Interactions Between Knowledge and Time in First-Order Logic for Multi-Agent Systems - Completeness Results

7/28/2019 Interactions Between Knowledge and Time in First-Order Logic for Multi-Agent Systems - Completeness Results

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Journal of Articial Intelligence Research 45 (2012) 1-45 Submitted 11/11; published 9/12

Interactions between Knowledge and Time ina First-Order Logic for Multi-Agent Systems:

Completeness Results

F. Belardinelli [email protected]. Lomuscio [email protected] of Computing Imperial College London, UK

AbstractWe investigate a class of rst-order temporal-epistemic logics for reasoning about multi-

agent systems. We encode typical properties of systems including perfect recall , synchronic-ity , no learning , and having a unique initial state in terms of variants of quantied inter-preted systems, a rst-order extension of interpreted systems. We identify several monodicfragments of rst-order temporal-epistemic logic and show their completeness with respectto their corresponding classes of quantied interpreted systems.

1. Introduction

While reactive systems (Pnueli, 1977) are traditionally specied using plain temporal logic,there is a well-established tradition in Articial Intelligence (AI) and, in particular, Multi-Agent Systems (MAS) research to adopt more expressive languages. Much of this traditionis inspired by the earlier, seminal work in AI by McCarthy (1979, 1990) and others aimed atadopting an intentional stance (Dennett, 1987) when reasoning about intelligent systems.

Specically, logics for knowledge (Fagin, Halpern, Moses, & Vardi, 1995), beliefs, desires,intentions, obligations, etc., have been put forward to represent the informational andmotivational attitudes of agents in the system. Theoretical explorations have focused onthe soundness and completeness of a number of axiomatisations as well as the decidabilityand computational complexity of the corresponding logics.

The great majority of work in these lines focuses on propositional languages. Yet, spec-ications supporting quantication are increasingly required in applications. For example,it is often necessary to refer to different individuals at different instances of time.

Quantied modal languages (Garson, 2001) have long attracted considerable attention.Early work included analysing the philosophical and logical implications of different se-tups for the quantication domains, particularly in combination with temporal concepts.More recently, considerable attention has been given to identifying suitable fragments thatpreserve completeness and decidability, and then studying the resulting computational com-plexity of the satisability problem. This article follows this direction.

In more detail, we investigate the meta-theoretical properties of monodic fragmentsof quantied temporal-epistemic logic where interactions between quantiers, time, andknowledge of the agents are present. There is a deep-rooted interest (Fagin et al., 1995;Meyden, 1994) in understanding the implications of interaction axioms in this context, asthey often express interesting properties of MAS, including perfect recall, synchronicity,

c 2012 AI Access Foundation. All rights reserved.


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Belardinelli & Lomuscio

and no learning. These features are well-understood at the propositional level (Fagin,Halpern, & Vardi, 1992; Halpern, van der Meyden, & Vardi, 2004) and are commonly usedin several application areas. The technical question this paper aims to resolve is whether asimilar range of results can be provided in the presence of (limited forms of) quantication.

As we shall demonstrate, the answer to this question is largely positive.

1.1 State of the Art

The analysis and application of temporal-epistemic logic in a rst-order setting has anestablished tradition in AI. One of the early contributions is the work of Moore (1990),which presents a theory of action that takes into consideration the epistemic preconditionsto actions and their effects on knowledge. More recently, a number of rst-order temporal-epistemic logics for reasoning about MAS were introduced by Wooldridge et al. (2002,2006, 1999), often in the context of the MABLE programming language for agents. Thesame authors introduced a rst-order branching time temporal logic for MAS (Wooldridge

& Fisher, 1992), and developed it in a series of papers (Wooldridge et al., 2002, 2006).First-order multi-modal logics also constitute the conceptual base of a number of otheragent theories, such as BDI logics (Rao & Georgeff, 1991), the KQML framework (Cohen& Levesque, 1995), and the LORA framework (Wooldridge, 2000b). All of these includeoperators for mental attitudes (e.g., knowledge, belief, intention, desire, etc.), as well astemporal and dynamic operators with some form of quantication. However, most of thecurrent literature has so far fallen short of a systematic analysis of the formal properties of these frameworks. Some of the frameworks above are so rich that they are unlikely to benitely axiomatisable, let alone decidable. Still, these earlier contributions are an inspirationto the present investigation, as they are among the few to have explicitly addressed thesubject of rst-order temporal-epistemic languages in a MAS setting.

At a purely theoretical level, rst-order temporal and epistemic logics have also re-ceived increasing attention with a range of contributions on axiomatisability (Degtyarevet al., 2003; Sturm et al., 2000; Wolter & Zakharyaschev, 2002), decidability (Degtyarevet al., 2002; Hodkinson et al., 2000; Wolter & Zakharyaschev, 2001), and complexity (Hod-kinson, 2006; Hodkinson et al., 2003). Wolter and Zakharyaschev (2001) introduced themonodic fragment of quantied modal logic, where the modal operators are restricted toformulas with at most one free variable, and they proved the decidability of various frag-ments. Similar results have been obtained for monodic fragments of rst-order temporallogic (Hodkinson, 2002; Hodkinson et al., 2000), and the computational complexity of theseformalisms have been analysed (Hodkinson, 2006; Hodkinson et al., 2003). Further, Wolterand Zakharyaschev (2002) provided a complete axiomatisation of the monodic rst-ordervalidities on the natural numbers. The monodic fragment of rst-order epistemic logic hasalso been explored (Sturm et al., 2000; Sturm, Wolter, & Zakharyaschev, 2002), and anaxiomatisation including common knowledge has been provided. These lines of researchconstitute the theoretical background against which this research is set.

The contributions discussed previously used plain Kripke models as the underlying se-mantics. However, it has been argued though that in applications a computationally-grounded semantics (Wooldridge, 2000a) is preferable, as this enables systems to be mod-elled directly. We introduced quantied interpreted systems (QIS) to ll this gap (Belar-

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Interactions between Knowledge and Time in a First-Order Logic for MAS

dinelli & Lomuscio, 2009). This enabled us to provide a complete axiomatisation of themonodic fragment of quantied temporal-epistemic logic on linear time (Belardinelli & Lo-muscio, 2011). However, no interaction between temporal and epistemic modalities wasstudied. Preliminary investigations into the interactions between temporal and epistemic

operators in a rst-order setting have already appeared (Belardinelli & Lomuscio, 2010). Inthis paper we extend the previous results and also consider epistemic languages containingthe common knowledge operator.

1.2 The Present Contribution

This paper extends the current state of the art in rst-order temporal-epistemic logic byintroducing a family of provably complete calculi for a variety of quantied interpreted sys-tems characterising a range of properties including perfect recall, no learning, synchronicity,and having a unique initial state. We prove the completeness of the presented rst-ordertemporal-epistemic logics via a quasimodel construction, which has previously been used(Hodkinson, Wolter, & Zakharyaschev, 2002; Hodkinson et al., 2000) to prove decidabilityfor monodic fragments of rst-order temporal logic ( FoTL ). Quasimodels have also beenapplied to rst-order temporal as well as epistemic logic (Sturm et al., 2000; Wolter & Za-kharyaschev, 2002). Wolter et al. (2002) present a complete axiomatisation for the monodicfragment of FoTL on the natural numbers; a similar result for a variety of rst-order epis-temic logics with common knowledge has also appeared (Sturm et al., 2000). However, theinteraction between temporal and epistemic modalities in a rst order setting has not beentaken into account yet, nor has the interpreted systems semantics. Nonetheless, both of these features are essential for applications to multi-agent systems and are the subject of analysis here.

1.2.1 Structure of the Paper.

In Section 2 we rst introduce the rst-order temporal-epistemic languages Lm and LC mwith common knowledge for a set Ag = {1, . . . , m } of agents. We then present the relevantclasses of QIS as well as the monodic fragments of Lm and LC m . In Sections 3 we introducethe axiomatisations for these classes of QIS, while the details of the completeness proofsare presented in Sections 4 and 5. Finally, in Section 6 we elaborate on the results obtainedand discuss possible extensions and future work.

2. First-Order Temporal-Epistemic Logics

Interpreted systems are a standard semantics for interpreting temporal-epistemic logics in amulti-agent setting (Fagin et al., 1995; Parikh & Ramanujam, 1985). We extend interpretedsystems to the rst-order case by enriching these structures with a domain of individuals.We rst investigated static quantied interpreted systems, where no account for theevolution of the system is given (Belardinelli & Lomuscio, 2008, 2009). Then, fully-edgedQIS on a language with also temporal modalities were introduced (Belardinelli & Lomuscio,2010, 2011). We follow the denition of QIS provided in the references.

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for r R , n N , I is a rst-order interpretation, that is, a function such that

for every constant c, I (c, r (n)) D , for every predicate constant P k , I (P k , r (n)) is a k-ary relation on D.

Further, for every r, r R , n, n N , I (c, r (n)) = I (c, r (n )) .

