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<ul><li><p>A Complete First-Order Logic of Knowledge and Time</p><p>Francesco BelardinelliScuola Normale Superiore, Pisa</p><p>F.Belardinelli@sns.it</p><p>Alessio LomuscioDepartment of ComputingImperial College London</p><p>A.Lomuscio@imperial.ac.uk</p><p>Keywords: Epistemic Logics: first-order temporalepistemic logic, message passing systems, completeness.</p><p>Abstract</p><p>We introduce and investigate quantified interpreted systems,a semantics to reason about knowledge and time in a first-order setting. We provide an axiomatisation, which we showto be sound and complete. We utilise the formalism to studymessage passing systems (Lamport 1978; Fagin et al 1995) ina first-order setting, and compare the results obtained to thoseavailable for the propositional case.</p><p>IntroductionThe area of modal logic (Blackburn, van Benthem, andWolter 2007; Chagrov and Zakharyaschev 1997) has re-ceived considerable attention in artificial intelligence overthe years. Research has pursued both fundamental theoret-ical investigations (completeness, decidability, complexity,etc), as well as the use of modal formalisms in specifica-tion and automatic system verification, as in model checking(Clarke, Grumberg, and Peled 1999).</p><p>Among the most well-known formalisms are proposi-tional modal logics for reasoning about knowledge, orpropositional epistemic logics (Fagin et al 1995; Meyer andHoek 1995). The typical epistemic language extends propo-sitional logic by adding n modalities Ki representing theknowledge of agent i in a group A = {1, . . . , n} of agents.For expressiveness purposes, epistemic logic has been ex-tended in several ways. In one direction, further modali-ties have been added to the formalism (distributed knowl-edge, common knowledge, belief, etc.) for representing theknowledge shared in a group of agents. In another one, theepistemic language has been enriched with temporal oper-ators under the assumption of a given model of time (e.g.,linear or branching, discrete or continuous, etc.). In allthese lines of work there is a tension between extending theexpressiveness of the language reflecting the system to bemodeled and retaining some useful theoretical properties ofthe formalism, such as decidability.</p><p>This tension is still present in the exercise conducted here,where we aim at extending a combination of epistemic and</p><p>Copyright c 2008, Association for the Advancement of ArtificialIntelligence (www.aaai.org). All rights reserved.</p><p>temporal logic to predicate level. We apply this result inthe modeling of a class of computational structures normallyreferred to as message passing systems (Lamport 1978). Wealso show that known metatheoretical properties of messagepassing systems (Fagin et al 1995) become validities in thepredicate logic here considered.</p><p>Our starting point is a number of results by Halpern, vander Meyden, and others regarding the combination of timeand knowledge at propositional level (Fagin, Halpern, andVardi 1992; Meyden 1994) together with studies by, amongothers, Hodkinson, Reynolds, Wolter, Zakharyaschev forfirst-order temporal logic including both positive (Hod-kinson, Wolter, and Zakharyaschev 2000; Reynolds 1996;Wolter and Zakharyaschev 2002) and negative results(Wolter 2000). In this note we also make use of our ini-tial work in this direction (Belardinelli and Lomuscio 2007a;2007b), where static (i.e., non-temporal) quantified epis-temic logics were axiomatised.</p><p>Our motivation for the above comes from an interest inreasoning about reactive, autonomous distributed systems,or multi-agent systems (MAS), whose high-level proper-ties may usefully be modeled by epistemic formalisms suit-ably extended to incorporate temporal logic. While tem-poral epistemic logics are well understood at propositionallevel (Fagin et al 1995; Meyer and Hoek 1995), their useful-ness has been demonstrated in a number of applications (se-curity and communication protocols, robotics), and modelchecking tools have been developed for them (Gammieand van der Meyden 2004; Raimondi and Lomuscio 2007;Dembinski et al 2003), still there is a growing need inweb-services, security, as well as other areas, to extendthese languages to first-order (see (Cohen and Dams 2007;Solanki, Cau, and Zedan 2006; Vigano 2007)). More-over, a number of formalisms, including BDI logics (Raoand Georgeff 1991), the KQML framework (Cohen, andLevesque 1995), and LORA (Wooldridge 2000), have putforward agent theories that include the power of first-orderquantification. However, most of these contributions do notaddress the issue of completeness, a core concern here.