Tableau-based decision procedure for the multi-agent epistemic logic with all coalitional operators for common and distributed knowledge

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by  Mai Ajspur

Tableau-based decision procedure for the
multiagent epistemic logic with all coalitional
operators for common and distributed knowledge
Mai Ajspur1, Valentin Goranko2, and Dmitry Shkatov3
1 Roskilde University, [email protected],
2 2 Technical University of Denmark and University of Johannesburg(cid:63),
1 [email protected]
0 3 University of the Witwatersrand, [email protected]
2

n January 16, 2012
a
J
5 Abstract. We develop a conceptually clear, intuitive, and feasible de-
2 cisionprocedurefortestingsatisfiabilityinthefullmultiagentepistemic
logicCMAEL(CD)withoperatorsforcommonanddistributedknowl-

] edgeforallcoalitionsofagentsmentionedinthelanguage.Tothatend,
O
weintroduceHintikkastructuresforCMAEL(CD)andprovethatsat-
L isfiability in such structures is equivalent to satisfiability in standard
. models. Using that result, we design an incremental tableau-building
s
c procedure that eventually constructs a satisfying Hintikka structure for
[ every satisfiable input set of formulae of CMAEL(CD) and closes for

every unsatisfiable input set of formulae.
1
v
6
Keywords: multi-agent epistemic logic, satisfiability, tableau, decision proce-
4
3 dure
5
.
1 1 Introduction
0
2
1 Over the last three decades, multiagent epistemic logics [9], [27] have been play-
: ing an increasingly important role in computer science and AI. The earliest
v
prominent applications have been to specification, design, and verification of
i
X distributedprotocols [23]and[24];anumberofotherapplicationsaredescribed
r in, among others, [9], [10], and [27]. The most recent, and perhaps more impor-
a
tantonesaretospecification,design,andverificationofmultiagent systems —a
researchareathathasemergedontheborderlinebetweendistributedcomputing,
AI, and game theory [36], [45], [47].
1.1 Multiagent epistemic logics and decision methods for them
Languagesofmultiagentepistemiclogicsconsideredintheliteraturecontainvar-
iousrepertoiresofepistemicoperators.Werefertothebasicmultiagentepistemic
(cid:63) Visiting professor
logic, containing only operators of individual knowledge for a finite non-empty
set Σ of agents, as MAEL (Multi Agent Epistemic Logic). Since all epistemic
operators of this logic are S5-type modalities, it is also referred to in the litera-
tureasS5 ,wherenisthenumberofagentsinthelanguage.Thelogicobtained
n
from MAEL by adding the operator of common knowledge among all agents in
Σ isthencalledMAEL(C).Thislogic,alongwithMAEL,wasstudiedin[25].
Analogously,if MAELisaugmentedwiththeoperatorofdistributedknowledge
for all agents, then the resulting logic will be called MAEL(D). It was studied
in [10] and [38]. MAEL augmented with operators of both common and dis-
tributed knowledge for the set of all agents, hereafter called MAEL(CD), was
studied in [39], and a tableau-based decision procedure for it was first presented
in [14]. Thus, all logics mentioned so far either do not have both operators of
common and distributed knowledge, or only have those operators for the whole
set of agents in the language.
Atthesametime,therehasrecentlybeenanincreasinginterestinthestudy
ofcoalitional multiagentlogics(see[30],[31],[32],[2],[40],[13]),i.e.logicswhose
languages refer to any groups (coalitions) of agents. These are important, inter
alia,inmultiagentsystems,whereagentsmay“cooperate”(i.e.,formacoalition)
in order to achieve a certain goal. Most of the so far studied logical formalisms
referring to coalitions of agents have only been concerned with formalizing rea-
soning about strategic abilities of coalitions. (A notable exception is [40], where
the Alternating-time Temporal Epistemic Logic ATEL was introduced, whose
language contains both common knowledge and strategic operators for coali-
tions of agents.) Clearly, real cooperation can only be achieved by communica-
tion, i.e., exchange of knowledge. Thus, it is particularly natural and important
to consider multiagent epistemic logics with operators for both common and
distributed knowledge among any (non-empty) coalitions of agents. This is the
logic under consideration in the present paper, hereby called CMAEL(CD)
(for Coalitional Multi-Agent Epistemic Logic with operators of Common and
Distributed knowledge). It subsumes all multiagent epistemic logics mentioned
above, except ATEL.
