MCMAS: an opensource model checker for the verification of multiagent systems
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Abstract
We present MCMAS, a model checker for the verification of multiagent systems. MCMAS supports efficient symbolic techniques for the verification of multiagent systems against specifications representing temporal, epistemic and strategic properties. We present the underlying semantics of the specification language supported and the algorithms implemented in MCMAS, including its fairness and counterexample generation features. We provide a detailed description of the implementation. We illustrate its use by discussing a number of examples and evaluate its performance by comparing it against other model checkers for multiagent systems on a common case study.
Keywords
Verification Multiagent systems Model checking1 Introduction
Model checking [15] is widely recognised as one of the leading logicbased techniques for the verification of reactive systems [54]. In this paradigm, a system \(S\) is encoded as a transition system, or model, \(M_S\) by means of a programme in a dedicated modelling language such as reactive modules [1] or NuSMV [13]. A specification \(P\) of the system is represented as a logical formula \(\phi _P\). Verifying whether the system \(S\) satisfies the specification \(P\) is encoded as the problem of checking whether the model \(M_S\) satisfies the logical formula \(\phi _P\), formally written as \(M_S \models \phi _P\). Several specifications of reactive systems, including liveness, safety and several specification patterns [20] of interest, can be encoded in discrete temporal logic, either in its linear variant LTL, or in its branching version CTL. Wellknown extensions to this approach include employing realtime and probabilistic specifications [40, 43].
The fundamental challenge in model checking is the socalled statespace explosion, i.e. the fact that the state space of a system grows exponentially with the number of variables employed to describe it. Various techniques have been developed over the years to tame this difficulty including Binary Decision Diagrams, abstraction, bounded model checking, induction, and assume–guarantee reasoning, thereby resulting in systems with state spaces of \(10^{25}\) and beyond to be verifiable.
While this approach has proven successful for a variety of systems, including reactive and embedded systems as well as hardware designs, multiagent systems (MAS) are often specified using languages strictly stronger than plain temporal logic. MAS are distributed systems whose components, or agents, act autonomously to meet their private and joint objectives [71]. MAS are typically specified by asserting the intended evolution of highlevel properties of the agents, including their knowledge [24], their beliefs [32], their intentions [18], and obligations [35]. This follows the successful tradition in MASbased approaches that involves ascribing highlevel attitudes to highly autonomous systems to better predict and specify their resulting behaviours [55]. Specifically, epistemic logic [24], or logic of knowledge, has provided a natural and intuitive but yet formally sound and computationally attractive, framework for reasoning about security protocols [30], agreement protocols [66], knowledgebases, etc. By adopting epistemic modalities as primitives, one can naturally express both private and collective (common, or distributed) knowledge of the agents in the system. For example, a security requirement concerning privacy or secrecy in a system run can be translated into a specification stating that no agent eventually knows the fact in question [64]. Similarly, mutual authentication, is naturally expressed by stating that a principal knows that another principal knows that some key is shared [5]. Similarly, an epistemic account can provide a natural set of specifications for cachecoherence protocols [4]. These are only some examples; we refer to the specialised literature for more examples [31, 64].
It is therefore compelling to extend traditional model checking approaches so that they can support specifications that include agentbased features. However, these are often sophisticated modal logics that may involve tailored fixedpoints computations. So, novel labelling algorithms evaluating agentbased modalities need to be defined and integrated with those for the temporal logics of interest. In addition, since agentbased logics traditionally come equipped with a semantics that is finergrained than plain Kripke models for temporal logic, input languages for the system description also need to be tuned to the needs of MAS. These include having an intuitive description of the states, actions, local protocols and local evolutions. These considerations suggest that adapting an existing model checking toolkit to the needs of the verification of MAS specified by epistemic and other logics inspired by MAS is not a simple exercise and may, in fact, result in more effort than building a dedicated one.
In this paper, we describe MCMAS, a model checking toolkit for the verification of MAS specified through a range of agentbased logics. MCMAS uses Ordered Binary Decision Diagrams (OBDDs, [7]) for the symbolic representation of state spaces and dedicated algorithms for the computation of a number of epistemic operators encoding private and group knowledge and Alternatingtime Temporal Logic (ATL) operators. MAS are described in MCMAS by means of Interpreted Systems Programming Language (ISPL) programs, whose semantics is close to interpreted systems, a popular framework in temporalepistemic logic [24]. MCMAS is equipped with an Eclipsebased plugin for coding support and offers a number of features typical of advanced model checkers, including the graphical representation of counterexamples and witnesses, as well as fairness support. MCMAS is released as open source [68] and has been used in several research projects worldwide. We here present MCMAS version 1.2.2, which extends a previous version [47] by means of more efficient algorithms to compute the state space and the labelling of formulas. Also, among other new features, more sophisticated treatment of uniform strategies, counterexamples and fairness constraints is now supported.
The rest of the paper is organised as follows. In Sect. 2, we provide the formal underpinnings of the technique the model checker implements by giving the syntax and semantics of the logics employed as well as the algorithms implemented in the checker. In Sect. 3 we describe ISPL, the input to the model checker. Section 4 describes the implementation and gives a few examples. Section 5 focuses on specific applications and reports experimental results. Section 6 describes related work and concludes the paper.
2 Symbolic model checking multiagent systems
In this section, we give the theoretical foundations of MCMAS. We succinctly describe the semantics of interpreted systems in Sect. 2.1. We give the syntax of ATLK in Sect. 2.2 and provide model checking algorithms in Sect. 2.3. We conclude in Sect. 2.4 by presenting the OBDDbased encodings for the algorithms.
2.1 Interpreted systems
At the heart of MCMAS and its modelling language is the notion of interpreted system as a formalisation of multiagent systems. Interpreted systems were popularised by Fagin et al. [24] as a semantics for reasoning about knowledge; they can be extended to incorporate game theoretic notions such as those provided by ATL modalities. Here we loosely follow the presentation given in [47], where global transitions are given as the composition of local transitions.
We assume \(AP\) to be a set of atomic propositions and a set of agents \(Ag=\{Ag_0, Ag_1,\ldots ,Ag_n\}\) for the system. We often refer to \(Ag_0\) as the environment of the system.
Definition 1