</p><p>In MAS applications the power of first-order logic is wel-come every time agents knowledge is concerned with: Relational statement, as in agent i knows that message </p><p>was sent by a to b, or formallyKiP Send(a, b, );</p><p>Proceedings, Eleventh International Conference on Principles of Knowledge Representation and Reasoning (2008)</p><p>705</p></li><li><p>(where P is the diamond for past time); Functional dependency and identity: at some future point</p><p>agent i will know that message is the encryption of mes-sage with key k, formally</p><p>F Ki( = enc(k, ));</p><p> An infinite domain of individuals, or a finite domainwhose cardinality cannot be bounded in advance: agenti has to read an e-mail before deleting it,</p><p>(Delete(i, ) P Read(i, ));</p><p> Quantification on agents (Lomuscio and Colombetti1996): the child of any process knows which processlaunched it</p><p>iKchild(i)P Launch(i, child(i))</p><p>Furthermore, in the context of logics for knowledge itis known that epistemic modalities can be combined withquantifiers to express concepts such as knowledge de re andde dicto (Fitting, and Mendelsohn 1999; Hughes and Cress-well 1996). For instance, an agent i might know that everycomputation will eventually produce an output, thus havingthe de dicto knowledge expressed by the following specifi-cation:</p><p>comp Ki F y Output(comp, y)but she might not know the actual output of every computa-tion. Therefore, the following de re specification:</p><p>comp y Ki F Output(comp, y)would not be satisfied. From the examples above we con-clude that quantification can significantly extend the expres-siveness of epistemic languages.</p><p>While the specifications above call for a first-order lan-guage, we need to consider why one should use an undecid-able language when a decidable one (propositional temporalepistemic logic in our case) does a reasonable job already.Although this is a sensible objection, we should stress that inmany practical applications, such as in model checking, weare typically not so much concerned with the validity prob-lem but with satisfaction in a given model, which is oftenan easier problem, particularly for some classes of formulas.Additionally, recent research, including among others (Hod-kinson, Wolter, and Zakharyaschev 2000; Sturm, Wolter,and Zakharyaschev 2000; 2002; Wolter and Zakharyaschev2001), has put forward useful decidable fragments of first-order modal logic, thereby opening the way for further ex-tensions.</p><p>We approach the problem by introducing quantified in-terpreted systems, an extension to first-order of standardinterpreted systems (Halpern, and Fagin 1989; Parikh andRamanujam 1985), which are used to interpret a languagefor temporal epistemic logic including distributed knowl-edge. First, a sound and complete axiomatisation is pre-sented. Second, message passing systems, a basic frame-work for reasoning about asynchronous systems (Lamport1978) are analysed in the light of the novel formalism, andthe results compared to the treatment in propositional logic.</p><p>A Quantified Temporal Epistemic LogicIn this section we extend to first-order the formalism of in-terpreted systems, a class of structures introduced to modelthe behaviour of multi-agent systems (Fagin et al 1995;Meyer and Hoek 1995). In what follows we assume a finiteset A = {i1, . . . , in} of agents.</p><p>SyntaxThe first-order modal language Ln contains individual vari-ables x1, x2, . . ., n-ary functors fn1 , f</p><p>n2 , . . . and n-ary pred-</p><p>icative letters Pn1 , Pn2 , . . ., for n N, the identity predi-</p><p>cate =, the propositional connectives and, the universalquantifier , the epistemic operators Ki, for i A, the dis-tributed knowledge operators DG, for non-empty G A,the future operator [F ], and the past operator [P ].Definition 1 Terms and formulas in the language Ln aredefined in the Backus-Naur form as follows:</p><p>t ::= x |fk(~t) ::= P k(~t) | t = t | | |Ki |DG | [F ] | [P ] |x</p><p>The formula Ki means agent i knows , while DGrepresents is distributed knowledge among the agentsin G, and [F ] (respectively [P ]) stands for will al-ways be true (respectively has always been true). Thesymbols , , , , , F (sometime in the future), P (sometime in the past) are defined as standard. The temporaloperators [F ]+ (every future time including the present) and[P ]+ (every past time including the present) can be definedas [F ] and [P ] respectively.</p><p>We refer to 0-ary functors as individual constantsc1, c2, . . . A closed term v is a term where no variable ap-pears; closed terms are either constants or terms obtained byapplying functors to closed terms.</p><p>By t[~y] (resp. [~y]) we mean that ~y = y1, . . . , yn are allthe free variables in t (resp. ); while t[~y/~t] (resp. [~y/~t])denotes the term (resp. formula) obtained by substitutingsimultaneously some, possibly all, free occurrences of ~y int (resp. ) with ~t = t1, . . . , tn, renaming bounded variablesif necessary.</p><p>Quantified Interpreted SystemsInterpreted systems are widely used to model the behaviourof MAS, in this subsection we extend these structures tofirst-order. This extension can be performed in several ways,all leading to different results. For instance, we could intro-duce a domain of quantification for each agent and/or foreach computational state (see (Belardinelli and Lomuscio2007a; 2007b) for a discussion of the static case). In this pa-per we consider the simplest extension, obtained by addinga single quantification domain D common to all agents andstates. We present further options in the conclusions.