In order to be practically useful for such tasks as specification and design of
distributedormultiagentsystems,therespectivelogicneedtobeequippedwith
algorithmssolving(constructively)itssatisfiabilityproblem,i.e.testingwhether
a given input formula ϕ of that logic is satisfiable and, if so, providing enough
information for the construction of a model for ϕ. Decidability of modal log-
ics, including epistemic logics, is usually proved by establishing a ‘small model
property’, which provides a brute force decision procedure consisting of exhaus-
tive search for a model amongst all those whose size is within the theoretically
prescribed bounds. The two most common practically feasible general methods
for satisfiability checking of modal logics are based on automata [41] and on
tableaux (see e.g., [33], [3], [11], [46], [8], [19], [12]).
There are various styles of tableau-based decision procedures; see [11], [19]
and [12] for detailed exposition and surveys. An easy to describe but some-
whatlessefficientandpracticallyunfeasibleapproach,thatwewillcallmaximal
2
tableau (also called top-down in [8]), consists in trying to build in one step a
‘canonical’finitemodelforanygivenformulaoutofall maximalconsistentsub-
setsoftheclosureofthatformula.Thismethodalwaysworksin(atleast)expo-
nentialtimeandusuallyproducesawastefullylargemodel,ifanyexists.Amore
flexible and more practically applicable version, adopted in the present paper,
is a so called incremental (aka, ‘bottom-up’) tableau building procedure. While
in all known cases, the worst-case time complexity for maximal andincremental
tableaux are the same, the crucial difference is that maximal tableaux always
require the amount of resources predicted by the theoretical worst-case time
estimate, while incremental tableaux work on average much more efficiently4.
1.2 Related work and comparison
The present work is part of a series of papers ([17],[14],[18],[15]) where we have
embarkedontheprojectofdevelopingpracticallyefficientyetintuitiveandcon-
ceptually clear incremental-tableau-based satisfiability checking procedures for
arangeofmultiagentlogics.Thispaperbuildsontheconferencepapers[14]and
[18] by substantially extending, revising, and improving them.
Therearethreeinherentcomplicationsaffectingtheconstructionofatableau
procedure for the logic CMAEL(CD), arising respectively from the common
knowledge(fixpoint-definableoperator),thedistributedknowledge(withassoci-
ated epistemic relation being the intersection of the individual knowledge epis-
temic relations), and the interactions between the knowledge operators over dif-
ferent coalitions of agents.
Several tableau-based methods for satisfiability-checking for modal logics
with fixpoint-definable operators have been developed and published over the
past 30 years, all going back to the tableau-based decision methods developed
for the Propositional Dynamic Logic PDL in [34], for the branching-time tem-
poral logics UB in [3] and CTL in [8, Section 5] and [7]. In terms of handling
eventualities arising from the fixed-point operators our tableau method follows
morecloselyontheincrementaltableauxforthelineartimetemporallogicLTL
in [46] and for CTL in [8, Section 7].
A particular complication arising in the tableau for CMAEL(CD) stems
from the fact that the epistemic operators, being S5 modalities, are symmetric,
and thus the epistemic boxes have global effect on the model, too. This requires
a special mechanism for propagating their effect backwards when occurring in
states of the tableau. In the present paper we have chosen to implement such
mechanism by using analytic cut rules, going back to Smullyan [37] and Fitting
[11], see also [19] and [28]. More recently, tableaux with analytic cut rules for
modal logics with symmetric relations have been developed in [21], [20], [5].
