Propositional Abduction in Modal Logic

14 Pages·2007·0.21 MB·English

(cid:3)
Propositional Abduction in Modal Logic
2 1
Marta Cialdea Mayer Fiora Pirri
1
DipartimentodiInformaticaeSistemistica,Universita(cid:18)diRoma\LaSapienza"
viaSalaria113,00198Roma,Italia
2
DipartimentoD.S.Informatica,TerzaUniversit(cid:18)adegliStudidiRoma
viaSegre2,00146Roma,Italia
e-mail: fcialdea,[email protected]
Abstract
In this work, the problem of performing abduction in modal logics
is addressed, along the lines of [3], where a proof theoretical abduction
methodforfull(cid:12)rstorderclassicallogicisde(cid:12)ned,basedontableauxand
Gentzen-type systems. This work applies the same methodology to face
modalabduction. The non-classical contextenforces the valueof analyt-
ical proof systems as tools to face the meta-logical and proof-theoretical
questionsinvolvedin abductivereasoning.
The similarities and di(cid:11)erences between quanti(cid:12)ers and modal oper-
ators are investigated and proof theoretical abduction methods for the
modalsystemsK,D,T andS4arede(cid:12)ned,thataresoundandcomplete.
The construction of the abductive explanations is in strict relation with
theexpansionrulesforthemodallogics,inamodularmannerthatmakes
localmodi(cid:12)cationspossible. Themethodgiveninthispaperisgeneral,in
thesensethatitcanbeadaptedtoanypropositionalmodallogicforwhich
analytic tableaux are provided. Moreover, the way towards an extension
to(cid:12)rst order modallogic is straightforward.
(cid:3)ThisworkhasbeenpartiallysupportedbyBasicResearchAction,MedlarII
1
1 Abductive reasoning and modal logics
Manyformsofcommonsensereasoningarenotdeductiveinnature. Reasonable
inferences may be performed even if leading to conclusionsthat are not certain
but only plausible. Abduction is one of such inference schemata that re(cid:13)ect
unsound forms of inference. Abductive reasoning is a way to solve problems
where an observed event ' is not explained by the presently adopted theory (cid:2)
and an explanation for ' has to be looked for.
Aprecisede(cid:12)nitionofabductiveexplanationsisobviouslydependentonthe
logicunderlyingthetheory(cid:2): somefact(cid:11)couldexplain'inclassicallogic,but
notinintuitionisticorlinearlogic. So,letusassumethatalogicLisgiven,with
acorrespondingrelationoflogicalconsequencej=L. Thenanabduction problem
in L consists of a background theory (cid:2) and a formula ' such that (cid:2) 6j=L '.
A solution of the problem given by the pair h(cid:2);'i is to be chosen among the
formulae (cid:11) such that (cid:2)[f(cid:11)gj=L '.
Ingeneral,itisnotevenrequiredthatj=Lsatis(cid:12)esthetarskianpropertiesfor
a consequence relation. For example, it makes sense to think of (and it would
be worth exploring) non-monotonic explanations, i.e. cases where (cid:11) explains
' but (cid:11) ^ (cid:12) does not [2], so that the notion of explanation is related to a
non-monotonicconsequencerelation. Inthisperspective,abductioninthelogic
programmingcontext, with its tricks(like constraints),canbe seenasthe form
of abduction that is carried out in the logic induced by the semantics of the
arrowand negation-as-failureoperators.
Recently, much attention has been paid to abductive reasoning in di(cid:11)erent
(cid:12)elds of Arti(cid:12)cial Intelligence, such as diagnosis [4, 5, 13, 12], interpretation of
naturallanguagesentences[7,15],planrecognition[1],sceneinterpretation[14].
This work is a contribution to the characterization of abduction as a form of
inferencewhosecontextisonlythelogicitdependsupon. Abstractionfromany
particularproblemhelps tofocus ongeneraltheoreticalquestions,while results
can be obtained that bene(cid:12)t the applicative (cid:12)elds.
In this work, the problem of performing abduction in modal logics is ad-
dressed. The meaningfulness of such a problem can be recognized by consid-
ering two simple examples: the (cid:12)rst is from Arti(cid:12)cial Intelligence, the second
from program veri(cid:12)cation.