\(L_i\) is a finite set of possible local states for agent \(i\).

\(Act_i\) is a finite set of possible actions for agent \(i\).

\(P_i: L_i \rightarrow 2^{Act_i\!\setminus \!\emptyset }\) is a local protocol function for agent \(i\) returning possible actions at a given local state.

\(\tau _i: L_i \times Act_0 \times \cdots \times Act_n \rightarrow L_i\) is a deterministic local transition function returning the local state for agent \(i\) resulting from the execution of a joint action at a given local state; we assume that every action is protocol compliant, i.e. if \(l_i^{\prime }=\tau _i(l_i, a_0, \ldots , a_i, \ldots , a_n)\), then \(a_i \in P_i(l_i)\) for all \(i \in Ag\).

\(I \subseteq L_0 \times L_1 \times \cdots \times L_n\) is the set of initial global states.

\(h \subseteq L_0 \times \cdots \times L_n \times AP\) is a labelling relation encoding which atomic propositions are true in which state.
We say that \(G=L_0 \times \cdots \times L_n\) is the set of possible global states for the system and \(ACT=Act_0 \times \cdots \times Act_n\) the set of possible joint actions. For a global state \(g=(l_0,\ldots ,l_n) \in G\) and any \(i \in Ag\), we consider the function \(l_i:G \rightarrow L_i\) such that \(l_i(g)=l_i\), returning the local state of agent \(i\) in the global state \(g\).
Observe that interpreted systems describe finitestate systems composed of agents possibly synchronising with each other and the environment via joint actions. Also note that the local protocols implement the agents’ decision making and state the conditions for the transitions in the system. We refer to [24] for more details.
Interpreted systems naturally induce Kripke models which can be used to interpret our specification language. These are defined as follows.
Definition 2