</p><p>More formally, for each agent i A in a multi-agentsystem we introduce a set Li of local states li, li, . . ., anda set Acti of actions i, i, . . .. We consider local statesand actions for the environment e as well. The set S LeL1 . . .Ln contains all possible global states of theMAS, while Act Acte Act1 . . . Actn is the set of</p><p>706</p></li><li><p>all possible joint actions. Note that some states may neverbe reached and some joint actions may never be performed.We also introduce a transition function : Act (S S).Intuitively, ()(s) = s encodes that the agents can ac-cess the global state s from s by performing the joint action Act. The transition function defines the admissibleevolutions of the MAS. We say that the global state s isreachable in one step from s, or s s, iff there is Actsuch that ()(s) = s; while s is reachable from s iffs + s, where + is the transitive closure of relation .</p><p>To represent the temporal evolution of the MAS we con-sider the flow of time T = T, defined as n &gt; m iffm &lt; n, and the partial order such that n m iff n &lt; m or n = m.</p><p>Let be an assignment from the variables in Ln to theindividuals in D, the valuation I(t) of a term t is definedas (y) for t = y, and I(t) = I(fk)(I(t1), . . . , I(tk)),for t = f(~t). A variant </p><p>(xa</p><p>)of an assignment assigns</p><p>a D to x and coincides with on all the other variables.Definition 3 The satisfaction relation |= for Ln,(r,m) P , and an assignment is defined as follows:</p><p>(P, r,m) |= P k(~t) iff I(t1), . . . , I(tk) I(P k, r,m)(P, r,m) |= t = t iff I(t) = I(t)(P, r,m) |= iff (P, r,m) 6|= (P, r,m) |= iff (P, r,m) 6|= or (P, r,m) |= (P, r,m) |= Ki iff ri(m)=ri(m)implies(P, r,m) |=(P, r,m) |= DG iff ri(m) = ri(m) for all i G,</p><p>implies (P, r,m) |= (P, r,m) |= [F ] iff m &lt; m implies (P, r,m) |= (P, r,m) |= [P ] iff m &gt; m implies (P, r,m) |= (P, r,m) |= x iff for all a D, (P(</p><p>xa), r,m) |= </p><p>The truth conditions for , , , , , F , and P are defined from those above. In particular, the temporaloperators [F ]+ and [P ]+ respect the intended semantics:(P, r,m) |= [F ]+ iff m m implies (P, r,m) |= (P, r,m) |= [P ]+ iff m m implies (P, r,m) |= </p><p>A formula Ln is said to be true at a point (r,m) iff itis satisfied at (r,m) by every ; is valid on a QIS P iff itis true at every point in P; is valid on a class C of QIS iffit is valid on every QIS in C.</p><p>The present definition of QIS is based on two assump-tions. Firstly, the domainD of individuals is the same for ev-ery agent i, so all agents reason about the same objects. Thischoice is consistent with the external account of knowledgeusually adopted in the framework of interpreted systems: ifknowledge is ascribed to agents by an external observer, i.e.,the specifier of the system, it seems natural to focus on theset of individuals assumed to exist by the observer. Sec-ondly, the domain D is assumed to be the same for everyglobal state, i.e., no individual appears nor disappears inmoving from one state to another. This also can be justi-fied by the external account of knowledge: all individualsare supposed to be existing from the observers viewpoint.However, either assumption can be relaxed to accommodateagent-indexed domains as well as individuals appearing anddisappearing in the flow of time. We discuss further optionsin the conclusions. Finally, it can be the case that A D:this means that the agents can reason about themselves, theirproperties, and relationships.</p><p>ExpressivenessClearly, the language Ln is extremely expressive. We canuse it to specify the temporal evolution of agents knowl-edge, as well as the knowledge agents have of temporal factsabout individuals. Both features are exemplified in the fol-lowing specification: agent i will know that someone senthim a message when he receives it,</p><p>j, [F ] (Rec(i, j, ) Ki P Send(j, i, )) (1)In Ln we can also express that if agent i receives a mes-</p><p>sage, then he will know that someone sent it to him:</p><p> [F ](j Rec(i, j, ) Ki j P Send(j, i, )) (2)The latter specification is weaker than the former: (2) says</p><p>nothing about the identity of the sender, while (1) requiresthat the receiver knows the identity of the sender. Further, wecan express the fact that the existence of a sender is assumedonly at the time the message is sent:</p><p> [F ](j Rec(i, j, ) Ki P j Send(j, i, ))</p><p>707</p></li><li><p>In the section on message passing systems we providefurther examples of the expressiveness of Ln. Most impor-tantly, we will show that this expressiveness is attained whileretaining completeness.</p><p>We conclude this paragraph by considering some relevantvalidities on the class of QIS. Given that the domain of quan-tification is the same in every global state, both the Barcanformula and its converse are valid on the class of all QIS forall primitive modalities:</p><p>QIS |= xKi KixQIS |= xDG DGxQIS |= x[F ] [F ]xQIS |= x[P ] [P ]x</p><p>Also, these validities are in line with the birds eye ap-proach usually adopted in epistemic logic. However, shouldwe wish to do so, we can drop them by in...</p></li></ul>