We develop an incremental-tableau-based decision procedure for the multi-agent epistemic logic MAEL(CD) (aka S5_n (CD)), whose language contains operators of individual knowledge for a finite set Ag of agents, as well as operators of distributed and common knowledge among all agents in Ag. Our tableau procedure works in (deterministic) exponential time, thus establishing an upper bound for MAEL(CD)-satisfiability that matches the (implicit) lower-bound known from earlier results, which implies ExpTime-completeness of MAEL(CD)-satisfiability. Therefore, our procedure provides a complexity-optimal algorithm for checking MAEL(CD)-satisfiability, which, however, in most cases is much more efficient. We prove soundness and completeness of the procedure, and illustrate it with an example.
Deep Dive into Tableau-based decision procedure for the multi-agent epistemic logic with operators of common and distributed knowledge.
We develop an incremental-tableau-based decision procedure for the multi-agent epistemic logic MAEL(CD) (aka S5_n (CD)), whose language contains operators of individual knowledge for a finite set Ag of agents, as well as operators of distributed and common knowledge among all agents in Ag. Our tableau procedure works in (deterministic) exponential time, thus establishing an upper bound for MAEL(CD)-satisfiability that matches the (implicit) lower-bound known from earlier results, which implies ExpTime-completeness of MAEL(CD)-satisfiability. Therefore, our procedure provides a complexity-optimal algorithm for checking MAEL(CD)-satisfiability, which, however, in most cases is much more efficient. We prove soundness and completeness of the procedure, and illustrate it with an example.
Over the last two decades, multi-agent epistemic logics ( [2,8]) have played a significant role in computer science and artificial intelligence. The main application seems to have been to design, specification, and verification of distributed protocols ( [6]), but a plethora of other applications are described in, among others, [3], [2] and [8].
Languages of multi-agent epistemic logics considered in the literature contain various repertoires of modal operators. In the present paper, we consider the “full” multi-agent epistemic logic, which we call MAEL(CD), whose language contains operators of individual knowledge for a non-empty, finite set Σ of agents as well as operators of common (C) and distributed (D) knowledge among all agents in Σ. (Since all modal operators of MAEL(CD) are S5-modalities, the logic is also referred to in the literature as S5 n (CD)). To be used for such tasks as designing protocols conforming to a given specification, MAEL(CD), needs to be equipped with an algorithm checking for MAEL(CD)-satisfiability. The first step in that direction was taken in [10], where the decidability of MAEL(CD) has been established by showing that it has a finite model property. This result was proved in [10] via filtration; therefore, the decision procedure suggested by that argument is based on an essentially brute-force enumeration of all finite models for MAEL(CD), which suggest a satisfiability-checking algorithm that is theoretically important, but of limited practical value. Our tableau procedure has, in comparison, the following advantages:
It establishes a (deterministic) ExpTime upperbound for MAEL(CD)-satisfiability, which matches the lower-bound that follows from the results of [7].
It provides an algorithm for checking MAEL(CD)satisfiability that is not only provably complexityoptimal, but which in the vast majority of cases requires much less resources than what is predicted by the worst-case upper bound. This is one of the hallmarks of incremental tableaux ( [11]) as opposed to the top-down tableaux in the style of [1], which always require the amount of resources predicted by the worst-case complexity estimate. Top-down tableaux for the fragment of MAEL(CD) not containing the operator of distributed knowledge have been presented in [7].
The type of incremental tableau developed herein originates in [11]; tableaux in a similar style were recently developed for the multi-agent logic ATL and some of its variations in [5]. Thus, the present paper continues the enterprize of designing complexity-optimal decision procedures for logics used in design, specification and verification of multi-agent systems ( [2,12]). The particular style of the tableaux presented here is meant to be compatible with the tableaux from [5], so that we can in the future build tableaux for more sophisticated logics for multiagent systems.
The main reason for the restriction of the distributed and common knowledge operators only to be (implicitly) parameterized by the whole set of agents referred to in the language, adopted in this paper, is to be able to present the main ideas and features of the tableaux in sufficient detail, while avoiding some additional technical complications arising in the case of several such operators, each one associated with a non-empty subset of the set of all agents. This, more complicated, case will be treated in a follow-up paper.
2 Syntax and semantics of MAEL(CD)
The language L of MAEL(CD) contains a (possibly, countably-infinite) set AP of atomic propositions, typically denoted by p, q, r, . . .; a finite, non-empty set Σ of (names of) agents, typically denoted by a, b . . .; a sufficient repertoire of the Boolean connectives; and the modal operators K a (“the agent a knows that . . . “), D (“it is distributed knowledge among Σ that . . . “) and C (“it is common knowledge among Σ that . . . “). Thus, the formulae of L are defined as follows:
where p ranges over AP and a ranges over Σ. The other boolean connectives can be defined in the usual way. We omit parentheses in formulae whenever it does not result in ambiguity. We denote arbitrary formulae of L by ϕ, ψ, χ, . . . (possibly with decorations). We write ϕ ∈ L to mean that ϕ is a formula of L. Formulae of the form ¬Cϕ are called eventualities.
Formulae of L are interpreted over multi-agent epistemic models, based on multi-agent epistemic frames. We will also need a more general notion of multi-agent epistemic structure.
Notice that in (pseudo-)frames condition 4 of definition 2.1 is equivalent to the requirement that R C is the transitive closure of a∈Σ R a . Also notice that, as in any MAEF each R a is an equivalence relation, R C is also an equivalence relation.
is the set of all atomic propositions that are declared true at s.
F is a multi-agent epistemic pseudo-frame, then M is a multi-agent epistemic pseudo-model (pseudo-MAEM).
The satisfaction relation between (pseudo-)MAEMs and formulae is defined in the st
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