Notice that we assume a unique domain of interpretation , as well as a xed interpretation for individual constants; so we simply write I (c). Following standard notation (Fagin et al.,1995), for r R and n N , a pair ( r, n ) is a point in P . If r (n) = le , l1, . . . , l m is theglobal state at point ( r, n ) then r e(n) = le and r i (n) = li are the environments and agentis local state at ( r, n ) respectively. Further, for i Ag the epistemic equivalence relation

i is dened such that ( r, n ) i (r , n ) iff r i (n) = r i (n ). Clearly, each i is an equivalencerelation. Two points ( r, n ) and ( r , n ) are said to be epistemically reachable , or simplyreachable, if ( r, n ) (r , n ) where is the transitive closure of i Ag i .

In this paper we consider the following classes of QIS.

Denition 3. A quantied interpreted system P satises

synchronicity iff for every i Ag, for all points (r, n ), (r , n ),(r, n ) i (r , n ) implies n = n

perfect recall for agent i iff for all points (r, n ), (r , n ), if (r, n ) i (r , n ) and n > 0then either (r, n 1) i (r , n ) or there is k < n such that (r, n 1) i (r , k) and for all k , k < k n implies (r, n ) i (r , k )

no learning for agent i iff for all points (r, n ), (r , n ), if (r, n ) i (r , n )then either (r, n + 1) i (r , n ) or there is k > n such that (r, n + 1) i (r , k) and for all k , k > k n implies (r, n ) i (r , k )

unique initial state iff for all r , r R , r (0) = r (0)

These conditions have extensively been discussed in the literature (Halpern et al., 2004)together with equivalent formulations. Intuitively, a QIS is synchronous if time is part of the local state of each agent. A QIS satises perfect recall for agent i if is local state recordseverything that has happened to him (from the agents point of view) so far in the run. Nolearning is dual to perfect recall: agent i does not acquire any new knowledge during a run.Finally, a QIS has a unique initial state if all runs start from the same global state.

A QIS P satises perfect recall (resp. no learning) if P satises perfect recall (resp. nolearning) for all agents. We denote the class of QIS with m agents as QIS m ; the superscripts pr, nl , sync , uis denote the subclasses of QIS m satisfying perfect recall, no learning,synchronicity, and having a unique initial state respectively. For instance, QIS sync,uism isthe class of all synchronous QIS with m agents and having a unique initial state.

We now assign an interpretation to the formulas in Lm and LC m by means of quantiedinterpreted systems. Let be an assignment from variables to individuals in D, the valuation

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I (t) of a term t is dened as (y) for t = y, and I (t) = I (c) for t = c. A variant xa of anassignment assigns a D to x and agrees with on all other variables.

Denition 4. The satisfaction of a formula L m at point (r, n ) P under an assignment

, denoted (P , r,n ) |= , is dened inductively as follows:

(P , r,n ) |= P k (t1, . . . , t k ) iff I (t1), . . . , I (tk ) I (P k , r (n))(P , r,n ) |= iff (P , r,n ) |= (P , r,n ) |= iff (P , r,n ) |= or (P , r,n ) |= (P , r,n ) |= x iff for all a D , (P

xa , r,n ) |=

(P , r,n ) |= iff (P , r,n + 1) |= (P , r,n ) |= U iff there is n n such that ( P , r,n ) |=

and ( P , r,n ) |= for all n n < n(P , r,n ) |= K i iff for all r , n , (r, n ) i (r , n ) implies (P , r , n ) |=

For L C m we have to consider also the case for the common knowledge operator:

(P , r,n ) |= C iff for allk N , (P , r,n ) |= E k

The truth conditions for , , , , G and F are dened from those above. Fromthe denition above it follows that ( P , r,n ) |= C iff for all (r , n ) reachable from ( r, n ),(P , r , n ) |= .

A formula is true at a point (r, n ) if it is satised at ( r, n ) by every assignment ; istrue on a QIS P if it is true at every point in P ; is valid on a class C of QIS if it is trueon every QIS in C . Further, a formula is satisable on a QIS P if it is satised at somepoint in P , for some assignment ; is satisable on a class C of QIS if it is satisable on

some QIS in C .By considering all combinations of pr , nl , sync and uis we obtain 16 subclasses of QIS mfor any m N . Not all of them are independent, nor axiomatisable. Indeed, some of theseare not axiomatisable even at the propositional level (Halpern & Moses, 1992; Halpern &Vardi, 1989). In the rst column of Table 1 we group together the classes of QIS that sharethe same set of validities on the languages Lm and LC m . The proofs of these equivalencesare similar to those of the propositional case (Halpern et al., 2004) and are not reportedhere. Further, we dene the languages PL m and PL C m as the propositional fragments of Lm and LC m respectively (formally, PL m and PL C m are obtained by restricting atomicformulas to 0-ary predicate constants p1, p2, . . .). Table 1 summarises the results by Halpernet al. (2004) concerning the axiomatisability of propositional validities in PL m and PL C m .Observe that, as regards the language PL m , for m = 1 all sets of validities on the variousclasses of QIS are axiomatisable, while for m 2 no axiomatisation can be given forQIS nl,uism and QIS nl,pr,uism (Halpern & Vardi, 1986, 1989). As to the language PL C m , werestrict to the case for m 2, as for m = 1 PL C m has the same expressive power asPL m . For m 2 no class of validities on PL C m has a recursive axiomatisation but QIS m ,QIS syncm , QIS uism , QIS sync,uism and QIS nl,sync,uism , QIS nl,pr,sync,uism .

In the next section we show that the axiomatisability results at the propositional levelcan be lifted to the monodic fragment of the languages Lm and LC m .

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QIS PL 1 PL m , m 2 PL C m , m 2QIS m , QIS syncm , QIS uism , QIS sync,uism QIS prm , QIS pr,uism QIS pr,syncm , QIS pr,sync,uism

QIS nlm QIS nl,syncm QIS nl,prm QIS nl,pr,syncm QIS nl,uism QIS nl,pr,uism QIS nl,sync,uism , QIS nl,pr,sync,uism

Table 1: Equivalences between classes of QIS and axiomatisability results for the proposi-tional fragments PL m and PL C m . The sign indicates that the set of validitieson a specic class is axiomatisable; while indicates that it is not.

2.3 The Monodic Fragment

In the rest of the paper we will show that a sufficient condition for lifting the results inTable 1 to the rst-order case is to restrict the languages Lm and LC m to their monodicfragments.

Denition 5. The monodic fragment L1m is the set of formulas L m such that any subformula of of the form K i, , or 1 U 2 contains at most one free variable. Similarly,the monodic fragment LC 1m is the set of formulas L C m such that any subformula of of the form K

i, C , , or

1 U

2contains at most one free variable.

The monodic fragments of a number of rst-order modal logics have been thoroughlyinvestigated in the literature (Hodkinson et al., 2000, 2003; Wolter & Zakharyaschev, 2001,2002). In the case of Lm and LC m these fragments are quite expressive as they containformulas like the following:

y C ( zAvailable (y, z ) U xRequest (x, y )) (1)

K i xyz (Request (x, y ) Available (y, z )) K i xyz (Request (x, y ) Available (y, z )) (2)

Formula (1) intuitively states that it is common knowledge that every resource y willeventually be requested by somebody, but until that time the resource remains available toeverybody. Notice that y is the only free variable within the scope of modal operators U and C . Formula (2) represents that if agent i knows that at the next step every resource isavailable whenever it is requested, then at the next step agent i knows that this is indeedthe case. However, note that the formula

xK i (Process (x) yF Access (x, y ))

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which intuitively means that agent i knows that every process will eventually try to accessevery resource, is not in L1m as both x and y occur free within the scope of modal operatorF . Still, the monodic fragments of Lm and LC m are quite expressive as they contain allde dicto formulas, i.e., formulas where no free variable appears in the scope of any modal

operator, as in (2). So, the limitation is really only on de re formulas.We stress the fact that the formulas above have no propositional equivalent in the

case that they are interepreted on quantied interpreted systems in which the domain of quantication is innite, or its cardinality cannot be bounded in advance.

Finally, observe that the Barcan formulas x x and K i x xK i are both true in all quantied interpreted systems, as each QIS includes a unique domainof quantication. This implies that the universal quantier commutes with the temporalmodality and the epistemic modality K i . Thus, it can be the case that for some formulas, L m , we have that is a validity, but L 1m and / L 1m . For instance,consider = xP (x, y ) and = x P (x, y). We will see that this remark does notinterfere with our results.

3. Axiomatisations

In this section we present sound and complete axiomatisations of the sets of monodic va-lidities for the classes of quantied interpreted systems in Section 2. First, we introducethe basic system QKT m that extends to the rst-order case the multi-modal epistemic logicS5m combined with the linear temporal logic LTL.

Denition 6. The system QKT m contains the following schemes of axioms and rules,where , and are formulas in L1m and = is the inference relation.

First-order logic Taut classical propositional tautologies MP , = Ex x [x/t ]Gen [x/t ] = x, where x is not free in

Temporal logic K ( ) ( )T1 T2 U ( ( U ))Nec = T3 = ( U )

Epistemic logic K K i ( ) (K i K i )T K i 4 K

i K

iK

i

5 K i K i K i Nec = K i

The operator K i is an S5 modality, while the next and until U operators are axioma-tised as linear-time modalities (Fagin et al., 1995). To this we add the classical postulatesEx and Gen for quantication, which are both sound as we consider a unique domain of individuals in the quantied interpreted systems.