4 This claim can not be made mathematically precise due to the lack of an a priori
probabilitydistributiononformulaeofalogic.Theinterestedreadermayconsult[17]
forcomparisonofefficiencyofthetwotypesoftableauxinthecontextofAlternating-
time temporal logic ATL.
3
We note that there is a natural tradeoff between conceptual clarity and sim-
plicity of (tableau-based) decision procedures on the one hand, and their tech-
nical sophistication and optimality on the other hand. We emphasize that the
main objective of developing the tableau procedure presented here is the con-
ceptual clarity, intuitiveness, and ease of implementation, rather than practical
optimality. While being optimal in terms of worst-case time complexity and in-
corporating some new and non-trivial optimizing features (such as restricted
applications of cut rules) this procedure is amenable to various improvements
and further optimizations. Most important known such optimizations are on-
the-fly techniques for elimination of bad states and one-pass tableau methods
developed for some related logics in [35], [1] and cut-free versions of tableau as
in [1] for MAEL(C), [22] for PDL with converse operators, [29] for the de-
scription logic SHI and of sequent calculi, in [26] for MAEL(C) and in [4] for
LTL and CTL. We discuss briefly the possible modifications of our procedure,
implementing such optimizing techniques in Section 6.
Here is a summary (in a roughly chronological order) of the more closely
related previous work, besides our own, on tableau-based decision procedures
for multiagent epistemic logics with common and/or distributed knowledge:
– the maximal tableaux for MAEL(C), presented in [25];
– the semantic construction used in [10, Appendix A1] to prove completeness
of an axiomatic system for MAEL(D);
– theproofofdecidabilityof MAEL(CD)basedonfinitemodelpropertyvia
filtration in [39];
– the maximal tableau-like decision procedure for ATL, presented in [44] and
extended to ATEL in [43];
– theexponential-timetableau-basedproceduredevelopedin[6]fortestingsat-
isfiabilityintheBDIlogic,thathassomecommonfeatureswithCMAEL(C);
– the optimized cut-free single-pass tableaux for the multi-agent logic of com-
mon knowledge MAEL(C), in [1]. on tableaux for multiagent logics using
global caching and analytic cuts in [5].
1.3 Structure of the paper
InSection2,weintroducethesyntaxandsemanticsofthelogicCMAEL(CD).
InSection3,weintroduceHintikkastructuresforCMAEL(CD)andshowthat
HintikkastructuresareequivalenttoKripkemodelswithrespecttosatisfiability
of formulae. Then, in Section 4, we develop the tableau procedures checking for
satisfiability of formulae of CMAEL(CD). In Section 5, we prove the correct-
nessofourprocedureinSection6weestimateitscomplexity,discussitefficiency
and indicate some possible technical improvements. We end with concluding re-
marks pointing out some directions for further development.
4
2 Syntax and semantics
2.1 Syntax of CMAEL(CD)
The language of CMAEL(CD) contains a fixed, at most countable, set AP
of atomic propositions, typically denoted by p,q,r,...; a finite, non-empty set
Σ of (names for) agents5, typically denoted by a,b,..., while sets of agents,
called coalitions, will be usually denoted by A,B,...; a sufficient repertoire of
theBooleanconnectives,say¬(“not”)and∧(“and”);and,foreverynon-empty
coalition A, the epistemic operators D (“it is distributed knowledge among A
A
that ...”)andC (“it is common knowledge among A that ...”).Theformulae
A
of CMAEL(CD) are thus defined by the following BNF expression:
ϕ:=p|¬ϕ|(ϕ ∧ϕ )|D ϕ|C ϕ,
1 2 A A
whereprangesoverAPandArangesoverthesetP+(Σ)ofnon-emptysubsetsof
Σ. The other Boolean connectives can be defined as usual. We denote formulae
of CMAEL(CD) by ϕ,ψ,χ,... and omit parentheses in formulae whenever it
does not result in ambiguity.
