In [17], Gelfond and Lifschitz propose an interesting methodology to
evaluatethe expressivityofaformalismforrepresenting incompleteknowledge. They
presentasimpletemporallanguageAforrepresentingincompletetemporalknowledge
andapplyitsuccessfully toanumberofwell-knownbenchmarkproblemsintemporal
reasoning. Due to its natural and well-understood semantics, A provides a direct
meanstoevaluatetheexpressivityofotherformalisms. Byprovidingatransformation
from A to extended logic programming [16] and proving its soundness, they show
the expressivity of this formalismfor representing temporal knowledge and, more in
general, incomplete knowledge.
Inthe past,another approachhas been explored fortemporalreasoning,based on
eventcalculus[13],[31],[25],[10]. [10]proposes solutionsforthesamebenchmarksas
in [17]. This approach makes use of the formalismof abductive logic programming.
Onemayinterpretanabductiveprogramasanopenlogicprograminthesense thatit
containsonly de(cid:12)nitions forthe non-abducible predicates. The completionsemantics
forabductive logicprogramsof[3]re(cid:13)ects this wellbytakingasthe completionofan
abductive program the set of completed de(cid:12)nitions of the non-abducible predicates
only. As a consequence, the completiondoes not imposeany restriction on the inter-
1
pretation of the abducible predicates and incomplete knowledge can be represented
via the abducible predicates. Observe that this treatment di(cid:11)ers totally from the
treatmentof a de(cid:12)ned predicate with emptyde(cid:12)nition: the latter predicate is always
false.
In general an abductive program willallow too much uncertainty. In most appli-
cations, the knowledge engineer has knowledge that implicitly restricts the possible
states of the unde(cid:12)ned abductive predicates. A declarative way of representing this
knowledgeisbyFirst-order Logic(FOL)axioms. Inorder to illustratethisapproach,
wepresent atransformationfromAdomaindescriptions toabductive logicprograms
with FOL axioms (section 3). Despite the fact that the benchmarks in [10] were
solved with abductive event calculus, it turns out to be di(cid:14)cult to transform A do-
main descriptions to event calculus programs: as situation calculus, A is based on
the branching time philosophy, whereas event calculus is based on a linear time phi-
losophy. However, the use of abductive logic programs is in no way restricted to
event calculus. The proposed transformation maps an A domain description to an
abductive situation calculus.
A second highlevel goalis to illustrate howan abductive procedure can be useful
for automated reasoning with abductive logic programs and FOL axioms. That an
abductive procedure can be used for explanation of some observation is well-known
from [29], [31]. It is less known that an abductive procedure can also be used de-
duction and for proving consistency of a theory. In section 4, we illustrate how an
abductive procedure can be used to prove the consistency of an abductive logic pro-
gram with FOL axioms and that an abductive procedure can be used for deduction
in an abductive logic program.
Intherestofthepaperwewillmostlyrefertoabductivelogicprogramsasopenlogic
programs,and to abducible predicates as unde(cid:12)ned predicates. We prefer these names
over the commonly used terminology because they re(cid:13)ect the declarative meaning
of the concepts: an unde(cid:12)ned predicate has no de(cid:12)nition; an open logic program
is open in the sense that it describes the de(cid:12)ned predicates in terms of a set of
unde(cid:12)ned predicates whose interpretation is completely left open. The terminology
abductive logic programand abducible predicate is somewhatmisleading,especially in
the context of this paper, whose goal it is to show explicitly how to perform other
formsof reasoning than abduction on abductive logic programs.
The paper is structured as follows. In section 2, we recall the language A and its
semantics. In section 3, the transformationfromA to situation calculus programs is
presented and the soundness and completeness are proved. In section 4, the use of
abduction for explanation and deduction is illustrated. Section 5 gives a comparison
between our transformation and the transformation in [17]. Section 6 compares our
work with another recent transformation from A proposed in [11]. In section 7, we
discuss other related work. A short paper on this subject has been published as [8].