Let us assume that an agent is reasoning about her own knowledge, rep-
resented by a modal theory (cid:2), and she observes that ' is consistent with her
knowledge, i.e. 3' is true. However, 3' is not derivable in (cid:2). Abduction
would produce what is needed to sanction 3'. Of course, the importance of
epistemic abduction is clearer when considering multiple agent contexts: for
example, both natural language understanding and user modeling require the
abilitytoabducethegoalsofthespeaker,andstudentmodelingincludestheab-
ductionofthestudent'sbeliefsfromtheobservationofheranswerstoquestions
and problems.
As a second example, let us assume that a program (cid:5) is described by a
2
theory (cid:2) in a temporal or dynamic logic L and that the termination property
for (cid:5) is represented by the formula '. Then, determining the class of input
data for which (cid:5) terminates can be seen as abducing a precondition (cid:11) such
that (cid:2)[f(cid:11)gj=L '.
In[3]analyticalproofsystemsareproposedassuitabletoolstofacethemeta-
logical and proof-theoretical questions involved in abductive reasoning and a
proof theoretical abduction method for full (cid:12)rst order classical logic is de(cid:12)ned.
The underlying idea is to solve the abduction problem h(cid:2);'i by building a
formula that closes the open branches of a tableau for (cid:2)[f:'g. The method
does not require any preliminary reduction of formulae into normal forms, but
dynamic skolemization is used. In the (cid:12)rst order case, quanti(cid:12)ers are to be
reintroduced following a sort of reverse skolemization. This work applies the
same methodology to face modal abduction. There are obvious similarities
betweenquanti(cid:12)ersandmodaloperators(see,amongothers,[11]forafunctional
translation method of modal formulae into (cid:12)rst order formulae). As far as the
tableau based abduction method is concerned, however, a main di(cid:11)erence lies
in that the structure of a quanti(cid:12)ed formula is essentially rebuildable from its
Skolem forms, while the trace of modal operators, once they are dropped, is
lost. This fact induces a recursive reformulation of the construction of the
abduced formulae, thus shifting the view from a meta-logical level to a logical
one. Theresultingconstructionisinstrict relationwith theexpansionrulesfor
the modal logic, in a modular manner that makes local modi(cid:12)cations possible,
so as to (cid:12)t the method to di(cid:11)erent modal systems. In fact, the abductive
methodgiveninthispaperisgeneral,inthesensethatitcanbeadaptedtoany
propositional modal logic for which analytic tableaux are provided. Moreover,
the way towards an extension to (cid:12)rst order modal logic is straightforward.
1.1 Basic notions about modal logics
Thelogicallanguageofmodalsystemsconsistsofalltheclassicallogicalsymbols
andthemodaloperator2(necessity). Modalformulaearede(cid:12)nedconsequently.
Formulae of the form 2(cid:11) are called boxed formulae. The modal operator 3 is
de(cid:12)ned as follows: 3(cid:11)(cid:17):2:(cid:11).
In the sequel we consider modal systems characterizable by a subset of the
following schemata:
(K) 2((cid:11) ! (cid:12)) ! (2(cid:11) ! 2(cid:12))
(T) 2(cid:11) ! (cid:11)
(D) 2(cid:11) ! 3(cid:11)
(4) 2(cid:11) ! 22(cid:11)
(cid:11)
and the necessitation rule: . In particular, the systems K (with axiom K),
2(cid:11)
D (with axioms K and D), T (with axioms K and T) and S4 (with axioms K,
T and 4) are considered.
3
Thesemanticsofmodallogicsisde(cid:12)nedasusualintermsofL-modalstruc-
tures M = hW;RL;Vi, where W is a non empty set (of possible worlds), RL
is a binary relation on W (accessibility relation), whose properties depend on
the logic L, and V is a truth assignment, that maps W (cid:2)P into ftrue;falseg,
whereP isthesetofpropositionalsymbolsofthelanguage. Thenotionoftruth
of a formula (cid:11) in a world w 2W, (M;w)j=(cid:11), is de(cid:12)ned as usual.
Thenotionoflogicalconsequencein amodallogicL isnotalwaysgiventhe
same de(cid:12)nition. Two versions are commonly used:
F
(F) (cid:6)j=L (cid:11) i(cid:11) for every L-modal structure M:
(for all w 2W and for all (cid:27) 2(cid:6), (M;w)j=(cid:27))
implies (for all w 2W, (M;w)j=(cid:11)).