The distributed knowledge operator D ϕ intuitively means that an “A-
A
superagent”, who knows everything that any of the agents in A knows, can
obtain ϕ as a logical consequence of their knowledge. For example, if agent a
knows that ψ and agent b knows that ψ →χ, then D χ is true even though
{a,b}
neitheranorbknowsχ.TheoperatorsofindividualknowledgeK ϕ(“the agent
a
a knows that ϕ”),fora∈Σ,canbedefinedasD ϕ,henceforthsimplywritten
{a}
(cid:86)
D ϕ. Then, we define K ϕ:= D ϕ.
a A a∈A a
The common knowledge operator C ϕ intuitively means that ϕ is “public
A
knowledge” among A, i.e., that every agent in A knows that ϕ and knows that
every agent in A knows that ϕ, etc. Formulae of the form ¬C ϕ are referred to
A
as (epistemic) eventualities, for the reasons given later on.
2.2 Coalitional multiagent epistemic models
Formulae of CMAEL(CD) are interpreted in coalitional multiagent epistemic
models.Inordertodefinethose,wefirstneedtointroducecoalitional multiagent
epistemic structures and frames.
Definition 1. Acoalitionalmultiagentepistemicstructure(CMAES)isatuple
G = (Σ,S,{RD} ,{RC} )
A A∈P+(Σ) A A∈P+(Σ)
where
5 The notion of agent used in the present paper is an abstract one; in the context
of distributed systems, for example, agents can be thought of as processes making
up the system; in the context of multiagent systems, they can be thought of as
independent software components of the system.
5
1. Σ is a finite, non-empty set of agents6;
2. S (cid:54)=∅ is a set of states;
3. for every A∈P+(Σ), RD is a binary relation on S;
A
4. for every A∈P+(Σ), RC is the reflexive, transitive closure of (cid:83) RD.
A B⊆A B
Definition 2. Acoalitionalmultiagentepistemicframe(CMAEF)isaCMAES
F=(Σ,S,{RD} ,{RC} ),
A A∈P+(Σ) A A∈P+(Σ)
where each RD is an equivalence relation satisfying the following condition:
A
(†) RD =(cid:84) RD
A a∈A a
(Here, and further, we write RD instead of RD , where a∈Σ.)
a {a}
If condition (†) above is replaced by the following, weaker one:
(††) RD ⊆RD whenever B ⊆A,
A B
then F is a coalitional multiagent epistemic pseudo-frame (pseudo-CMAEF).
Notethatinevery(pseudo-)CMAEFRD ⊆(cid:84) RD,andhence(cid:83) RD =
A a∈A a B⊆A B
(cid:83) RD.Hence,condition4ofDefinition1in(pseudo-)CMAEFsisequivalent
a∈A a
torequiringthatRC isthetransitiveclosureof(cid:83) RD,foreveryA∈P+(Σ).
A a∈A a
Also, note that each RC in a (pseudo-)CMAEF is an equivalence relation.
A
Definition 3. A coalitional multiagent epistemic model (CMAEM) is a tuple
M=(F,AP,L),whereFisaCMAEFwithasetofstatesS,APisasetofatomic
propositions, and L:S (cid:55)→P(AP) is a labeling function, assigning to every state
s the set L(s) of atomic propositions true at s.
If F is a pseudo-CMAEF, then M = (F,AP,L) is a multiagent coalitional
pseudo-model (pseudo-CMAEM).
The notion of truth, or satisfaction, of a CMAEL(CD)-formula at a state
of a (pseudo-)CMAEM is defined in the standard Kripke semantics style. In
particular:
– M,s(cid:13)D ϕ iff (s,t)∈RD implies M,t(cid:13)ϕ;
A A
– M,s(cid:13)C ϕ iff (s,t)∈RC implies M,t(cid:13)ϕ.
A A
Definition 4. Given a (pseudo-)CMAEM M, a CMAEL(CD)-formula ϕ is
satisfiable in M if M,s(cid:13)ϕ holds for some s∈M; ϕ is valid in M if M,s(cid:13)ϕ
holds for every s∈M.
A formula ϕ is satisfiable if it is satisfiable in some CMAEM; it is valid,
denoted (cid:13)ϕ, if it is valid in every CMAEM.