2 The temporal language A
The language A [17] allows to describe relationships between (cid:13)uents (= time depen-
dentproperties ofthe world)andactions. A isapropositionallanguage: both(cid:13)uents
andactionsarerepresented bypropositionalsymbols. Twotypesofexpressionsoccur.
2
A v-proposition describes the value of a (cid:13)uent after a (possibly empty) sequence of
actions. Its syntax is as follows:
f after a1;:::;an
Herea1;:::;anisasequence ofactionsymbolsandf isa(cid:13)uentexpression: apositive
or negative literal containing a (cid:13)uent. The expression means that f is true after
executing the sequence of actions a1;:::;an. If the sequence of actions is empty
(n=0), the v-proposition describes the initial situation. Instead of f after , one
usually writes:
Initially f
An e-proposition describes the e(cid:11)ect of actions on the (cid:13)uents. It has the form:
a causes f if p1;:::;pn
where f;p1;:::;pn are (cid:13)uent expressions and a is an action symbol. The expression
means that if p1;:::;pn are true, then the e(cid:11)ect of a on the current situation is that
f becomes true. p1;:::;pn are called preconditions. If n=0, one writes:
a causes f.
A domaindescription is a set of v- and e-propositions.
Example We recall the Yale Turkey Shooting problem(YTS) as formulatedin [17].
The (cid:13)uents are loaded, alive; the action names are shoot, wait and load. The
domaindescription DYTS contains the followingpropositions:
Initially alive
Initially :loaded
load causes loaded
shoot causes :alive if loaded
shoot causes :loaded
Example The Murder Mystery domainDMM is a variantofYTS, obtained by sub-
stituting
:alive after shoot;wait
for Initially :loaded in DYTS. This is a prototypical postdiction problem: we
want to obtain the conclusion that initiallythe gun is loaded.
The semantics for A is de(cid:12)ned as follows. A state is a set of (cid:13)uent names and
describes apossiblestateoftheworld. Givena(cid:13)uentsymbolf andastate(cid:27),f holds
in (cid:27) if f 2 (cid:27), otherwise :f holds in (cid:27). A transition function (cid:8) maps pairs (a;(cid:27))
of action symbols a and states (cid:27) into the set of states. (cid:8) describes how a situation
changes under application of an action a. A structure M is a pair ((cid:27)0;(cid:8)), where (cid:27)0
represents the initialstate and(cid:8)the transition function. Ma1;:::;am denotes the state
(cid:8)(am;(cid:8)(am(cid:0)1;:::;(cid:8)(a1;(cid:27)0):::)). A v-proposition
f after a1;:::;am
holds in a structure ((cid:27)0;(cid:8)) i(cid:11) f holds in Ma1;:::;am.
3
De(cid:12)nition2.1 A structure M =((cid:27)0;(cid:8)) is a model of a domain expression D if and
only if the following rules are satis(cid:12)ed:
(cid:15) Each v-proposition f after a1;:::;am holds in M.
(cid:15) Foranystate(cid:27),(cid:13)uent expressionf andactiona,ifthereexistsane-proposition
a causes f if p1;:::;pn
such that p1;:::;pn hold in (cid:27), then f holds in (cid:8)(a;(cid:27)). Otherwise, f holds in
(cid:8)(a;(cid:27)) i(cid:11) f holds in (cid:27).
A domain description is called consistent if it has a model. We introduce a new
notion,e-consistency. Adomaindescription ise-consistentifthe set ofe-propositions
of D is consistent. There is a simplenecessary and su(cid:14)cient condition for a domain
description to be e-consistent.
Lemma 2.1 A domain description D is e-consistent i(cid:11) for each pair of rules
a causes f if p1;:::;pn and a causes :f if pn+1;:::;pm
in D, there exists an i and j such that pi is the complement of pj.
Thisconditionissatis(cid:12)ed when the complementaryliteralsarefoundinthe bodies of
the two rules but also when they appear in the body of one rule, as in
shoot causes alive if loaded;:loaded
Such arule has aninconsistent body. It can never be applied andcan never cause an
inconsistency.