K
(K) (cid:6)j=L (cid:11) i(cid:11) for every L-modal structure M:
for all w 2W (for all (cid:27) 2(cid:6), (M;w)j=(cid:27) implies (M;w)j=(cid:11)).
Clearly, if (cid:6) is not empty,
^
K F
(cid:6)j=L (cid:11) i(cid:11) j=L ( (cid:27)) ! (cid:11)
(cid:27)2(cid:6)
If(cid:6)isempty,thetwonotionscollapse,sothatinparticularthenotionofvalidity
they induce is the same (this is the reason why any of the two versions can be
often used without discrepancies). The version (F) corresponds to the notion
ofderivabilityin axiomaticsystems(with necessitation),while (K)corresponds
to derivability of the corresponding sequent in sequent calculi. Consequently,
the main di(cid:11)erence between the two versions is that the following properties:
(i) if (cid:2)j=' then (cid:2)[f:'g is inconsistent
(ii) if (cid:2)[f(cid:11)gj=' then (cid:2)j=(cid:11) ! '
hold for(K) but notfor(F) (their conversesaretruein both cases). Thus, (K)
is closer to the classical notion of logical consequence.
In recent literature (F) is often preferred because of its greater generality.
However, in this work, we use (K) just because it de(cid:12)nes a stronger relation
and, consequently, it corresponds to a more general notion of abduction: if
K F
(cid:2)[f(cid:11)gj=L ' then (cid:2)[f(cid:11)gj=L ', but the converseis not true. In the context
ofabduction,theuseof(K)ismorereasonable(theredoesnotseemtobemuch
senseinassumingthat(cid:11)isagoodexplanationfor2(cid:11))anditallowstheduality
betweendeductionandabductiontobepreserved. Ofcourse,thesameconcepts
could be de(cid:12)ned by use of (F) and implication, but the notation would be far
more cumbersome.
K
In the following we shall simply use the symbol j=L instead of j=L.
4
1.2 Basic notions about modal abduction
Abductive explanations are in general required to respect some fundamental
conditions, in orderto be accepted as \interesting". If well settled consequence
relations j=L are considered, such as the classical, modal or intuitionistic ones,
thefollowingrestrictionsaretobereasonablyimposedonexplanations(cid:11)foran
abduction problem h(cid:2);'i:
(i) (cid:11) is consistent with (cid:2), i.e. (cid:2)6j=L :(cid:11).
(ii) (cid:11) is a minimal explanation for the abduction problem h(cid:2);'i, i.e. for any
formula (cid:12), if (cid:2)[f(cid:12)gj=L ' and (cid:11)j=L (cid:12), then (cid:12) j=L (cid:11).
Moreover,thesyntacticalformofexplanationsisusuallyrestricted;forexample,
in (cid:12)rst orderclassicallogic, an explanationis aprenex formulawhosematrixis
aconjunctionofliterals. Alogical reasonforthishasbeenpointedoutby[9]: if
(cid:2)=f(cid:18)1;:::;(cid:18)ngis(cid:12)nite, thenanyminimalexplanationforh(cid:2);'iisequivalent
to (cid:18)1 ^:::^(cid:18)n ! '. The following de(cid:12)nition is a straightforward adaptation
to the modal language of the classical syntactical restriction on explanations.
De(cid:12)nition 1 (modal C-formulae) Aformula (cid:11) is a modal C-formula if it is
built up from literals using only the modal operators 2 and 3 and conjunction.
Consequently:
De(cid:12)nition 2 (L-explanations for abduction problems) (cid:11) is an explana-
tion for the abduction problem h(cid:2);'i if it is a C-formula in the language of
(cid:2)[f'g and (cid:2)[f(cid:11)gj=L '.
Whenanexplanationisrequiredtobeeitherminimalorconsistentwiththe
theory, it will be explicitly stated. L-equivalent explanations are considered as
identical,sothatL-explanationsareinfactequivalenceclassesofformulae(the
terms L-equivalence, L-derivability, etc. are used in the obvious sense).
2 Modal abduction by use of analytic tableaux
Semantic tableaux are used as refutation systems. They are built by means of
a set of rules that preserve satis(cid:12)ability of sets of formulae. In this work we
make use of the unsigned versionof modal tableaux [6]. Modal expansion rules
are stated as follows, where set union is brie(cid:13)y denoted by the comma, in the
0 0
obvious sense; for example, (cid:11);(cid:6);(cid:6) is a short-cut for f(cid:11)g[(cid:6)[(cid:6). Moreover,
0
2(cid:6) is a set of boxed formulae and (cid:6) is a set of non boxed formulae.