6 Notice that we use the same symbol, “Σ”, both for the set of names of agents in
thelanguageandforthesetofagents inCMAES’s.Itwillalwaysbeclearfromthe
context which set we refer to.
6
The satisfaction condition for the operator C can be re-stated in terms of
A
reachability. Let M be a (pseudo-)CMAEM with state space S and let s,t∈S.
We say that t is A-reachable from s if either s = t or, for some n ≥ 1, there
existsasequences=s ,s ,...,s ,s =tofelementsofS suchthat,forevery
0 1 n−1 n
0 ≤ i < n, there exists a ∈ A such that (s ,s ) ∈ RD. It is then easy to see
i i i+1 ai
that the satisfaction condition for C is equivalent to the following one:
A
– M,s(cid:13)C ϕ iff M,t(cid:13)ϕ for every t that is A-reachable from s.
A
The following claim be easily verified.
Proposition 1. (cid:13)C ϕ↔(ϕ∧(cid:86) D C ϕ).
A a∈A a A
Remark If Σ = {a}, then D ϕ ↔ C ϕ is valid for all ϕ. Thus, the single-
a a
agent case is essentially trivialized and, therefore, we assume throughout the
remainder of the paper that the set Σ of names of agents in the language of
CMAEL(CD) contains at least 2 agents.
3 Hintikka structures for CMAEL(CD)
We are ultimately interested in (constructive) satisfiability of (finite sets of)
formulae in models. However, the tableau procedure we present in this paper
checks for the existence of a more general kind of semantic structure for the
input formula, namely a Hintikka structure. In Section 3.1, we introduce Hin-
tikkastructuresforCMAEL(CD).InSection3.2weshowthatsatisfiabilityin
Hintikkastructuresisequivalenttosatisfiabilityinmodels;consequently,testing
for satisfiability in a Hintikka structure can replace testing for satisfiability in a
model.
3.1 Fully expanded sets and Hintikka structures
There are two fundamental differences between (pseudo-)models and Hintikka
structuresforCMAEL(CD),whichmakeworkingwiththelattersubstantially
easier than working directly with models. First, while models specify the truth
valueofeveryformulaofthelanguageateachstate,Hintikkastructuresonlydo
so for the formulae relevant to the evaluation of a fixed formula θ (or, a finite
set of formulae Θ) at a distinguished state. Second, the relations in (pseudo-)
models have to satisfy certain conditions (see Definition 2), while in Hintikka
structures conditions are only placed on the labels of states. These labeling
conditions ensure, however, that every Hintikka structure generates, through
the constructions described in Section 3.2, a pseudo-model so that membership
of formulae in the labels is compliant with the truth in the resultant pseudo-
model. We then show how to convert a pseudo-model into a bona fide model in
a “truth-preserving” way.
To describe Hintikka structures, we need the concept of fully expanded set.
Suchsetscontainalltheformulaethathavetobesatisfiedlocallyatthestateun-
derconsideration.Wedividealltheformulaethatarenotelementaryinthesense
7
thattheirsatisfactionatthestatedoesnotimplysatisfactionofanyotherformu-
laeatthesamestate(suchasp∈APor¬D ϕ)intoα-formulasandβ-formulas.
A
The former are formulae of a conjunctive type, i.e. their truth implies the truth
of all their α-components at the same state, while the latter are of a disjunctive
type: their truth implies the truth of at least one of their β-components at the
same state. Table 3.1 shows the α- and β-formulas of CMAEL(CD) together
with their α- and β-components. The following claims are straightforward, the
cases of common knowledge using Proposition 1.
α-formula α-components β-formula β-components
¬¬ϕ {ϕ} ¬(ϕ∧ψ) {¬ϕ,¬ψ}
ϕ∧ψ {ϕ,ψ} ¬C ϕ {¬ϕ}∪{¬D C ϕ| a∈A}
A a A
D ϕ {D ϕ,ϕ}
A A
C ϕ {ϕ}∪{D C ϕ| a∈A}
A a A
Table 1. α- and β-formulas of CMAEL(CD) with their respective components
Lemma 1. 1. Everyα-formulaisequivalenttotheconjunctionofitsα-components.