In [11], an action a for which there exists a pair of rules
0 0
a causes f if p1;:::;pn and a causes :f if p1;:::;pm
such that no complement of a literal in the (cid:12)rst is contained in the second is called
self-contradicted. [11] contains a similarproposition as lemma2.1.
Proof Assume that for some pair of rules:
a causes f if p1;:::;pn and a causes :f if pn+1;:::;pm
nocomplementaryliteralsoccur infp1;:::;pmg. Thisset maycontainpositive
and negative literals. De(cid:12)ne the state (cid:27) consisting of only the positive literals
in fp1;:::;pmg. Then obviouslyeach pi holds in (cid:27). (cid:8)(a;(cid:27)) is not consistently
de(cid:12)ned because both f and :f should hold in (cid:8)(a;(cid:27)).
Viceversa,one easilyveri(cid:12)es thatwhenthe syntacticalconditioninthe lemma
issatis(cid:12)ed, the de(cid:12)nitionofa modelofa domaindescription givesa consistent
description of a transition function (cid:8). 2
4
De(cid:12)nition2.2 (Non-inertial) We say that a (cid:13)uent f is non-inertial under an ac-
tion a in a state (cid:27) i(cid:11) there exists an e-proposition
a causes f if p1;:::;pn or a causes :f if p1;:::;pn
such that p1;:::;pn hold in (cid:27). Otherwise, it is inertial under action a.
[17] observes that the e-propositions of a domain D completely determine the
transition function (cid:8). Below proposition 2.1 gives a precise and deterministic char-
acterisation of (cid:8), under the condition that D is e-consistent.
Proposition 2.1 Let D be e-consistent. If D has a model ((cid:27)0;(cid:8)) then (cid:8) is de(cid:12)ned
as follows: for any state (cid:27), action a and (cid:13)uent symbol f: f 2(cid:8)(a;(cid:27)) i(cid:11)
(cid:15) f holds in (cid:27) and f is inertial under a in state (cid:27), or
(cid:15) there exists an e-proposition a causes f if p1;:::;pn such that p1;:::;pn hold
in (cid:27).
The proof is straightforward.
Note that if the condition of e-consistency is not satis(cid:12)ed, then the description
of (cid:8) in the proposition does not correspond to the description of (cid:8) in de(cid:12)nition 2.1.
Indeed, for f, a and (cid:27) such that for e-propositions
0 0
a causes f if p1;:::;pn and a causes :f if p1;:::;pm
0 0
the (cid:13)uents p1;:::;pn;p1;:::;pm hold in (cid:27), the second item of de(cid:12)nition 2.1 requires
that both f 2 (cid:8)(a;(cid:27)) and f 2= (cid:8)(a;(cid:27)). This is a contradiction. On the other hand,
proposition 2.1 requires that f 2 (cid:8)(a;(cid:27)) since the second item applies. Because the
transformation proposed in next section implements the formulation of proposition
2.1ratherthanthatofde(cid:12)nition2.1,itwillbecompleteonlyfore-consistentdomains.
Av-propositionQis entailedbyadomaindescriptionD i(cid:11)Qholdsineachmodel
of D. A domain description is called complete if it has a unique model. The YTS
domainand Murder Mystery domainare examples of complete domaindescriptions.
Since they share their e-propositions, their models have an identical transition func-
tion(cid:8)whichmapstuples(wait;(cid:27))on(cid:27),(load;(cid:27))on(cid:27)[floadedg,andmaps(shoot;(cid:27))
on(cid:27)nfalive;loadedgifloaded2(cid:27),otherwiseon(cid:27). ThemodelM0ofDYTS hasinitial
situationfaliveg. ThemodelM1 ofDMM hasinitialsituationfalive;loadedg. Anin-
completedomaindescriptionisobtainedbydroppingthev-propositionInitially alive
fromthe Murder Mystery domain. One additionalmodel is the structure with tran-
sition function (cid:8) but with initialsituation fg.
A provides only restricted expressivity: the language is only propositional, no
relationships between (cid:13)uents can be de(cid:12)ned, no indeterminate events are allowed.
Nevertheless, A allows to formalise several interesting domains. This and its clear
semantics makes the language interesting for experiments as in [17], [11] and in this
paper.