0 0
:2(cid:11);2(cid:6);(cid:6) :2(cid:11);2(cid:6);(cid:6)
(cid:25)K-rule) (cid:25)4-rule)
:(cid:11);(cid:6) :(cid:11);(cid:6);2(cid:6)
0
2(cid:6);(cid:6) 2(cid:11);(cid:6)
(cid:23)D-rule) (cid:23)T-rule)
(cid:6) 2(cid:11);(cid:11);(cid:6)
5
The systems K, D, T and S4 are characterized by the addition of modal rules
to the classical ones, as the following table shows:
K { (cid:25)K-rule
D (cid:23)D-rule (cid:25)K-rule
T (cid:23)T-rule (cid:25)K-rule
S4 (cid:23)T-rule (cid:25)4-rule
In the sequel we shall generically speak of (cid:23)-rule and (cid:25)-rule; the version
actually referred to depends on the context.
If (cid:6) is a set of formulae, an L-tableau for (cid:6) is a tree whose root is labeled
by (cid:6) and every non-root node is obtained from a preceding node in the same
branch by means of the application of an expansion rule of the system L. A
branch B of a tableau is closed if both ' and :' occur in some node of B, for
someformula', orif the atomfalseoccursin B; otherwiseit iscalledopen. A
tableau is closed i(cid:11) all its branches are closed. A L-tableau refutation of (cid:6) is a
closed L-tableau for (cid:6). The L-tableau systems are sound and complete, i.e. (cid:6)
is L-unsatis(cid:12)able i(cid:11) (cid:6) has an L-tableau refutation [6].
Thestandardformulationofanalytictableausystemsdoesnotmakeexplicit
reference to structural rules. However, we assume that the following form of
weakening is among the rules of the systems:
0
2(cid:6);(cid:6)
2(cid:6)
0
where (cid:6) contains no boxed formulae. This rule is certainly redundant for the
aims of theorem proving, but it is needed to guaranteethe completeness of the
abduction procedure for the systems K, T and S4 (the reason can be found in
the proof given in the Appendix).
Inthe classicalcase,the constructionofanexplanationcanbedoneonlyon
the basisof the set of the open leavesof the tableau, without need to store and
work on the structure of the whole tableau: a literal is chosen for each open
leaf, that closes the corresponding branch; the set of literals is conjoined and
quanti(cid:12)ers are reintroduced. In fact, the elimination of quanti(cid:12)ers by means of
skolemization preserves the information about their mutual dependencies that
is needed to reintroducethem. In contrastto these, modal operatorscannotbe
correctly reintroduced without considering the structure of the tableau, where
theymaydisappearwithoutleavinganytrace. Thisfactisintuitivelyexplained
by the following simple example. Let us consider the more general problem
of (cid:12)nding a formula (cid:11) that is L-inconsistent with a set (cid:9). Let L = D, (cid:9) =
0
f2'_2 g and (cid:9) =f2('_ )g; the following treesare D-tableauxfor (cid:9) and
0
(cid:9), respectively.
2'_2 2('_ )
2' 2 '_
' '
6
Both tableaux have the same open leaves. Nevertheless, while the (cid:12)rst one
closes with the addition of (cid:11) = 3:'^3: , the closure of the latter requires
0
the stronger (cid:11) =3(:'^: ).
2.1 The set of abductive explanations
Leth(cid:2);'ibeanabductionprobleminL. Then,as(cid:2) 6j= ',thereisnoL-tableau
refutation for (cid:2)[f:'g. A solution for the abduction problem h(cid:2);'i can be
found among the formulae that force the closure of a tableau for (cid:2)[f:'g. Of
course,it wouldbemeaninglesstocloseanunexpandedtableauxbyaddingthe
atomfalseortheformula'. So,ifonlynontrivialexplanationsarelookedfor,
the tableaux should in principle be developed as far as possible. However, also
tableaux that can still be expanded are to be considered, because (i) (cid:23)-rules
may be reapplicable, and (ii) (cid:25)-rules may cancel some formulae. The following
de(cid:12)nitions cope with the determination of the point where the construction
of a tableau can be interrupted in order to perform abduction. Here and in
the following, tableaux for (cid:2)[f:'g will be called tableaux for the abduction
problem h(cid:2);'i.