\(Ag\) is the set of agents of \(IS\);

\(ACT \subseteq Act_0 \times \cdots \times Act_n\) is the set of joint actions for the system \(IS\);

\(S\subseteq L_0 \times \cdots \times L_n\) is the set of global states reachable from \(I\) via \(T\);

\(T \subseteq S \times ACT \times S\) is a transition relation representing the temporal evolution of the system. We assume that \(T\) is decomposable and define the transition relation as \((s,a,s') \in T\) iff for all \(i \in Ag\) we have \(\tau _i(l_i(s),a))=l_i(s')\);

\(\{\sim _i\}_{i \in Ag \!\setminus \! \{Ag_0\}} \subseteq S \times S\) is the set of equivalence relations, one for each agent but not the environment, encoding the epistemic accessibility relations. For any \(i \in Ag \!\setminus \! \{Ag_0\}\), we assume that \((s, s') \in \sim _i\) iff \(l_i(s)=l_i(s')\).
We use the notation \(s~\mathop {\rightarrow }\limits ^{a}~s'\) as a shortcut for \((s,a,s') \in T\). We use the term path to denote any sequence of states \(\pi = (s^0,s^1,\ldots ,s^n,\ldots )\) such that, for all \(i \ge 0\), we have \((s^i,a,s^{i+1}) \in T\) for some action \(a \in ACT\). Given a path \(\pi \), we denote with \(\pi (k)\) the state at position \(k\). Given a set of agents \(\varGamma \subseteq Ag\) and a joint action \(a = (a_0, a_1,\ldots ,a_n)\), we denote with \(S_{\varGamma }\) the projection of \(S\) on the local states of the agents in \(\varGamma \) and with \(a_{\varGamma }\) the tuple consisting of the elements in \(a\) restricted to the agents in \(\varGamma \). In this case, we say that \(a\) is a completion for \(a_{\varGamma }\). For instance, if \(a=(a_0, a_1, a_2, a_3, a_4)\) and \(\varGamma = \{1,3\}\), \(a_{\varGamma } = (a_1,a_3)\) and \(a_{Ag \backslash \varGamma } = (a_0,a_2,a_4)\). Given \(\varGamma \subseteq Ag\) and \(ACT\), the set \(ACT_{\varGamma }\) denotes the set of all tuples \(a_{\varGamma }\) as above.
We say that a joint action \(a \in ACT\) is enabled in a state \(s \in S\) if there exists a state \(s' \in S\) such that \((s,a,s') \in T\). Similarly, given a group \(\varGamma \), we say that \(a_{\varGamma }\) is enabled in a state if there exists a completion of \(a_{\varGamma }\) to a joint action \(a \in ACT\) such that \(a\) is enabled in that state. Note that, by Definition 1, each component of an enabled action is locally protocol compliant. Further note that since a local action is always possible at any local state, the models considered are serial, i.e. there are no deadlocks.
A strategy for agent \(i\) is a function \(\sigma _i: L_i\rightarrow 2^{ACT_i}\!\setminus \! \{\emptyset \}\) such that if \(a_i\in \sigma _i(l_i)\), then \(a_i \in P_i(l_i)\). Given a group of agents \(\varGamma \), a strategy for \(\varGamma \) is a function \(\sigma _{\varGamma }: S_{\varGamma }\rightarrow 2^{ACT_\varGamma }\!\setminus \! \{\emptyset \}\) such that \(\sigma _{\varGamma } (l_{x_1}, \ldots , l_{x_k}) = (\sigma _{x_1}(l_{x_1}),\) \( \ldots , \sigma _{x_k}(l_{x_k}))\), where \(\sigma _{x_1}, \ldots , \sigma _{x_k}\) are strategies for the agents \(x_1, \ldots , x_k \in \varGamma \).
The strategies defined above are analogous to the nonuniform, incomplete information, memoryless strategies in [2]. Note that an agent (or a group of agents) adhering to memoryless strategies with incomplete information may perform different actions in different global states whose local component is the same. This allows for an element of action “guessing” that is not considered useful when reasoning in terms of strategic abilities. In these cases, it is more meaningful to consider, still under incomplete information and memoryless assumptions, deterministic uniform strategies [37] of the form \(\sigma _i: S_i \rightarrow ACT_{\varGamma }\!\setminus \! \emptyset \), which restrict the definition of strategy above by assuming that the same action is performed in the global states in which agent \(i\) has the same local state.