2. Every β-formula is equivalent to the disjunction of its β-components.
Definition 5. The closureoftheformulaϕisthesmallestsetofformulaecl(ϕ)
such that:
1. ϕ∈cl(ϕ);
2. cl(ϕ)isclosedwithrespecttoα-andβ-componentsofallα-andβ-formulae,
respectively;
3. for any formula ψ and coalition A, if ¬D ψ ∈cl(ϕ) then ¬ψ ∈cl(ϕ).
A
(cid:83)
Definition 6. For any set of formulae ∆ we define cl(∆):= {cl(ϕ)|ϕ∈∆}.
A set of formulae ∆ is closed if ∆=cl(∆).
Remark 1. Intuitively, the closure of a set of formulae Γ consists of all formulae
that may appear in the tableau whose input is the set of formulae Γ.
Definition 7. A set of formulae is patently inconsistent if it contains a con-
tradictory pair of formulae ϕ and ¬ϕ.
Definition 8. A set ∆ of CMAEL(CD)-formulae is fully expanded if it sat-
isfies the following conditions:
– ∆ is not patently inconsistent;
– if ϕ is an α-formula and ϕ∈∆, then all α-components of ϕ are in ∆.
– if ϕ is a β-formula and ϕ∈∆, then at least one β-component of ϕ is in ∆.
8
Intuitively,anon-patentlyinconsistentsetisfullyexpandedifitisclosedun-
derapplicationsofalllocal (pertainingtothesamestateofastructure)formula
decomposition rules.
Definition 9. The procedure FullExpansion applies to a set of formulae Γ
and produces a (possibly empty) family of sets FE(Γ), called the family of full
expansions of Γ, obtained as follows: start with the singleton family {Γ}; if Γ
is patently inconsistent, halt and return FE(Γ)=∅; otherwise repeatedly apply,
until saturation, the following set replacement operations, each time to a non-
deterministically chosen set Φ from the current family of sets F and a formula
ϕ ∈ Φ; though, we prioritize the eventualities in Γ so that these formulae are
processed first:
1. If ϕ is an α-formula with α-components ϕ and ϕ , then replace Φ by Φ∪
1 2
{ϕ ,ϕ }.
1 2
2. If ϕ is a β-formula such that none of its β-components is in Φ, then replace
Φ with the family of extensions
{Φ∪{ψ}|ψ is a β-component of ϕ}
3. If ϕ = ¬C ψ and ¬ψ ∈/ Φ, but some of the other β-components of ϕ is in
A
Φ, then add to F the set Φ∪{¬ψ}
Thefollowingprovisoappliestotheprocedureabove:ifapatentlyinconsistent
set is added to F as a result of such application, it is removed immediately
thereafter.
Saturation occurs when no application of a set replacement operation can
change the current family F. At that stage, the family FE(Γ) of sets of formulae
is produced and returned. Reaching a stage of saturation is guaranteed to occur
because all sets of formulae produced during the procedure FullExpansion are
subsets of the finite set cl(Γ).
NoticethattheprocedureFullExpansionallowsaddingnotmorethanone
β-component of a formula ϕ=¬C ψ to the initial set, besides ¬ψ.
A
In what follows, we will need the following proposition.
Proposition 2. For any finite set of formulae Γ:
(cid:94) (cid:95)(cid:110)(cid:94) (cid:111)
(cid:13) Γ ↔ ∆|∆∈FE(Γ) .
Proof. By Lemma 1, every set replacement operation applied to a family F
(cid:87) (cid:86)
preserves the formula { ∆ | ∆ ∈ FE(Γ)} up to logical equivalence. At the
(cid:86)
beginning, that formula is Γ, hence the claim follows.