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3.1 Kripke Models

To prove the completeness results in Theorem 2, we rst introduce an appropriate class of Kripke models as a generalisation of QIS and prove completeness for these models. Thenwe apply a correspondence result between Kripke models and QIS to obtain the desiredresults.

Denition 8 (Kripke model) . A Kripke model is a tuple M = W, R W , { i }i Ag , D, I such that

W is a non-empty set of states;

R W is a non-empty set of functions r : N W ;

for every agent i Ag, i is an equivalence relation on W ;

D is a non-empty set of individuals;

for every w W , I is a rst-order interpretation, that is, a function such that

for every constant c, I (c, w) D , for every predicate constant P k , I (P k , w) is a k-ary relation on D.

Further, for every w, w W , I (c, w) = I (c, w ).

Notice that Def. 8 differs from other notions of Kripke model in that it includes theset R W of functions to guarantee that the correspondence between Kripke models andQIS is one-to-one. We also assume a unique domain of interpretation , as well as a xed interpretation for individual constants, so also in this case we simply write I (c). Kripkemodels are a generalisation of QIS in that they do not specify the inner structure of the statesin W . Also for Kripke models we introduce points as pairs ( r, n ) for r R W and n N .A point derives its properties from the corresponding state; for instance, ( r, n ) i (r , n ) if r (n) i r (n ).

We consider Kripke models satisfying synchronicity, perfect recall, no learning, andhaving a unique initial state. The denition of these subclasses is analogous to Def. 3.

Denition 9. A Kripke model M satises synchronicity iff for every i Ag, for all points (r, n ), (r , n ),

(r, n ) i (r , n ) implies n = n

perfect recall for agent i iff for all points (r, n ), (r , n ), if (r, n ) i (r , n ) and n > 0then either (r, n 1) i (r , n ) or there is k < n such that (r, n 1) i (r , k) and

for all k , k < k n implies (r, n ) i (r , k ).no learning for agent i iff for all points (r, n ), (r , n ), if (r, n ) i (r , n )

then either (r, n + 1) i (r , n ) or there is k > n such that (r, n + 1) i (r , k) and for all k , k > k n implies (r, n ) i (r , k ).

unique initial state iff for all r , r R W , r (0) = r (0) .

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Notice that if M satises synchronicity, or perfect recall, or no learning, or has a uniqueinitial state, then also g(M ) satises the same property. This follows from the fact that(r, n ) i (r , n ) iff (r ri , n ) i (r r , n ). Thus, g denes a map from each of the 16 subclassesof Km outlined in Def. 9 to the corresponding subclass of QIS m and we obtain the following

corollary to Lemma 1.Corollary 1. Let X be any subset of { pr, nl, sync, uis }. For every monodic formula L 1m(resp. LC 1m ), if is satisable in KX m , then is satisable in QIS X m .

For reasoning about the monodic fragments of Lm and LC m when we are dealing with nolearning and perfect recall, we introduce the following class of monodic friendly Kripkemodels. These structures are motivated by the fact that KT1 and KT3 are too weakto enforce either perfect recall or no learning on Kripke models when these axioms arerestricted to monodic formulas. However, they suffice for monodic friendly structures. Inthe following, we also prove that satisability in Kripke models is equivalent to satisfabilityin monodic friendly structures when we restrict our languages to monodic formulas.

Denition 11 (mf-model) . A monodic friendly Kripke model is a tuple M mf = W, R W , { i,a }i Ag,a D , D, I such that

W , R W , D and I are dened as for Kripke models;

for i Ag, a D , i,a is an equivalence relation on W .

We can dene synchronicity, perfect recall, no learning, and having a unique initial statealso for mf-models by parametrising Def. 9 to each relation i,a . For instance, an mf-modelsatises perfect recall for agent i if for all points (r, n ), (r , n ), for all a D , whenever(r, n ) i,a (r , n ) and n > 0 then either ( r, n 1) i,a (r , n ) or there is k < n such that(r, n 1) i,a (r , k) and for all k , k < k n implies (r, n ) i,a (r , k ). As regards thesubclasses of the class MF m of all mf-models with m agents, we adopt the same namingconventions as for QIS and Kripke models. Notice that Kripke models can be seen asmf-models such that for all i Ag, a, b D , i,a is equal to i,b .

Finally, the satisfaction relation |= for L 1m (resp. LC 1m ) in a mf-model M mf isdened in the same way as in Kripke models, except for the epistemic operators:

(M mf , r,n ) |= K i [y] iff for allr , n , (r, n ) i, (y) (r , n ) implies (Mmf , r , n ) |= [y]

where at most y appears free in . Notice that if is a sentence, then ( M mf , r,n ) |= K i iff (r, n ) i,a (r , n ) implies (M mf , r , n ) |= for all a D . The case for the commonknowledge operator C is straightforward by denition of E k . In particular, two points ( r, n )and ( r , n ) are epistemically reachable for a D , or simply reachable, if ( r, n ) a (r , n ),where a is the transitive closure of i Ag i,a .

We remark that the converse of the Barcan formula, or CBF , K i x xK i holdsin all mf-models; while the Barcan formula, or BF , xK i K i x does not. To checkthis consider the mf-model M = W, R W , { i,a }i Ag,a D , D, I in Fig.1(a) such that

- W = {w, w , w }

- R W = {r, r , r } and r (0) = w, r (0) = w , and r (0) = w

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w

P (a)

...

w

P (a), P (b)

...

w

P (b)

...

i,a i,b

(a) The mf-model M .

v

Q(c)

...

v

Q(c)

...

v

...

i,c i,d

(b) The mf-model M .

Figure 1: Arrows represent the system runs; while epistemically related states are groupedtogether.

- D = {a, b}

- I (P 1, r (0)) = {a, b}, I (P 1, r (0)) = {a} and I (P 1, r (0)) = {b}

- i,a and i,b are equivalence relations such that ( r, 0) i,a (r , 0) and ( r, 0) i,b (r , 0).

We can see that ( M , r, 0) |= xK iP (x), but ( M , r, 0) |= K i xP (x) as (r, 0) i,a (r , 0) and(M , r , 0) |= P (x) for (x) = b.

Furthermore, the K axiom K i ( ) (K i K i ) is not valid on mf-modelseither. In fact, consider the mf-model M = W , R W , { i,a }i Ag,a D , D , I in Fig.1(b)such that

- W = {v, v , v }

- R W = {q, q , q } and q (0) = v, q (0) = v , and q (0) = v

- D = {c, d}

- I (Q1, q (0)) = {c}, I (Q1, q (0)) = {c} and I (Q1, q (0)) =

- i,c and i,d are equivalence relations such that ( q, 0) i,c (q , 0) and ( q, 0) i,d (q , 0).

Finally, let (x) = c. We can check that ( M , q, 0) |= ( Q(x) xQ(x)) Q(x) and(M , q , 0) |= ( Q(x) xQ (x)) Q(x). Thus, ( M , q, 0) |= K i (Q(x) xQ(x)) K i Q(x).But ( M , q , 0) |= xQ (x), so (M , q, 0) |= K i xQ(x).

We now prove the following lemma, which will be used in the completeness proof forsystems satisfying perfect recall or no learning. The lemma states that, when we deal withsatisfability of monodic formulas, mf-models suffice.

Lemma 2. Let MF K,BF m be the class of all mf-models validating the formulas K and BF .For every monodic formula L 1m (resp. LC 1m ),

Km |= iff MF K,BF m |=

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Proof. The implication from right to left follows from the fact that the class Km of allKripke models is isomorphic to the subclass of monodic friendly Kripke models such thatfor all i Ag, a, b D , i,a is equal to i,b . In other words, given a Kripke model M =W, R W , { i }i Ag , D, I we can dene the mf-model M = W, R W , { i,a }i Ag,a D , D, I ,

where for every a D , i,a is equal to i . It is straightforward to see that M validatesboth K and BF (in particular, the counterexamples in Fig. 1 are ruled out). Further, if M |= then M |= . Thus, if MF K,BF m |= , then Km |= .

For the implication from left to right, assume that M mf = W, R W ,{ i,a }i Ag,a D , D, I is an mf-model validating K and BF such that ( M mf , r,n ) |= for some point ( r, n ) and some assignment . We can then build a Kripke model M =W , R W , { i }i Ag , D , I from M mf such that ( M

, r,n ) |= as follows. We start byassuming W = W , R = R and D = D. Further, for each i Ag, dene i as thetransitive closure of a D i,a . Finally, set I = I . We now have to check that the Kripkemodel M is well dened and does not validate .

First of all, we point out the following issue associated with the construction above: it

can be the case that for some point ( q, k) and some monodic formula [x], it happens that(M mf , q ,k) |= K i[x], (q, k) i, (x ) (q , k ) and ( q, k) i, (y) (q , k ) for some (x) = (y).Further, suppose that ( M mf , q , k ) |= [x], while we obviously have that ( M

mf , q , k ) |=

[x]. Now by the denition of i we have that ( M , q ,k) |= K i [x]; so the two models donot satisfy the same formulas. We can solve this problem by modifying the interpretation I according to the structure of the monodic formula [x], while keeping the same truth valuefor [x] at point ( q, k). We consider the relevant cases according to the structure of [x];the induction hypothesis consists of the fact that we are able to nd such an interpretationI for all subformulas of [x].