The model induced by an interpreted system is said to be nonuniform, i.e. along its paths the agents may pick different actions, compatibly with their protocols, in the same local state at different global states. To evaluate an interpreted system under the uniformity assumption, we consider the various (uniform) models derived from the induced nonuniform model in which along any path the agents select the same action whenever they are in the same local state. As we will see later, MCMAS supports the verification of both uniform and nonuniform models derived from an interpreted system.
2.2 Syntax of ATLK and satisfaction
We use ATLK as the specification language for the agents in the system. ATLK combines the logic ATL [2] with modal operators to reason about the knowledge of the agents in the system. As it is known, ATL extends the logic CTL [15] by replacing CTL temporal operators with strategic cooperation modalities expressing what state of affairs a coalition of agents can bring about in a system, irrespective of the actions of the other agents. A large amount of work in Artificial Intelligence and MultiAgent Systems routinely employs rich specifications using the concepts of what an agent, or a collection of agents, knows about the system and each other (e.g. see [24] for an introduction to the area) and what a group of agents can collectively enforce in the system (e.g. see [65]). We illustrate this through some scenarios in Sect. 5.
Definition 3
Since, differently from [2], here we work with incomplete information, the meaning of the modalities depends on the uniformity assumption made on the system. If we do not assume uniformity, the reading of the ATL modalities is as follows. The formula \(\langle \!\langle \varGamma \rangle \!\rangle X \phi \) expresses that the agents in \(\varGamma \) can ensure that \(\phi \) holds at the next state irrespective of the actions of the agents in \(Ag\!\setminus \!\varGamma \). In other words, the agents in \(Ag\!\setminus \!\varGamma \) cannot ensure that \(\phi \) is false at the next state.
The formula \(\langle \!\langle \varGamma \rangle \!\rangle G \phi \) conveys that it is possible that the actions of the agents in \(\varGamma \) result in \(\phi \) being true forever in the future, irrespective of the actions of the agents in \(Ag \!\setminus \! \varGamma \). As above, this means that the agents outside \(\varGamma \) cannot ensure \(\phi \) is not uniformly realised. Similarly, the formula \(\langle \!\langle \varGamma \rangle \!\rangle [\phi U \psi ]\) signifies that the agents in \(\varGamma \) may be able to realise \(\psi \) at some point in the future and to ensure that \(\phi \) holds till then.
While the combination of incomplete information, knowledge and ATL modalities give rises to the readings above, we are generally interested in evaluating what the agents have the power to enforce by means of ATL formulas [2]. In our setting, this is the reading of the ATL operators under the assumption of uniformity.
In this case, the formula \(\langle \!\langle \varGamma \rangle \!\rangle X \phi \) is read as “group \(\varGamma \) has a strategy to enforce \(\phi \) in the next state (irrespective of the actions of the agents not in \(\varGamma \))”; \(\langle \!\langle \varGamma \rangle \!\rangle G \phi \) represents “group \(\varGamma \) has a strategy to enforce \(\phi \) forever in the future”; and \(\langle \!\langle \varGamma \rangle \!\rangle [\phi U \psi ]\) means “group \(\varGamma \) has a strategy to enforce that \(\psi \) holds at some point in the future and can ensure that \(\phi \) holds until then”.
The remaining operators are used to characterise epistemic states of agents as in [24]. In particular, \(K_i \phi \) is read as “agent \(i\) knows \(\phi \)”, \(E_{\varGamma } \phi \) as “everybody in group \(\varGamma \) knows \(\phi \)”, \(D_{\varGamma } \phi \) as “\(\phi \) is distributed knowledge in \(\varGamma \)”, and \(C_{\varGamma } \phi \) as “\(\phi \) is common knowledge in \(\varGamma \). It is known that the standard branchingtime temporal operators \(EX, EG\), and \(EU\) can be expressed by considering the “grand coalition of all agents”, e.g. \(EX \phi \equiv \langle \!\langle Ag \rangle \!\rangle X \phi \).
The satisfaction of ATLK specifications on induced models is defined recursively as follows (recall that given an action tuple \(a_{\varGamma }\) for the agents only in \(\varGamma \), we write \(a\) for any of the completions of \(a_{\varGamma }\)):
Definition 4