We now define Hintikka structures for CMAEL(CD):
Definition 10. AcoalitionalmultiagentepistemicHintikkastructure(CMAEHS)
is a tuple
(Σ,S,{RD} ,{RC} ,AP,H)
A A∈P+(Σ) A A∈P+(Σ)
such that:
9
– (Σ,S,{RD} ,{RC} ) is a CMAES (recall Definition 1);
A A∈P+(Σ) A A∈P+(Σ)
– AP is a set of atomic propositions;
– H is a labeling of the elements of S with sets of CMAEL(CD)-formulae
that satisfy the following constraints, for every s,s(cid:48) ∈S:
CH1 H(s) is fully expanded;
CH2 If ¬D ϕ∈H(s), then (s,t)∈RD and ¬ϕ∈H(t), for some t∈S;
A A
CH3 If (s,s(cid:48))∈RD, then D ϕ∈H(s) iff D ϕ∈H(s(cid:48)), for every B ⊆A;
A B B
CH4 If ¬C ϕ∈H(s), then (s,t)∈RC and ¬ϕ∈H(t), for some t∈S.
A A
Definition 11. Let H be a CMAEHS with state space S. A CMAEL(CD)-
formula θ is satisfiable in H if θ ∈ H(s), for some s ∈ S. Likewise, a set of
CMAEL(CD)-formulae Θ is satisfiable in H if Θ ⊆H(s), for some s∈S.
3.2 Equivalence of Hintikka structures and models for CMAEL(CD)
Here we show that satisfiability in Hintikka structures is equivalent to satisfia-
bility in models. For brevity, we only deal with single formulae; the extension to
finite sets of formulae is straightforward. The main complications in the proofs
below arise due to the presence of distributed knowledge operators in the lan-
guage of a logic.
HerewewillprovethataCMAEL(CD)-formulaθissatisfiableinaCMAEM
iffitissatisfiableinaCMAEHS.First,weshowthatsatisfiabilityinaCMAEM
impliessatisfiabilityinaCMAEHS.Then,weshowthatsatisfiabilityinaCMAEHS
impliessatisfiabilityinapseudo-CMAEM,whichinturnimpliessatisfiabilityin
a CMAEM.
ThatsatisfiabilityinaCMAEMimpliessatisfiabilityinaCMAEHSisalmost
immediate. Given a CMAEM M with a set of states S, define the extended
labeling function L+ from S to the power-set of CMAEL(CD)-formulae as
M
follows: L+ (s)={ϕ| M,s(cid:13)ϕ}. It is then routine to check the following.
M
Lemma 2. Let M=(Σ,S,{RD} ,{RC} ,AP,L) be a CMAEM
A A∈P+(Σ) A A∈P+(Σ)
satisfyingθandletL+ betheextendedlabelingonM.Then,(Σ,S,{RD} ,
M A A∈P+(Σ)
{RC} ,AP,L+ ) is a CMAEHS satisfying θ. Therefore, satisfiability in a
A A∈P+(Σ) M
CMAEM implies satisfiability in a CMAEHS.
Fortheconversedirectionweneedtwosteps,doneinLemma3andLemma4.
Lemma 3. Letθ beaCMAEL(CD)-formulasatisfiableinaCMAEHS.Then,
θ is satisfiable in a pseudo-CMAEM.
Proof. LetH=(Σ,S,{RD} ,{RC} ,AP,H)beanCMAEHSfor
A A∈P+(Σ) A A∈P+(Σ)
θ. We construct a pseudo-CMAEM M(cid:48) satisfying θ out of H as follows.
First, for every A ∈ P+(Σ), let R(cid:48)D be the reflexive, symmetric, and tran-
A
sitive closure of (cid:83) RD and let R(cid:48)C be the transitive closure of (cid:83) R(cid:48)D.
A⊆B B A a∈A a
Thus, both R(cid:48)D and R(cid:48)C are equivalence relations and RD ⊆ R(cid:48)D and RC ⊆
A A A A A
R(cid:48)C, for every A ∈ P+(Σ). Second, let L(s) = H(s) ∩ AP, for every s ∈ S.
A
10