For [x] = P (x) we simply assume that (x) I (P, q (k )), so that ( M , q , k ) |=[x] and (M , q ,k) |= K i [x]. Note that this does not change the truth value of anyepistemic formula in ( q, k) as we assumed that ( q, k) i, (x ) (q , k ) (otherwise [x] wouldbe satised in ( q , k )). The cases for propositional connectives and modal operators aresimilarly dealt with by applying the induction hypothesis. For [x] = y[x, y ] we havethat ( M mf , q , k ) |= [x], therefore there exists b D such that ( M

ybmf , q , k ) |= [x, y ].

Now we have to consider 4 different cases depending on whether ( q, k) satises any of these4 formulas:

K i x [x, y] (3)K i x [x, y] (4)K i x [x, y] (5)K i x [x, y] (6)

By using the axioms and inference rules in QKT m for Formula (3) we can show whatfollows (where is used for entailment):

(M mf , q ,k) |= K i y[x, y ] K i x[x, y](M mf , q ,k) |= xK i y[x, y] yK i x[x, y ] by Ex(M mf , q ,k) |= K i x y[x, y] K i y x[x, y ] by zK i K i z(M mf , q ,k) |= K i ( x y[x, y ] y x[x, y]) by K i ( ) K i K i

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suby {, C ( zAv (y, z ) U xReq (x, y)), EC ( zAv (y, z ) U xReq (x, y )),{K i C ( zAv (y, z ) U xReq (x, y))}i Ag , zAv (y, z ) U xReq (x, y), zAv (y, z ), xReq (x, y),, C ( zAv (y, z ) U xReq (x, y)), EC ( zAv (y, z ) U xReq (x, y )),{K i C ( zAv (y, z ) U xReq (x, y))}i Ag , zAv (y, z ) U xReq (x, y), zAv (y, z ), xReq (x, y ),

, C ( zAv (y, z ) U xReq (x, y)), EC ( zAv (y, z ) U xReq (x, y )),{ K i C ( zAv (y, z ) U xReq (x, y ))}i Ag , zAv (y, z ) U xReq (x, y), zAv (y, z ),

xReq (x, y), , C ( zAv (y, z ) U xReq (x, y)), EC ( zAv (y, z ) U xReq (x, y )),{ K i C ( zAv (y, z ) U xReq (x, y ))}i Ag , zAv (y, z ) U xReq (x, y), zAv (y, z ), xReq (x, y)}

Table 3: the set suby for equal to Formula (1).

t {, C ( zAv (y, z ) U xReq (x, y )), EC ( zAv (y, z ) U xReq (x, y)),{K i C ( zAv (y, z ) U xReq (x, y))}i Ag , zAv (y, z ) U xReq (x, y), zAv (y, z ), xReq (x, y), , C ( zAv (y, z ) U xReq (x, y)), EC ( zAv (y, z ) U xReq (x, y)),

{ K i C ( zAv (y, z ) U xReq (x, y))}i Ag , zAv (y, z ) U xReq (x, y), zAv (y, z ), xReq (x, y)}

Table 4: a type t in cl0, for equal to Formula (1).

We dene ad() as the greatest number of alternations of distinct K i s along any branchin s parse tree (Halpern et al., 2004). Further, an index is any nite sequence = i1, . . . , i kof agents such that in = in +1 , for 1 n < k ; the length of is denoted by | |. Also, i isthe absorptive concatenation of indexes and i such that i = if ik = i. Finally, K is ashorthand for K i1 . . . K ik . Now let be an index such that | | ad(). If is the emptysequence then cl = clad (). If = i , then cl = clk,i for k = ad() | |. We now

introduce types for quasimodels, which intuitively can be seen as individuals described bymaximal and consistent sets of formulas.

Denition 13 (Type) . A -type t for is any maximal and consistent subset of cl , i.e., for every monodic formulas and in cl ,

(i) t iff / t ;

(ii) t iff , t .

Two -types t , t are said to agree on sub0 if t sub0 = t sub0, i.e., if they sharethe same sentences. Given a -type t for and a constant c con, t , c is an indexed type for , abbreviated as t c . In Table 4 we report a type t in cl0, for equal to Formula (1).

We now introduce state candidates , which intuitively represent the states of a quasi-model.

Denition 14 (State candidate) . A -state candidate for is a pair C = T, T con such that

(i) T is a set of -types for that agree on sub0;

(ii) T con is a set containing for each c con an indexed type t c such that t T .

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We also introduce the notion of point , which describes a state candidate from the per-spective of a particular type.

Denition 15 (Point) . A -point for is a pair P = C, t such that

(i) C = T, T con is a -state candidate for ;

(ii) t T is a -type.

Note that, by a slight abuse of notation, we call points both the pairs ( r, n ) in QIS andthe pairs P = C, t . This is to be consistent with previous work (Fagin et al., 1995; Halpernet al., 2004); the context will disambiguate. Also, we write t C for C = T, T con andt T . Similarly for t P . Given a -state candidate C = T, T con and a point P = C, t we dene the formulas C and P as follows:

C :=

t T

xt [x] x

t T

t [x]

tc

T con

t [x/c ]

P := C t

where we do not distinguish between a type t and the conjuction of formulas it contains.A -state candidate C is S -consistent if the formula C is consistent w.r.t. the system S ,

i.e., if S C. Similarly, a -point P is S -consistent if the formula P is consistent w.r.t. S .We refer to plain consistency whenever the system S of reference is understood. Consistentstate candidates represent the states of our quasimodels. We now dene the relations of suitability that constitute the relational part of quasimodels.

Denition 16. A 1-type t 1 and a 2-type t 2 are -suitable , or t 1 t 2, iff 1 = 2and t 1 t 2 is consistent. They are i-suitable , or t 1 i t 2, iff 1 i = 2 i and t 1 K i t 2is consistent.

A 1-state candidate C1 and a 2-state candidate C2 are -suitable , or C1 C2, iff 1 = 2 and C1 C2 is consistent. They are i-suitable , or C1 i C2, iff 1 i = 2 iand C1 K i C2 is consistent.

A 1-point P 1 and a 2-point P 2 are -suitable , or P 1 P 2, iff 1 = 2 and P 1 P 2 is consistent. They are i-suitable , or P 1 i P 2, iff 1 i = 2 i and P 1 K i P 2

is consistent.

A 1-point P 1 = C1, t 1 and a 2-point P 2 = C2, t 2 are -suitable for a constantc con, or P 1 c P 2, iff P 1 P 2, t c1 T con1 and t c2 T con2 . They are i-suitable

for c, or P 1 ci P 2, iff P 1 i P 2, t c1 T con1 and t c2 T con2 .

By using the axioms T , 4 and 5 it can be shown that the relation i is reexive,transitive and symmetric, that is, an equivalence relation. Also, the relation is serial.In the following lemma we list some properties of relations and i that will be useful inwhat follows.

Lemma 3. (i) Let subx , if t 1 t 2 then t 1 iff t 2.

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(ii) Let K i subx and let t be a -type, K i t iff for all -types t , t i t implies t . Moreover, let | i | ad(), then K i t iff for all i -types t , t i t implies t .

Proof.(i) The proof is similar to the one for Lemma 9(i) in the work of Wolter et al. (2002). If

t 1 and / t 2 then t 2 and since t 1 t 2 is consistent, then also is consistent, which is a contradiction. From right to left, if t 2 and / t 1 then t 1. Since t 1 t 2 is consistent, then also is consistent, whichis a contradiction.

(ii) From left to right, if K i t and / t then t and since t K i t is consistent,then also K i K i is consistent, which is a contradiction. From right to left, if K i / t then we can extend the set {} { | K i t } to a -type t . In particular,t i t and t . Moreover, if | i | ad() then we can similarly prove that K i t

iff for all i -types t , t i t implies t .We now present the frame underlying the quasimodel for .

Denition 17 (Frame) . A frame F is a tuple R , D, { i,a }i Ag,a D , f where

(i) R is a non-empty set of indexes r, r , . . . ;

(ii) D is a non-empty set of individuals;

(iii) for every i Ag, a D , i,a is an equivalence relation on the set of points (r, n ) for r R and n N ;

(iv) f is a partial function associating to each point (r, n ) a consistent state candidate f(r, n ) = Cr,n such that

(a) the domain of f is not empty;

(b) if f is dened on (r, n ), then it is dened on (r, n + 1) ;(c) if f is dened on (r, n ) and (r, n ) i,a (r , n ), then f is dened on (r , n ).

The function f is partial to take into consideration the case of synchronous systems. Also,it is straightforward to introduce frames satisfying perfect recall, no learning, synchronicity,or having a unique initial state, by following the same denitions given for mf-models. Next,we provide the denition of objects , which correspond to the runs of Gabbay et al. (2003).We choose this terminology to avoid confusion with the runs in QIS.

Denition 18 (Object) . Given an individual a D , an object in the frame F is a map aassociating a type a (r, n ) T r,n to every (r, n ) Dom (f) with f(r, n ) = Cr,n = T r,n , T conr,nsuch that

1. a (r, n ) a (r, n + 1)

2. if (r, n ) i,a (r , n ) then a (r, n ) i a (r , n )

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3. U a (r, n ) iff there is n n such that a (r, n ) and for all n , n n < nimplies a (r, n )

4. if a (r, n ) i t are -types, then for some (r , n ), (r, n ) i,a (r , n ) and a (r , n ) = t

5. if C a (r, n ) then there exists a point (r , n ) reachable from (r, n ) such that a (r , n )

An object + satises (1), (2), (3), (5) above and (4 ) below instead of (4).