\((\mathcal {M},s^0)\models p\) iff \(p\in h(s^0)\);

\((\mathcal {M},s^0)\models \lnot \phi \) iff it is not the case that \((\mathcal {M},s^0) \models \phi \);

\((\mathcal {M},s^0)\models \phi _1 \vee \phi _2\) iff \((\mathcal {M},s^0)\models \phi _1\) or \((\mathcal {M},s^0)\models \phi _2\);

\((\mathcal {M},s^0)\models \langle \!\langle \varGamma \rangle \!\rangle X \phi \) iff there exists a strategy \(\sigma _{\varGamma }\) and an action \(a_{\varGamma } \in \sigma _{\varGamma }(s^0_{\varGamma })\) such that for all states \(s^1\) such that \(s^0~\mathop {\rightarrow }\limits ^{a}~s^1\), we have \((\mathcal {M},s^1)\models \phi \);

\((\mathcal {M},s^0)\models \langle \!\langle \varGamma \rangle \!\rangle G \phi \) iff there exists a strategy \(\sigma _{\varGamma }\) and an action \(a^1_{\varGamma } \in \sigma _{\varGamma }(s^0_{\varGamma })\) such that all states \(s^1\) with \(s^0\mathop {\rightarrow }\limits ^{a^1} s^1\) are such that there is an action \(a^2_{\varGamma } \in \sigma _{\varGamma }(s^1_{\varGamma })\) such that all states \(s^2\) with \(s^1\mathop {\rightarrow }\limits ^{a^2} s^2\) are such that, etc., and we have that \((\mathcal {M},s^i)\models \phi \), for all \(i\ge 0\).

\((\mathcal {M},s^0)\models \langle \!\langle \varGamma \rangle \!\rangle [\phi _1 U \phi _2]\) iff there exists a strategy \(\sigma _{\varGamma }\) and an action \(a^1_{\varGamma } \in \sigma _{\varGamma }(s^0_{\varGamma })\) such that all states \(s^1\) with \(s^0\mathop {\rightarrow }\limits ^{a^1} s^1\) are such that there is an action \(a^2_{\varGamma } \in \sigma _{\varGamma }(s^1_{\varGamma })\) such that all states \(s^2\) with \(s^1\mathop {\rightarrow }\limits ^{a^2} s^2\) are such that, etc., and we have \((\mathcal {M},s^j)\models \phi _2\), for some \(j\ge 0\), and \((\mathcal {M},s^i)\models \phi _1\) for all \(0\le i < j\).

\((\mathcal {M},s^0)\models K_i \phi \) iff for all \(s^1\in S\) we have that if \(s^0\sim _i s^1\) then \((\mathcal {M},s^1)\models \phi \).

\((\mathcal {M},s^0)\models E_{\varGamma } \phi \) iff for all \(s^1\in S\) we have that if \(s^0 \left( \bigcup \limits _{i\in \varGamma }\sim _i \right) s^1\) then \((\mathcal {M},s^1)\models \phi \).

\((\mathcal {M},s^0)\models D_{\varGamma } \phi \) iff for all \(s^1\in S\) we have that if \(s^0\left( \bigcap \limits _{i\in \varGamma }\sim _i \right) s^1\) then \((\mathcal {M},s^1)\models \phi \).