5
3 Translation to Open Logic Programs
Inthissectionwepresent ageneraltranslationfromanAdomaindescriptionD toan
openlogicprogramwithFOLaxioms. Anopenlogicprogramisapairofasetofnor-
malclauses, Horn clauses augmentedwith negation as failureand aset ofpredicates,
called unde(cid:12)ned or abducible, such that the unde(cid:12)ned predicates occur only in the
bodyoftherules. Intuitively,anopenlogicprogramallowstorepresent incompletein-
formationbecauseitleavesopentheinterpretationoftheunde(cid:12)nedpredicates. Tosee
this, consider the completionsemantics of Console, Theseider Dupr(cid:19)e and Torasso for
abductivelogicprograms[3],anextensionofClark'scompletionsemantics. According
to this completion semantics, the declarative meaningof an open logic program P is
givenbythetheory,denoted compa(P),whichconsists oftheaxiomsofFree Equality
(FEQ,alsocalledClarkEqualityTheoryorUniquenamesaxioms)andthecompleted
de(cid:12)nitions of all de(cid:12)ned predicates, i.e. all predicates of the language which are not
unde(cid:12)ned. Otherwise said,compa(P)isobtainedbytakingP'sClarkcompletionand
dropping for each unde(cid:12)ned predicate p=n its completed de(cid:12)nition p(X)$false.
Notethatcompa(P)containsonlyequivalenceswithde(cid:12)nedpredicates attheleft.
Intuitively,suchanequivalencede(cid:12)neswhenthepredicateattheleftistrueintermsof
thesituationsdescribed attheright. Sinceunde(cid:12)nedpredicates havenosuchde(cid:12)ning
1
equivalence,the logicprogramleaves their interpretation open to agreat extend . In
general, an abductive logic program allows too much freedom to the interpretation
of unde(cid:12)ned predicates. Most often, the program P must be augmented with a set
T of First-order Logic (FOL) axioms representing other informationwhich restricts
the state ofthe unde(cid:12)ned predicates. The declarative semantics ofa couple (P;T) is
given by the FOL theory compa(P)+T. This de(cid:12)nes indirectly a model semantics
for (P;T): in the sequel, when we talk about a model of (P;T) we mean a classical
modelof compa(P)+T.
That the resulting formalismis adequate for representing incomplete information
is now obvious: for P = fg with all predicates unde(cid:12)ned, the theory compa(P)+T
collapsestoFEQ+T,i.e. classicallogicwithunique namesaxioms. The expressivity
of this formalismfor representing incomplete informationis widely accepted.
OurtransformationfromAtotheaboveformalismproducesprogramsinsituation
calculusstyle. Traditionally,twooptionsareavailabletorepresent a(cid:13)uentf inalogic
formalism: by a predicate f(s) or by Holds(f;s) where s is a state argument. Then
:f istranslatedto:f(s) or:Holds(f;s). Thetwoapproaches areequivalentbutthe
meta-approachhastheadvantagethattheframeaxiomcanbestatedforall(cid:13)uentsat
once,whereasinthe(cid:12)rstapproachoneframeaxiomper(cid:13)uentpredicateisneeded. As
1Assooftenwithcompletionsemantics,examplescanbeformulatedwhichcontradictthisintu-
ition. Forexample,the2-valuedcompletionoftheprogram:
p:-:p;:a
withunde(cid:12)nedpredicateaentailsthataistrue. Theproblemis causedbytheloopovernegation
ofp. Forprogramswhichdonotcontainsuchloops,itcanbeshownthattheunde(cid:12)nedpredicates
can have any interpretation. In other semantics such as the 3-valued completion semantics for
abductiveprograms[4], the justi(cid:12)cationsemanticsfor abductiveprograms[7] and the generalised
well-foundedsemanticsforabductivelogicprograms[28],evenforprogramswithloopsovernegation,
the interpretationof the unde(cid:12)ned predicatescan be any. Despite these problems with 2-valued
completionsemantics,we useit herebecauseof itsdeclarativesimplicityandits closerelationship
withFirst-orderLogic.