(4 ) if a (r, n ) is a -type, t is a i -type and a (r, n ) i t , then for some ( r , n ) i,a (r, n ),a (r , n ) = t .

Intuitively, an object identies the same individual, here represented by types, acrossdifferent state candidates. Now we have all the elements to give the denition of quasimodel.

Denition 19 (Quasimodel) . A quasimodel for is a tuple Q = R , O, { i, }i Ag, O , fsuch that R , O, { i, }i Ag, O , f is a frame, and

1. t for some t T r,n and T r,n Cr,n

2. Cr,n Cr,n +1

3. if (r, n ) i, (r , n ) then (r, n ) i (r , n )

4. for every t T r,n there exists an object O such that (r, n ) = t

5. for every c con, the function c such that c(r, n ) = t c T conr,n is an object in O.

A quasimodel + is dened as a quasimodel but where clauses (4) and (5) refer to objects +

rather than objects. We can dene quasimodels (resp. quasimodel + ) satisfying perfect recall,no learning, synchronicity, or having a unique initial state, by assuming the correspondingcondition on the underlying frame. The difference between objects (resp. quasimodel) andobjects + (resp. quasimodel + ) is purely technical. In particular, the latter are needed forsystems satisfying perfect recall and no learning as it will become apparent in Section 5.In the following lemma we list some properties of quasimodels that will be useful in whatfollows.

Lemma 4. In every quasimodel Q , for every object O ,

(i) K i (r, n ) iff for all (r , n ), (r , n ) i, (r, n ) implies (r , n ).

(ii) C (r, n ) iff for all points (r , n ) reachable from (r, n ) we have that (r , n ).

Proof.

(i) The implication from left to right follows from the fact that ( r , n ) i, (r, n ) implies(r, n ) i (r , n ). For the implication from right to left, if K i / (r, n ) then byLemma 3(ii) there is a -type t such that (r, n ) i t and t . By Denition 18 forsome (r , n ), (r, n ) i, (r , n ) and (r , n ) = t .

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(ii) The implication from left to right is proved by induction on the length of the pathfrom (r, n ) to (r , n ). Both the base case and the inductive step follow by axiom C1.The implication from right to left follows from Denition 18.

We now state the main result of this section, that is, satisfability in quasimodels impliessatisfability in mf-models. In what follows a quasimodel Q validates a formula if belongsto every type in every state-candidate in Q .

Theorem 3. If there is a quasimodel (resp. quasimodel + ) Q for a monodic formula , then is satisable in a mf-model M mf . Moreover, if Q validates the formulas K and BF , then so does M mf . Finally, if Q satises perfect recall, or no learning, or synchronicity, or has a unique initial state, then so does M mf .

Proof. This proof is inspired by those of Lemmas 11.72 and 12.9 in the work of Gabbayet al. (2003), but here we consider monodic friendly Kripke models rather than standardKripke models. First, for every monodic formula of the form K i , C , or 1 U 2we introduce a k-ary predicate constant P k for k equal to 0 or 1, depending whether thereare 0 or 1 free variables in . The formula P k (x) is called the surrogate of . Given amonodic formula we denote by the formula obtained from by substituting all its modalsubformulas which are not within the scope of another modal operator with their surrogates.Since every state candidate C in the quasimodel Q is consistent and all the system S of rst-order temporal-epistemic logic considered in Section 3 are based on classical rst-orderlogic, the formula C is consistent with respect to rst-order (non-modal) logic. By G odelscompleteness theorem there is a rst-order structure I = I, D , where D is a non-empty setof individuals and I is a rst-order interpretation on D, that satises C, i.e., I |= C forsome assignment of the variables to the elements in D. We intend to build an mf-modelby joining all these rst-order structures. However, it is possible that these structures havedifferent domains with different cardinalities. To solve this problem, we consider a cardinalnumber 0 greater than the cardinality of the set O of all objects in Q and dene

D = { , | O , < }

Then, for ( r, n ) Q , for any -type t T r,n we have that

|{ , D | (r, n ) = t }| =

By the method described in Claim 11.24 by Gabbay et al. (2003), we can expand eachrst-order structure to obtain a structure I r,n = I r,n , D with domain D such that I r,nsatises Cr,n and

|{a D | (x) = a and I r,n |= t [x]}| =

So, we can assume without loss of generality that all rst-order structures I r,n share thesame domain D, and for every t T r,n , , D , we have

(r, n ) = t iff I r,n |= t [x]

for (x) = , . Equivalently, for t T r,n , (x) = , D ,

(r, n ) = { cl | I r,n |= [x]} (7)

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Moreover, I r,n (c) = c , 0 for every c con.We dene the mf-model M mf as the tuple W, R , { i,a }i Ag,a D , D, I such that W is

the set of points ( r, n ) for r in R Q and n N ; R is the set of runs from N to W such thatr (n) = ( r, n ); D is dened as above; for i Ag and , D , i, , is dened as i, ;

and the interpretation I is obtained by joining the various rst-order interpretations I r,n ,i.e., I (P, r (n)) = I r,n (P ) for every predicate constant P . We can now prove the followingresult for M mf .

Lemma 5. If the mf-model M mf is obtained from a quasimodel Q as described above, then for every subx ,

I r,n |= iff (M mf , r,n ) |=

Moreover, if Q is a quasimodel + , f(r, n ) is a -state candidate and ad(K ) ad() then

I r,n |= iff (M mf , r,n ) |=

Proof. The proof is similar to Lemma 12.10 in the work of Gabbay et al. (2003). Webegin with the rst part. The base case of induction follows by denition of the interpre-tation I in the mf-model. The step for propositional connectives and quantiers follows bythe induction hypothesis and equations 1 2 = 1 2, 1 = 1, x1 = x1.

Now let = [x] and assume that (x) = , , then we have:

I r,n |= [x] iff [x] (r, n ) (8)iff [x] (r, n + 1) (9)iff I r,n +1 |= [x] (10)iff (M mf , r,n + 1) |= [x] (11)iff (M mf , r,n ) |= [x]

Steps (8) and (10) follow by Equation (7). Step (9) is motivated by Lemma 3(i), andstep (11) follows by the induction hypothesis.

Let = ( U )[x] and (x) = , , then we have:

I r,n |= ( U )[x] iff ( U )[x] (r, n ) (12)iff there is n n such that [x] (r, n )

and [x] (r, n ) for all n n < n (13)iff there is n n such that I r,n |= [x]

and I r,n |= [x] for all n n < n (14)

iff there is n n such that ( M mf , r,n ) |= [x]and ( M mf , r,n ) |= [x] for all n n < n (15)

iff (M mf , r,n ) |= U [x]

Steps (12) and (14) follow by Equation (7). Step (13) is motivated by Def. 18, and step(15) follows by the induction hypothesis.

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5. Dealing with each System

In this section we consider the completeness proof for each system in Theorem 2. In par-ticular, we show that if a monodic formula is consistent with respect to a system S , thenwe can build a quasimodel (or a quasimodel + in specic cases) for based on a frame forS . In the following sections the symbol represents provability in the appropriate systemS . We start with some lemmas that are useful for the construction of the quasimodel forany system.

Lemma 6. (i) For any consistent monodic formula there is a consistent -state can-didate C = T, T con such that t for some t T .

(ii) Let P = C, t be a consistent -point for such that C = T, T con , and let c con.Then,

(a) if C C then there exists a -point P = C , t such that P P .

(b) if t c

T con

and C C then there exists a -point P = C , t such that Pc

P .(c) if 1 U 2 t then there is a sequence of -points P j = C j , t j for j k that realises 1 U 2, i.e., P = P 0 . . . P k , 2 t k and 1 t j for j < k .

(d) if 1 U 2 t c then there is a sequence of -points P j = C j , t j for j k that c-realises 1 U 2, i.e., the sequence realises 1 U 2 and P 0 c . . . c P k .

(e) if K i t then there is a -point P = C , t such that P i P and t .(f) if K i t c then there is a -point P = C , t such that P ci P and t .(g) if C t then there is a sequence of -points P j = C j , t j for j k such that

P = P 0 i0 . . . ik 1 P k and t k .(h) if C t c then there is a sequence of -points P j = C j , t j for j k such that

P = P 0 ci0 . . . cik 1 P k and t k .

Proof. The proof is similar to the one for Claims 11.75, 11.76 and 12.13 in the workof Gabbay et al. (2003), but here we consider -state candidates and -points. Let bethe disjunction of all formulas P for all -points P for . Consider the formula , whichis obtained by substituting all subformulas of of the form K i , C , or 1 U 2 thatare not within the scope of another modal operator with their surrogates. We can checkthat is true on all (non-modal) rst-order structures. Since both QKT m and QKTC mextend rst-order logic, by the semantical completeness of rst-order logic we have that

(24)

(i) Notice that, by the previous remark, also for = {P |P is a -point for } P .Moreover, is consistent and by (24) also is consistent. Therefore, there is adisjunct P of such that P is consistent. So, t for P = C, t .