De(cid:12)nition 3 (Compulsory expansion of a branch) The expansion of a
branch B with leaf (cid:6) is compulsory if one or more of the following conditions
hold:
(i) (cid:6) contains no literals and can be expanded.
(ii) (cid:6) contains some formula that can be expanded by means of a classical
logical rule;
(iii) (cid:6) contains some 2(cid:11) that can be expanded but has not yet been expanded
in B;
In other words, the expansion of B is compulsory whenever it can be per-
formedwithoutthe lossofinformationthat isusefulforabductionandwithout
cycling. Note that the application of a (cid:25)-rule is not always compulsory.
De(cid:12)nition 4 (Fundamental tableaux) Let T be a tableau. A branch B of
T is fundamental if its expansion is not compulsory. A node is fundamental if
it is the leaf of a fundamental branch. T is fundamental if all its branches are
fundamental.
Notethatafundamentalnodealwayscontainssomeliterals,unlessthelogic
considered is K. In this case, a node that contains only boxed formulae is
fundamental.
We next turn to the characterizationof the set ofabductive explanationsin
L built up from a fundamental L-tableau T for an abductive problem. It will
be denoted by EL(T). The following de(cid:12)nition is relative to the logics T and
S4. Its variants for K and D are stated later on.
7
De(cid:12)nition 5 (Construction of explanations in T and S4) If L is either
T or S4 and T is a fundamental L-tableau, the set EL(T) is de(cid:12)ned recursively
as follows.
1. If T consists of the single fundamental node (cid:6), then:
1a. If (cid:6) contains either false or a pair of complementary literals, then
EL (T) = ftrueg.
1b. Otherwise EL(T) = f:(cid:21) j (cid:21)2(cid:6) and (cid:21) is a literalg. Note that, as (cid:6)
is fundamental, EL(T) is not empty.
2. Otherwise, di(cid:11)erent cases are to be considered, according to the form of
the (cid:12)rst expansion rule applied in T.
(cid:6)
2a. T has the form (=) , where = is anapplication of a classical non-
T0
branching rule (weakening included). In this case EL(T) = EL(T0).
(cid:6)
2b. T has the form (=) , where = is an application of a (cid:12)-rule.
T0 T1
In this case EL(T) = f'0^'1 j '0 2EL(T0); '1 2EL(T1)g.
2(cid:11);(cid:6)
2c. T has the form (=) , where = is an application of the (cid:23)-rule
T0
to 2(cid:11). We distinguish two cases:
(i) If either (cid:6) is empty or it contains only boxed formulae, then
EL(T) = f3' j '2EL(T0)g;
(ii) otherwise EL(T) = EL(T0).
0
:2(cid:11);2(cid:6);(cid:6)
2d. T has the form (=) , where = is an application of the
T0
(cid:25)-rule to :2(cid:11). Then EL(T) = f2' j '2EL(T0)g.
As an example, let (cid:2) = f2(p^2q ! r)g and ' = 23r. The following
(leftmost)tree,where2(cid:6)=f2(p^2q!r);2:rg,isafundamentalS4-tableau
for (cid:2)[f:'g. The tree at its right side is an intuitive view of the construction
of a particular explanation, 23(p^2q) (the tree is to be read bottom up).
2(p^2q !r);32:r 23(p^2q)
2(cid:6);p^2q !r 3(p^2q)
2(cid:6);p^2q !r;:r p^2q
2(cid:6);:(p^2q);:r 2(cid:6);r;:r p^2q true
2(cid:6);:p;:r 2(cid:6);:2q;:r closed p 2q
open 2(cid:6);(cid:6);:q q
open
Suitablemodi(cid:12)cation tothe de(cid:12)nition of theset EL(T) areeasyto statefor
other logics. As far as the system D is considered, clauses 1, 2a, 2b and 2d
remain the same, while clause 2c is replaced by the following:
8
0
2(cid:6);(cid:6)
0
2c. T hastheform(=) ,where=isanapplicationofthe(cid:23)-rule. Then
T0
ED(T)=f3' j '2E(T0)g.
Considering system K, it must be noted that K-tableaux may have funda-
mental nodes containing only boxed formulae and literals, so, besides deleting
clause 2c, item 1b has to be replaced by:
0 0
1b. Otherwise, (cid:6) has the form 2(cid:6);(cid:3), where (cid:3) is a (possibly empty) set of
literals; then
EK(T) = f:(cid:21) j (cid:21)2(cid:3)g
0 0 0
[f3'j '2E(T ) for somefundamental K-tableauT for(cid:6)g.