6
in [17],we use Holds=2. The (cid:12)rst order language LD contains the predicate symbols
Holds=2, Noninertial=3 and Initially=1. Each (cid:13)uent and action symbol occurs in
LD as a constant. In addition, there is a constant s0 to denote the initial state and
a functor Result=2: Result(a;s) denotes the state obtained by applying action a on
state s. In the sequel, we will use Result[a1;:::;an;s] as a shorthand notation for
Result(an;Result(an(cid:0)1;:::Result(a1;s):::)). For n=0, this denotes s.
A allows uncertainty on the initial state. Correspondingly, the open program
comprises one unde(cid:12)ned predicate, Initially=1. The translation maps a domain de-
scriptionD to atheory (cid:25)D consisting ofan open logicprogramPD anda set ofFOL
axiomsICD. PD is de(cid:12)ned as follows:
(cid:15) Initialisation:
(3.1) Holds(F;s0) :- Initially(F)
(cid:15) Law of Inertia:
(3.2) Holds(F;Result(A;S)) :- Holds(F;S);:Noninertial(F;A;S)
0 0 0
(cid:15) Foreach e-proposition a causes f if p1;:::;pm;:p1;:::;:pn withf, pi and pj
positive literals:
(3.3) Holds(f;Result(a;S)) :- Holds(p1;S);:::;Holds(pm;S);
0 0
:Holds(p1;S);:::;:Holds(pn;S)
0
As in [17], we introduce the convention that when f is a negative literal :f ,
0
Holds(f;t) is used as a textual denotation for :Holds(f ;t). This handsome
convention allows us to say that a causes f if p1;:::;pn is translated to the
clause:
Holds(f;Result(a;S)) :- Holds(p1;S);:::;Holds(pn;S)
without considering the sign of the literals pi. Be aware that a programshould
never containliteralsoftheformHolds(:f;S), andthatwhen these literalsare
found in this and the followingsection, they always stand for :Holds(f;S).
(cid:15) For each e-proposition a causes f if p1;:::;pn with f a positive or negative
(cid:13)uent literal:
(3.4) Noninertial(jfj;a;S) :- Holds(p1;S);:::;Holds(pn;S)
where j:j maps a (cid:13)uent expression to the comprised (cid:13)uent symbol, i.e. for a
(cid:13)uent symbolf, both jfj and j:fj denote f.
The set of FOL axiomsICD is de(cid:12)ned as follows:
(cid:15) For each v-proposition f after a1;:::;an (n (cid:21) 0):
(3.5) Holds(f;Result[a1;:::;an;s0])
7
with the same syntactic convention on Holds=2 as above.
Example The domaindescription DYTS for the YTS problemis transformed to:
Holds(F;s0):-Initially(F)
Holds(F;Result(A;S)):-Holds(F;S);:Noninertial(F;A;S)
Holds(loaded;Result(load;S)):-
Noninertial(loaded;load;S):-
Noninertial(loaded;shoot;S):-
Noninertial(alive;shoot;S):-Holds(loaded;S)
Holds(alive;s0)
:Holds(loaded;s0)
TheclauseNoninertial(loaded;load;S):- maybedroppedfromthisprogramwithout
e(cid:11)ect on the semantics of Holds/2. In general, all Noninertial=3 rules for initiating
e(cid:11)ects of actions maybe dropped, without e(cid:11)ect on the semantics of Holds/2.
(cid:25)DYTS strongly resembles the YTS solution in [1] and in [14]. For example, [1]
proposes a Prolog program analogousto (cid:25)DYTS:
Holds(alive;s0):-
Holds(F;Result(A;S)):-Holds(F;S);:Noninertial(F;A;S)
Holds(loaded;Result(load;S)):-
Noninertial(loaded;shoot;S):-
Noninertial(alive;shoot;S):-Holds(loaded;S)
Note that this program entails the two FOL axioms. [1] proves that the program is
acyclic. The same holds for (cid:25)DYTS, and in fact for all transformed domaindescrip-
tions:
Proposition 3.1 The translation (cid:25)D of any domain description D is acyclic.