(a) By (24) and Nec we have . So, P is consistent and there must be a-point P such that P P is also consistent.

(b) The proof is similar to (a).

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(c) The proof is by contradiction. Let U be the set of all -points P such that there exist -points P j = C j , t j for j < k and P = P 0 . . . P k = P . Let = {P |P U } P .We can show that

2 (25)

otherwise, we would have a sequence realising 1 U 2. Moreover, by the denition of U ,

(26)

From (25) we obtainG G2

and together with (25) and (26) we derive

(2 G2) (27)

Now consider any P 1 U such that P P 1. By (27) we have

P 1 (2 G2) P 1 G2( P P 1 ) G2 (28)

On the other hand, since 1 U 2 t we have

( P P 1 ) F 2 (29)

but (28) and (29) contradict the fact that P P 1.

(d) The proof is similar to (c).

(e) First we remark that P K i ( ) is consistent. Thus, there exists a -pointP = C , t such that P K i ( P ) is consistent. Hence, P i P and t .

(f) The proof is similar to (e).

(g) The proof is by contradiction. Let V be the minimal set of -points D such that (i)P V ; (ii) if D V and D i D for some i Ag, then D V . Let = {D |D V } D .We can show

(30)If (30) did not hold, we would have a sequence as specied in the lemma. Moreover,by the denition of V ,

K i (31)for all i Ag. From (30) and (31) we obtain

( E )

and by axiom C2, C (32)

but by denition of P , P C

which contradicts (32).

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(h) The proof is similar to (g).

By the following result it is always possible to extend the -suitability relationbetween types to -suitability between points.

Lemma 7. Suppose that t and t are -types such that t t , then there are -points P = C, t and P = C , t such that P P . In particular, for any c con, there are -points P = C, t and P = C , t such that P c P .

Proof. By Lemma 6 we have that and for = {P |P is a -point for } P .Since t t , then t ( t ) is consistent. Thus, there must be -points P andP such that P t ( P t ) is consistent. Then, it is the case that P = C, t andP = C , t for some -state candidates C and C . As a result, P P . The second part of the lemma is proved similarly to the rst by observing that if t t then t [x/c ] t [x/c ]is consistent. Hence, also t [x/c ] ( t [x/c ]) is consistent. Thus, there must be-points P and P such that P t [x/c ] ( P t [x/c ]) is consistent. So, t c T con andt c T con , that is, P c P .

According to Lemma 7 we can always extend a possibly innite sequence of -types t 0t 1 . . . to a possibly innite sequence of -points P 0 P 1 . . . such that P k = Ck , t k .

Denition 20. Let a -sequence be a possibly innite sequence C0 C1 . . . of -state candidates. A -sequence is acceptable if for all k 0,

(i) if 1 U 2 t k , for t k Ck , then 1 U 2 is realised in a sequence of -points P j =C j , t j for k j n;

(ii) if 1 U 2 t ck , for t ck Ck , then 1 U 2 is c-realised in a sequence of -points P j =

C j , t j for k j n.

The following lemma entails the completeness result.

Lemma 8. Every nite -sequence of -state candidates can be extended to an innite acceptable -sequence.

Proof. Assume that C0 . . . Cn is a nite -sequence and 1 U 2 t k Ck forsome k n. Either 1 U 2 is realised in C0 . . . Cn , or by Lemma 6(ii)(c) we can extend to a -sequence that realises 1 U 2. This procedure is repeated for all formulas of the form 1 U 2 appearing at some point in the -sequence. Thus, we obtain a (possiblyinnite) -sequence C0 C1 . . . such that property (i) in Denition 20 is satised. Toalso satisfy property (ii) we reason similarly by using Lemma 6(ii)(d) instead.

Now let X be a new object, a sequence X , . . . ,X , Cn , Cn +1 , . . . is acceptable from n if it starts with n copies of X and Cn , Cn +1 , . . . is an acceptable -sequence. We can nowconsider the completeness proof for each single class of QIS.

5.1 The Classes QIS m , QIS syncm , QIS uism and QIS sync,uismWe start with the completeness proof for the systems QKT m and QKTC m , where there isno interaction between temporal and epistemic operators.

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The completeness of QKT m and QKTC m with respect to the classes QIS m and QIS syncmof quantied interpreted systems directly follows from Lemma 10 together with Theorem 3.Thus, we obtain the following item in Theorem 2.

Theorem 4 (Completeness) . The system QKT m (resp. QKTC m ) is complete w.r.t. the classes QIS m and QIS syncm of QIS.

To prove completeness for QIS uism and QIS sync,uism we use the next result, which is anextension from the propositional case (Halpern et al., 2004).

Remark 2. Suppose X is a subset of { pr, sync }. If L 1m (resp. LC 1m ) is satisable in QIS X m then it is also satisable in QIS X,uism .

Thus, the system QKT m (resp. QKTC m ) is also complete w.r.t. the classes QIS uism andQIS sync,uism of QIS.

5.2 The Classes QIS prm and QIS pr,uismWe now begin to investigate systems where interactions between time and knowledge arepresent. The completeness proof for QKT 1m with respect to QIS prm and QIS pr,uism relies onthe following lemma.

Lemma 11. For all -points P 1 = C1, t 1 , P 2 = C2, t 2 and i -type t 2, if P 1 P 2 and t 2 i t 2 then there is a i -point P 2 = C2, t 2 and a -sequence S 1 . . . S n = P 2of i -points such that S k = D k , sk , s1 i t 1 and sk i t 2 for 1 < k n. Further, if P 1 c P 2 then sck T

conD k for k n.

Proof. We extend the proof of Halpern et al. (2004, Lemma 5.5) to deal with statecandidates and monodic friendly Kripke frames. By the cited result we can prove that if

t 1 t 2 and t 2 i t 2 then there is a sequence of i -types s1 . . . sn = t 2 such that s1 i t 1 and sk i t 2 for 1 < k n. Now by Lemma 7 we can extend this sequence of i -types to a sequence of i -points S 1 . . . S n such that S k = D k , sk and thelemmas statement is satised. In particular, if P 1 c P 2 also by Lemma 7 we can assumewithout loss of generality that sck T

conD k for k n.

For any consistent L 1m we dene a quasimodel + for to establish the completenessof QKT 1m with respect to QIS prm . Let R be the set of all acceptable -sequences, and denef such that f(r, k ) = Ck if r is the -sequence C0, C1, . . . . Finally, let O be the set of allfunctions associating every ( r, n ) Dom (f) to a type (r, n ) T r,n such that conditions(A) and (B) above are satised and

(C) if (r, n ) is a -type, t is a i -type and (r, n ) i t , then for some ( r , n ), (r , n ) = t .

(E) if (r, n ) i (r , n ) and n > 0 then either (a) (r, n 1) i (r , n ) or (b) there isk < n such that (r, n 1) i (r , k) and for all k , k < k n implies (r, n ) i(r , k ).

Finally, for i Ag, O , we dene (r, n ) i, (r , n ) iff (r, n ) i (r , n ).The following lemma shows that the set O is non-empty. In particular, conditions (C)

and (E) are satised by the functions in O.

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Lemma 15. For -state candidates C1, C2 and i -state candidate C2, there is a i -state candidate C1 such that

if C1 C2 and C2 i C2 then C1 i C1 and C1 C2.

for c con, for P 1 = C1, t 1 , P 2 = C2, t 2 and P 2 = C2, t 2 , if P 1 c P 2 and P 2 ci P 2 then for P 1 = C1, t 1 , P 1 ci P 1 and P 1 c P 2.

Proof. if C1 C2 and C2 i C2 then there exist t 1 C1, t 2 C2 and t 2 C2 suchthat t 1 t 2 and t 2 i t 2. Moreover, without loss of generality we can assume that forsome c con, t c1 T con1 , t c2 T con2 and t c2 T con2 . Following the proof by Halpern etal. (2004, Lemma 5.8) we can nd a i -type t 1 such that t 1 i t 1 and t 1 t 2. Dene T 1as the set of all such t 1 and T con1 as the set of t c1 . We can show that C1 = T 1, T con1 is aconsistent i -state candidate such that C1 i C1, C1 C2, and for c con, P 1 ci P 1and P 1 c P 2.

For any consistent L 1m we dene a quasimodel + for to establish the complete-ness of QKT 2m with respect to QIS pr,syncm . Let R be the set of all -sequences accept-able from n, for some n N , and dene f such that f(r, k ) = Ck if r is the -sequenceX , . . . ,X , Cn , Cn +1 , . . . acceptable from n and k n, and undened otherwise. Finally, letO be the set of all functions associating every ( r, n ) Dom (f) to a type (r, n ) T r,nsuch that conditions (A) and (B) in Section 5.1 are satised and

(C) if (r, n ) is a -type, t is a i -type and (r, n ) i t , then for some ( r , n ), (r , n ) = t .

(F) if (r, n ) i (r , n ) and n > 0 then (r, n 1) i (r , n 1).

Finally, for i Ag, O , we dene (r, n ) i, (r , n ) iff (r, n ) i (r , n ) and n = n .The following remark shows that the set O is non-empty. In particular, conditions (C)

and (F) are satised by the functions in O.