The methods are sound and complete, in the sense stated by the following
theorems, that hold for L = K;D;T;S4.
Theorem 1 (Soundness) Let T be a fundamental L-tableau for (cid:6) and 2
0
EL(T). Then there exists a closed L-tableau T for (cid:6)[f g.
Theorem 2 (Completeness) Let (cid:6) be any set of modal formulae and an
L-consistent C-formula such that (cid:6)[f g is L-unsatis(cid:12)able. Then there exists
(cid:3)
a fundamental L-tableau T* for (cid:6) and a C-formula ' 2 EL(T ) such that
j=L '.
TheproofofTheorem1isastraightforwardinductiononT. Thesigni(cid:12)cant
steps in proving completeness can be found in the Appendix.
3 Existence of minimal explanations
If h(cid:2);'i is an abductive problem, let (cid:5)((cid:2);') denote the set of all the fun-
damental L-tableaux for (cid:2)[f:'g. By the completeness theorem, the set of
minimal explanations for h(cid:2);'i is included in the set
[
EL((cid:2);')= EL(T)
T2(cid:5)((cid:2);')
The question whether min(EL((cid:2);')) may be empty is addressed here.
In the case of (cid:12)rst order logic, there are abduction problems that admit
no minimal explanation [9], because there exist in(cid:12)nite descending chains of
formulae. This is not the case for modal logic.
For the non transitive logics K, D and T, the existence of minimal expla-
nations follows from the fact that the number of fundamental tableaux for a
given problem h(cid:2);'i is (cid:12)nite, therefore EL((cid:2);') is (cid:12)nite. In fact, at any step
in the expansion of a tableau there is only a (cid:12)nite number of choices and the
constructionof anytableau terminates, because eachapplication of a rule of K
9
and D decreases the number of logical operators; in T the number of logical
operatorsmaynotbereducedbya(cid:23)-rule,butitisassoonasa(cid:25)-ruleisapplied
(if a node contains no (cid:25)-formulae, reapplications of (cid:23)-rules can be avoided).
ThecaseofthelogicS4isnotsosimple. Infact,thereexistin(cid:12)nitetableaux,
i.e. there may be an in(cid:12)nite number of fundamental tableaux for a given ab-
ductive problem. The fact that in(cid:12)nite tableaux cycle does not help, because
every cycle could generate weaker explanations and in S4 there is an in(cid:12)nite
number of distinct modal functions, even of a single propositional variable. So,
in principle, a given problem may lack minimal explanations.
However,the following theorem solvesthe problem of the existence of mini-
mal explanations for S4 too.
Theorem 3 Let h(cid:2);'i be an abductive problem in a propositional modal logic
L that enjoys the (cid:12)nite model property. Then:
(i) there exists a minimal explanation of h(cid:2);'i, and
(ii) for any explanation of h(cid:2);'i there exists a minimal explanation min
such that j= min.
A proof of this theorem is given in the Appendix.
4 Concluding remarks
Abduction in a logic may be approached in several ways, at least one for every
deductive method de(cid:12)ned for the logic. However abduction in modal logic has
never been addressed so far. Obviously, the results of this work are only a (cid:12)rst
step towards a good characterization of modal abduction and leave a number
of problems to be solved.
Animprovementofthemethodcanbeobtainedbyuseofpre(cid:12)xedtableaux,
thatinmodaldeductionreducesthenumberoftableauxtobeconsidered. Apart
from a gain in e(cid:14)ciency, pre(cid:12)xes could help in storing important information
on the \history" of the modal operators in the same way as the use of Skolem
functionscodethepositionandqualityofquanti(cid:12)ersthatareeliminated. How-
ever, commonly used pre(cid:12)xes [6] do not allow the abductive construction to be
based only on the leaves of the tree, like in classical logic. In fact, those pre-
(cid:12)xes keep track of the application of modal rules whatsoever, without storing
any information about which rules have been applied. A careful rede(cid:12)nition of
pre(cid:12)xestobe usedforabduction isthereforetobedone,followingtheintuition
given in [10, 11, 8].
Acknowledgments The authors are indebted to the anonymous referees for
useful suggestions, which helped to improve the preliminary version.
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