Proof A slight modi(cid:12)cationof the level mappingproposed in [1] for the YTS pro-
gram applies for all domain descriptions. For all ground terms t, let jtjResult
denote the number of occurrences of the functor Result=2 in t. We de(cid:12)ne j:j
for all ground terms t;a and s as follows:
jInitially(t)j =0
jHolds(t;s)j=2(cid:2)jsjResult+1
jNoninertial(t;a;s)j=2(cid:2)jsjResult+2
One easily veri(cid:12)es that j:j is a level mapping. 2
Several types of semantics have been de(cid:12)ned for open logic programs: 2-valued
completionsemantics[3],generalisedstablesemantics[20],thegeneralisedwell-founded
semantics [28], 3-valued completion semantics and 3-valued (direct) (partial) justi(cid:12)-
cationsemantics with FEQ [7]. Due to the fact that (cid:25)D is acyclic and in each clause
ofPD,the variablesofthe body occur in the head,allthese semanticscoincide inthe
(weak)sense that the set ofallgroundatomsimpliedby(cid:25)D under anyofthe seman-
tics is identical. This extension of results of [1] is proven formallyin [4]. Because of
8
these results, we can investigate the soundness and completeness of the transforma-
tion under the simplest semantics for open logicprograms,the completionsemantics
of [3].
The translation (cid:25)D of a domain description contains two de(cid:12)ned predicates,
Holds=2 and Noninertial=3. The completed de(cid:12)nition of Holds=2 is of the form:
(3.6) 8F;T :Holds(F;T)$E1_E2_:::_Em
with: E1 = T =s0^Initially(F)
E2 = 9A;S :T =Result(A;S)^
Holds(F;S)^:Noninertial(F;A;S)
Ei(i>2)= 9S :F =f ^T =Result(a;S)^
Holds(p1;S)^:::^Holds(pn;S)
such that there is a correspondence between the disjuncts Ei (i > 2) and the e-
propositions a causes f if p1;:::;pn with f a (cid:13)uent symbol.
If we map S on state (cid:27), the Result functor on (cid:8) and Holds(F;S) to F 2(cid:27) then
the completed de(cid:12)nition of Holds=2 is similarto proposition 2.1.
The completed de(cid:12)nition of Noninertial=3 is of the form:
(3.7) 8A;F;S :Noninertial(F;A;S)$E1_:::_Em
with: Ei = A=a^F =jfj^Holds(p1;S)^:::^Holds(pn;S)
such thatprecisely foreach e-propositiona causes f if p1;:::;pn (f positiveor neg-
ative), there exists one corresponding disjunct in the completed de(cid:12)nition. This
formulais the counterpart of de(cid:12)nition 2.2: the formulationis almostidentical apart
fromthe fact that Holds(pi;S) should be replaced by "pi holds in S".
Atransformationsuchas(cid:25)canbeconsideredcorrect ifthesetofentailedformulas
are equivalent. In [17] a translation (cid:25) is de(cid:12)ned to be sound i(cid:11) for each domain D
and v-expression Q, if(cid:25)D j=(cid:25)Q then D entails Q. (cid:25) is de(cid:12)ned to be complete ifthe
reverse holds: if D entails Q then (cid:25)D j=(cid:25)Q.
Theorem 3.1 (soundness) Let D be a domain description. For any v-proposition
Q, if (cid:25)D j=(cid:25)Q then D entails Q.
For the proof of the theorem, we introduce the concept of a state function.
De(cid:12)nition3.1 Let M be any (possibly non-Herbrand) interpretation of LD, x an
element of the domain of M.
stateM(x)=ff j f is a (cid:13)uent symbol and M j=Holds(f;x)g
The notation in which domain elements of a model appear in a term or formula,
as in Result(a;x) and Holds(f;x), is not conventional. Such a term or formula
can be interpreted as a pair of a term or a formula with free variables and a vari-
able assignment of these free variables. E.g. Holds(f;x) corresponds to the pair
(Holds(f;X);fX=xg). "M j= Holds(f;x)g" corresponds to the more conventional
"M;fX=xgj=Holds(f;X)".