Lemma 16. The set O of functions that satises condition (A), (B), (C) and (F) is non-empty.

Proof. Conditions (A) and (B) follow from Lemma 6(ii)(a) and the fact that r is anacceptable -sequence. As regards (C) and (F), assume that (r, n ) f(r, n ) is a -type,t is a i -type and (r, n ) i t . For each s f(r, n ) different from (r, n ) consider the setU = { | K i s}. We can check that U is consistent and it can be extended to a i -type ssuch that s i s . Now dene T as the collection of all these s . Further, for each sc T con ,we set s c T con . Let C = T , T con . Clearly, C i C and C, s ci C , s . By Lemma 15

we can construct a -sequence C0 . . . Cn such that Cn = C and f(r, k ) i Ck fork n. By Lemma 8 we can extend this -sequence to an innite acceptable -sequencer . In particular, the function can be extended so that for k n, (r, k ) i (r , k) and(r , n ) = t . Thus, both (C) and (F) are satised.

We can now show the following lemma.

Lemma 17. The tuple R , O, { i, }i Ag, O , f is a frame that satises perfect recall and synchronicity.

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Proof. By Lemmas 6(i), 8 and 16 the sets R and O are non-empty. Also, f satises theconditions in Denition 17. Finally, each i, is an equivalence relation by denition, andit satises perfect recall and synchronicity by denition of the functions in O.

Now we prove the main result.

Lemma 18. The tuple R , O, { i, }i Ag, O , f is a quasimodel + for with perfect recall and synchronicity, and it validates the formulas K and BF .

Proof. By the previous lemma R , O , { i, }i Ag, O , f is a frame satisfying perfectrecall and synchronicity; so we are left to prove that the functions in O are objects + .Conditions (1)-(4) on objects + are sased by the remarks (A)-(F) and the denitionof i, . Furthermore, conditions (1), (2) and (3) on quasimodels + are satised by thedenitions of R , f and i, . As regards condition (4), we can make use of Lemma 15 toshow that it holds. Additionally, (5) holds by Lemma 6(ii)(b), (d), (f) and (h) and Lemma15. Finally, Q validates both the formulas K and BF , as all t C, for all C Q , areconsistent with QKT m .

This completes the proof for QKT 2m . Thus, we obtain the following item in Theorem 2.Theorem 6 (Completeness) . The system QKT 2m is complete w.r.t. the class QIS pr,syncm of QIS.

The completeness of QKT 2m with respect to QIS pr,sync,uism follows again by Remark 2.

5.4 The Class QIS nlmFirst, we give the following denitions, which will be used in the completeness proof.

Denition 21. If t is a -type, then t ,i is the conjunction of all -types t such that t i t .Similarly, if P is a -point, then P ,i is the set of -points P such that P i P .

Denition 22. Two sequences of types and are i -concordant if there is some n N(or n may be ) and non-empty consecutive intervals 1, . . . , n of and 1, . . . , n of such that for all s j and s j we have s i s for j n.

Two sequences and of state candidates are i -concordant if for all t C, for either C or C , there are two sequences and of types in and respectively that are i -concordant.

To prove the completeness of QKT 3m with respect to QIS nlm we need the following lemma,which is dual to Lemma 11.

Lemma 19. For all -points P 1 = C1, t 1 , P 2 = C2, t 2 and i -type t 1, if P 1 P 2 and

t 1 i t 1 then there exists a i -point P 1 = C1, t 1 and a -sequence P 1 = S 1 . . . S nof i -points such that S k = D k , sk , sk i t 1 for k < n , and t 2 i sn . Further, if P 1 c P 2then sck T

conD k for k n.

Proof. By adapting the result of Halpern et al. (2004, Lemma 5.11) to types we canprove that if t 1 t 2 and t 1 i t 1 then there is a sequence of i -types t 1 = s0 . . . snsuch that sk i t 1 for k < n and sn i t 2. Now by Lemma 7 we can extend this sequenceof i -types to a sequence of i -points S 1 . . . S n such that S k = D k , sk . So, the

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2. Let P = C, t and P = C , t be in . If t i t then f (t ) and f (t ) are i -concordant;

3. for at least one P the sequence f (P ) has a length of at least 2.

Finally, for any constant c con, we say that cf whenever f and f (P ) =P 0 c . . . c P k .

Notice that given a k-tree of state candidates with root C and t C, we can obtain ak-tree of points such that P = C , t iff C . Also, if , are k-tree of statecandidates and f , then we also have f where and are k-trees of pointsbased on and respectively.

We now show how to obtain acceptable sequences of state candidates from sequencesof trees. Given two sequences of -state candidates = C0, . . . , Ck and = C0, . . . , where is nite, the fusion is dened as C0, . . . , Ck 1, C0, . . . only if Ck = C0. Further,given an innite sequence = 0 f 0 1 f 1 . . . of k-trees, we say that a sequence of -state candidates is compatible with if there exists some h N and -state candidatesCh , Ch +1 , . . . , with C j j for j h, such that = f h (Ch ) f h +1 (Ch +1 ) . . .. The sequence is acceptable if every -sequence compatible with is innite and acceptable.

The basic idea of the completeness proof is to dene the quasimodel + starting from anacceptable sequence . Next we introduce some denitions and lemmas that are essentialfor the completeness proof.

Given a k-tree and a -point P we inductively dene the formula tree ,P thatdescribes the k-tree from the viewpoint of P .

Denition 25. If P is a -point, then tree ,P ::= P . If P is a i -point with = ithen

tree ,P = P{ point P | t i t}

K i tree ,P

If and are k-trees, P and P , then we write ( , P ) + ( , P ) if there is asequence of k-trees 0, . . . , l and functions f 0, . . . , f l 1 such that (a) = 0 f 0 . . . f l 1 l = ; (b) f j (P ) = P for j l 2 and f l 1(P ) = ( P , P ). Similarly, ( , P ) c+ ( , P )if (, P ) + ( , P ) and (a) = 0 cf 0 . . .

cf l 1 l = .

We prove the following lemma, which extends a result by Halpern et al. (2004, Lemma 5.12)to points.

Lemma 20. Suppose is a k-tree of points and P = C, t is a -point with | | = k,

(a) If t is a -type and tree ,P (t ) is consistent, then there is a k-tree and a -point P = C , t such that ( , P ) + ( , P ) and tree ,P is consistent.Further, if t c T con then ( , P ) c+ ( , P ).

(b) tree ,P {( ,P ) |( ,P ) + ( ,P )} tree ,P

(c) if tree ,P U is consistent, then there is a sequence 0, . . . , l of k-trees and points P 0, . . . , P l such that (i) P j j for j l; (ii) ( 0, P 0) = ( , P ); (iii)( j , P j ) + ( j +1 , P j +1 ) for j < l ; (iv) tree j ,P j is consistent for j < l ; (v)tree l ,P l is consistent. Further, if t c T con then (iii) ( j , P j ) c+ ( j +1 , P j +1 ) for j < l .

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exists a -point P = C , t such that t i t u and

tree ,P ( u K i ({ |( ,P ) + ( ,P )}

tree ,P )) (34)

is consistent. By part (a) there exists a k tree and P such that ( , P ) + ( , P )and tree ,P u K i ( { |( ,P ) + ( ,P )} tree ,P ) is consistent. But this means thatP = u. Thus we have a contradiction, since tree ,u K i tree ,P is inconsistent.

The general case for (a) follows from (b). Part (c) also follows from (b).The following lemma is the correspondent of Lemma 8 for k-trees.

Lemma 21. If L 1m is consistent with QKT 3m , then there exists an acceptable sequence of ad()-trees of state candidates such that belongs to the root of the rst tree.

Proof. As in Lemma 8 the key part of this proof consists of showing that, given a nitesequence 0 f 0 . . . f l 1 l of d-trees of points and a -point P = C, t l such that

U t (resp. t ), by Lemmas 19 and 20 we can extend the sequence of trees tosatisfy acceptability. Specically, suppose that U t . Let include P and the -pointsP = C , t l with | | k = | |. Note that is a k-tree. Further, by Lemma 20 wecan nd a sequence 0, . . . , n of k-trees and points P 0, . . . , P n such that (i) P j j for j n; (ii) (0, P 0) = ( , P ); (iii) ( j , P j ) + ( j +1 , P j +1 ) for j < l ; (iv) tree j ,P j is consistent for j < l ; and (v) tree n ,P l is consistent. By using Lemma 19 we canextend this to a sequence of ad()-trees starting with l that satises U as in the proof of Lemma 20(a). For t the argument is similar. Since is consistent, there must besome tree with root C such that t for some t C; we can then extend as above tocomplete the proof.

For any consistent L 1m we dene a quasimodel + for to establish the completenessof QKT 3m with respect to QIS nlm . Let R consist of all acceptable -sequences compatiblewith the ad()-tree , while the function f is given by f(r, k ) = Ck if r is the acceptable

-sequence C0, C1, . . . . Further, let O be the set of functions associating every ( r, n )Dom (f) to a type (r, n ) T r,n such that conditions (A), (B) and (C) given previously aresatised and the following holds:

(G) if (r, n ) i (r , n ) then either (r, n + 1) i (r , n ) or there exists k > n such that(r, n + 1) i (r , k) and for all k , k > k n implies (r, n ) i (r , k ).

Finally, for i Ag, O , (r, n ) i, (r , n ) iff (r, n ) i (r , n ).As in the previous cases we have the following.

Lemma 22. The set O of functions that satises conditions (A), (B), (C) and (G) is

non-empty.Proof. Conditions (A) and (B) are guaranteed by Lemma 6(ii) and by the fact that r

is an acceptable -sequence respectively. As regards (C) and (G), assume that (r, n ) isa -type, t is a i -type and (r, n ) i t . By using the proofs of Lemmas 20 and 19 we cannd an acceptable -sequence r compatible with the d-tree such that t f(r , 0) and(G) is satised.

We can now show the following lemma.

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For any c con, the relation c syncf is dened similarly. We dene a sync -acceptablesequence of trees as an acceptable sequence where the relation is substituted by therelation sync , that is, the sequence is acceptable if every sync -sequence compatible with is innite and acceptable. Similarly, given the relations + and c+ for c con, the

denitions of sync, + and c sync, + are straightforward. We now state the following result,which is a simplied version of Lemma 20. The proof is analogous to that of Lemma 20, inwhich Lemma 25 is used instead of Lemma 19.

Lemma 26. Let be a k-tree of points and P is a -point with | | = k,

(a) If t is a -type and tree ,P (t ) is consistent, then there exists a k-tree and a -point P = C , t such that ( , P ) sync, + ( , P ) and tree ,P is consistent. Further, if t c T con then ( , P ) c sync, + ( , P ).

(b) tree ,P {( ,P ) |( ,P ) sync, + ( ,P )} tree ,P

(c) if tree ,P

U is consistent, then there exists a sequence 0, . . . ,

lof k-trees and

points P 0, . . . , P l such that (i) P j j for j l; (ii) ( 0, P 0) = ( , P ); (iii)( j , P j ) sync, + ( j +1 , P j +1 ) for j < l ; (iv) tree j ,P j is consistent for j < l ;and (v) tree l ,P l is consistent. Further, if t c T con then (iii) ( j , P j ) c sync, +( j +1 , P j +1 ) for j < l .

Further, we make use of Lemma 26 to adapt Lemma 21 and obtain the following result.

Lemma 27. If L 1m is consistent with QKT 4m , then there exists a sync -acceptable sequence of ad()-trees of state candidates such that belongs to the root of the rst tree.

For any consistent L 1m we dene a quasimodel + for to establish the com-pleteness of QKT 4m with respect to QIS nl,syncm . Let X be a new object, a sequenceX , . . . ,X , Cn , Cn +1 , . . . is sync -acceptable from n if it starts with n copies of X and Cn , Cn +1 , . . .is a sync -acceptable -sequence compatible with the ad()-tree . Let R consist of all -sequences sync -acceptable from n, for some n N . The function f is dened as f(r, k ) = Ck if r is the -sequence X , . . . ,X , Cn , Cn +1 , . . . sync -acceptable from n and k n; f(r, k ) is un-dened otherwise. Further, Let O be the set of functions associating every ( r, n ) Dom (f)to a type (r, n ) T r,n such that conditions (A), (B) and (C) are satised and the followingholds:

(H) if (r, n ) i (r , n ) then (r, n + 1) i (r , n + 1).

Finally, for i Ag, O , (r, n ) i, (r , n ) iff (r, n ) i (r , n ) and n = n .Similarly to Lemma 22, we can show the following.

Lemma 28. The set O of functions that satises conditions (A), (B), (C) and (H) above is non-empty.

Moreover, the following result follows from Lemmas 6(i), 27 and 28.

Lemma 29. The tuple R , O , { i, }i Ag, O , f is a frame that satises no learning and synchronicity.

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Finally, by adapting the proof for Lemma 24 we can state the following result.

Lemma 30. The tuple R , O, { i, }i Ag, O , f is a quasimodel + for with no learning and synchronicity, and it validates the formulas K and BF .

This completes the proof for QKT 4m . Thus, we obtain the following item in Theorem 2.Theorem 8 (Completeness) . The system QKT 4m is complete w.r.t. the class QIS nl,syncm of QIS.

5.6 The Classes QIS nl,prm and QIS nl,pr,uis1

To obtain the completeness proof for QIS nl,prm we combine the results shown for QIS prm andQIS nlm .

If L 1m is consistent with QKT2,3m then by Lemma 21 there exists an acceptable

sequence of ad()-trees such that belongs to the root of the rst tree. Let R be theset of all acceptable -sequences that have a suffix that is compatible with , while the

function f is dened as in Section 5.2. Further, O is the set of all functions associatingevery ( r, n ) Dom (f) to a type (r, n ) T r,n that satises the conditions (A), (B), (C),(E) and (G). Finally, for i Ag, O , (r, n ) i, (r , n ) iff (r, n ) i (r , n ).

Lemma 31. The set O of functions that satises conditions (A), (B), (C), (E) and (G)is non-empty.

Proof. We can show that all conditions but (C) are satised similarly to the cases of QIS prm and QIS nlm . As to (C), suppose that (r, n ) is a -type in f(r, n ) and t is i -type.Also, is the sequence of ad()-trees 0 f 0 1 f 1 . . . of state candidates. The run ris derived by denition from a -sequence C0, C1, . . . that has a suffix CN , CN +1 , . . . that iscompatible with , and f(r, n ) = Cn . We consider two cases.

If n N , then there exists some kN

such that Cn k . By Lemma 11 there existsa -sequence S 0 . . . S h of i -state candidates such that t S h and S 0, . . . , S h is i -concordant with C0, . . . , Cn . Further, we can assume that S h k and let S h , S h +1 , . . .be the sequence compatible with . Now consider the -sequence S 0 S 1 . . ..By construction the run r derived from this sequence is in R and we can assume that(r , h) = t .

If n < N , then by Lemma 11 there exists a -sequence S 0 . . . S h of i -statecandidates such that t S h and S 0, . . . , S h is i -concordant with C0, . . . , Cn . By Lemma 19we can extend this sequence to a -sequence S 0 . . . S k that is i -concordantwith C0, . . . , CN . Since CN M for some M N , we can assume that S k M aswell. Let S h , S h +1 , . . . be the sequence compatible with , and consider the -sequenceS 0 S 1 . . .. As in the previous case, the run r derived from this sequence is in R byconstruction and we can assume that (r , h) = t .

By Lemmas 6(i), 21 and 31 we obtain the next result.

Lemma 32. The tuple R , O, { i, }i Ag, O , f is a frame that satises perfect recall and no learning.

Finally, we state the following lemma, whose proof follows the lines of the correspondingproofs for QIS prm and QIS nlm and Lemma 31.

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Lemma 33. The tuple R , O, { i, }i Ag, O , f is a quasimodel + for that satises perfect recall and no learning, and validates the formulas K and BF .

This establishes the completeness of QKT 2,3. Thus, we obtain the following item in

Theorem 2.

Theorem 9 (Completeness) . The system QKT 2,3m is complete w.r.t. the class QIS nl,prm of QIS.

The completeness of QKT 2,31 with respect to QIS nl,pr,uis1 follows from the following

remark, whose proof is analogous to the propositional case.

Remark 3. A formula L 11 is satisable in QIS nl,pr1 (resp. QIS

nl,pr,sync1 ) iff it is

satisable in QIS nl,pr,uis1 (resp. QIS nl,pr,sync,uis1 ).

5.7 The Class QIS nl,pr,syncm

To prove the completeness of QKT 1,4m with respect to QIS nl,pr,syncm we combine the resultsobtained for QIS nl,prm in the previous section with those for QIS nl,syncm and QIS pr,syncm .Specically, if L 1m is consistent with QKT

1,4m by Lemma 27 we can construct a sync -

acceptable sequence of ad()-trees such that belongs to the root of the rst tree. Let Rbe the set of all sync -acceptable -sequences with suffixes that are compatible with ; andthe function f is dened as in Section 5.2. Further, O is the set of all functions associatingevery ( r, n ) Dom (f) to a type (r, n ) T r,n that satises the conditions (A), (B), (C),(F) and (H). Finally, for i Ag, O , (r, n ) i, (r , n ) iff (r, n ) i (r , n ) and n = n .

By adapting the proof of Lemma 31 by means of Lemmas 15 and 25 we can show thefollowing result.

Lemma 34. The set O of functions that satises conditions (A), (B), (C), (F) and (H)is non-empty.

By Lemmas 6(i), 27 and 34 we obtain the following result.

Lemma 35. The tuple R , O, { i, }i Ag, O , f is a frame that satises perfect recall, nolearning and synchronicity.

Finally, we state the following lemma whose proof follows the lines of the correspondingproofs for QIS pr,syncm , QIS nl,syncm and Lemma 34.

Lemma 36. The tuple R , O, { i, }i Ag, O , f is a quasimodel + for that satises perfect recall, no learning and synchronicity, and validates the formulas K and BF .

This completes the proof for QKT 1,4m . Thus, we obtain the following item in Theorem 2.

Theorem 10 (Completeness) . The system QKT 1,4m is complete w.r.t. the class QIS nl,pr,syncmof QIS.

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Interactions Between Knowledge and Time in First-Order Logic for Multi-Agent Systems - Completeness Results