NotethatstateM(x)denotesastateasinsection2,andatransitionfunction(cid:8)can
be applied on it. Note also that f 2stateM(x) is equivalent with M j=Holds(f;x).
9
Proof of Theorem 3.1.
Nothing is to be proved when D is inconsistent. So, assume D is consistent.
It su(cid:14)ces to show that for each model Ma = ((cid:27)0;(cid:8)) of D and for each v-
proposition Q, there exists a model M of (cid:25)D such that Q holds in ((cid:27)0;(cid:8)) i(cid:11)
M j= (cid:25)Q. If then (cid:25)Q holds in all models of (cid:25)D, then also in these models
corresponding to models of D. Hence, Q holds in all models of D.
We construct a Herbrand model M. HU(LD) denotes the Herbrand uni-
verse. The basic idea is simple: we de(cid:12)ne M such that for each state term
s = Result[a1;:::;an;s0], stateM(s) = Ma1;:::;an and such that the atom
Noninertial(a;f;s) is true whenever f is non-inertial under a in the state
stateM(s). Things are complicated due to the fact the (cid:25)D is not a sorted
program and ill-sorted atoms may occur in the model. For that reason, we
0
extend D to D by allowing each term t 2 HU(LD) as a (cid:13)uent symbol and
as an action symbol. Note that non-original (cid:13)uents tf (symbols of HU(LD)
which are not (cid:13)uent symbols in D) and non-original actions ta do not occur
0
in the e-propositions. This implies that a non-original (cid:13)uent tf of D always
remainsas in the initialstate; a non-originalaction ta has no e(cid:11)ect on a state
(like the action wait in the YTS DYTS). ((cid:27)0;(cid:8)) can easily be extended to a
0 0 0 0 0 0
model ((cid:27)0;(cid:8)) of D . De(cid:12)ne (cid:27)0 = (cid:27)0. Let (cid:27) be any state of D , consisting of
0
original(cid:13)uents (cid:27) and new (cid:13)uents (cid:27)n. Extend (cid:8) to (cid:8) in the followingway:
0 0 0
(cid:8)(ta;(cid:27) )=(cid:27) for ta a non-originalaction
0 0
(cid:8)(ta;(cid:27) )=(cid:8)(ta;(cid:27))[(cid:27)n for ta an originalaction
0 0 0
Oneeasilyveri(cid:12)esthat((cid:27)0;(cid:8))isamodelofD . Moreover,foranyv-proposition
0 0
QbasedontheoriginallanguageofD,Qholdsin((cid:27)0;(cid:8))i(cid:11)Qholdsin((cid:27)0;(cid:8)).
0 0 0
Next we associate to ((cid:27)0;(cid:8)) a Herbrand model M of (cid:25)D = (cid:25)D . With any
0
termts 2HU(LD) we associate a speci(cid:12)c state of D . Below we call a termts
an empty-state term if ts 6=s0 and ts 6=Result(t1;t2) for some t1;t2. For any
term ts 2HU(LD), stateM(ts) is constructed as follows:
ts =s0 )stateM(ts)=(cid:27)0
ts is an empty-state term )stateM(ts)=(cid:30)
0
ts =Result[t1;:::;tn;s0] )stateM(ts)=(cid:8)[t1;:::;tn;(cid:27)0]
ts =Result[t1;:::;tn;t0], t0 an empty-state term
0
)stateM(ts)=(cid:8)[t1;:::;tn;(cid:30)]
M is de(cid:12)ned as follows:
fHolds(tf;ts) j tf 2stateM(ts)g[
fNoninertial(tf;ta;ts) j tf is non-inertial under ta
in stateM(ts)g[
fInitially(tf) j tf 2(cid:27)0g
Clearly for any v-proposition Q using original symbols of D, it holds that Q
holds in ((cid:27)0;(cid:8)) i(cid:11) M j= (cid:25)Q. A direct consequence is that M is a model of
ICD. It remains to be proven that M is a modelof compa(PD).
10