\((\mathcal {M},s^0)\models C_{\varGamma } \phi \) iff for all \(s^1\in S\) we have that if \(s^0 \left( \bigcup \limits _{i\in \varGamma }\sim _i \right) ^{+} s^1\), then \((\mathcal {M},s^1)\models \phi \), where \(^{+}\) denotes the transitive closure of the relation.
We say that an interpreted system \(IS=(\{L_i, Act_i, P_i,\) \( \tau _i\}_{i \in Ag}, I, h)\) satisfies an ATLK specification \(\phi \) if and only if \((\mathcal {M}_{IS},s^i) \models \phi \), for all \(s^i \in I\). We say that that interpreted system \(IS\) satisfies an ATLK specification \(\phi \) under uniformity iff at least one of the models \(\mathcal {M}_{IS}\) induced from \(IS\) under uniformity is such that \((\mathcal {M}_{IS},s^i) \models \phi \), for all \(s^i \in I\).
The semantics above is observational. In other words, the agents’ local states do not necessarily encode all the local states encountered by the agent in a run. Note that a bounded form of perfect recall can still be encoded in the semantics. Observational semantics is commonly regarded as the standard treatment of epistemic modalities [24] and it does not increase the complexity of the model checking problem when combined with CTL or ATL [52]. Perfect recall semantics with ATL leads to an undecidable model checking problem [2].
2.3 Symbolic model checking ATLK
We now define model checking algorithms for the logic ATLK; these extend the corresponding ones for CTL [15]. The approach here presented is symbolic in that it uses Ordered Binary Decision Diagrams (OBDDs) [7] as basic data structures to encode sets of states and transitions.
The procedures \(SAT_K, SAT_E\), and \(SAT_D\) for the epistemic operators are described in Algorithms 2, 3, and 4. These take as input the subformula to be checked and return the set of states satisfying the epistemic formula.
They compute the existential preimage of the set \(SAT(\lnot \phi _1)\) with respect to the appropriate epistemic relation, i.e. the set of states not satisfying the epistemic formula. The complement of this set with respect to the set of reachable states \(S\) is the set of states satisfying the formula in question.
Algorithm 5 iteratively calculates the set of states that can access a state not satisfying \(\phi \) via a finite sequence of epistemic relations for the agents in \(\varGamma \). The complement of this set with respect to \(S\) is equal to the set of states satisfying \(C_{\varGamma } \phi \) [60].
The algorithms for the ATL operators depend on the auxiliary procedure \(ATLPRE(\varGamma ,X)\) (see Algorithm 6), which computes the set of states \(Y \subseteq S\) from which there exists a joint action \(a_{\varGamma }\) for the agents in \(\varGamma \) such that all action completions \(a\) of \(a_{\varGamma }\) enabled at a state in \(Y\) generate a transition to a state in \(X\). Algorithm 7 employs this procedure directly to compute \(SAT_{ATLX}\), while Algorithms 8 and 9 implement the standard fixpoint algorithms.
2.4 Symbolic model checking and OBDDs
To verify interpreted systems against specifications in ATLK we use the algorithms above in which operations on sets are implemented as operations on Boolean formulae, appropriately encoded as Ordered Binary Decision Diagrams (OBDDs). As an example, consider the Boolean formula \(f_1(x_1,x_2,x_3) = \lnot x_1 \vee (x_1 \wedge \lnot x_2 \wedge \lnot x_3)\), where \(x_1, x_2, x_3\) are Boolean variables. The truth table of this formula is eight lines long. Alternatively, \(f_1\) can be represented by means of a binary tree with root node \(x_1\), as in Fig. 1, where the leaves represent the truth value of \(f_1\). This tree can be simplified as in Fig. 2. Notice that the simplified tree only has 5 nodes instead of 15. In general, the reduced tree can be orders of magnitude smaller than the truth table for a given Boolean formula.
Symbolic model checking exploits the compression capabilities of OBDDs to represent large state spaces efficiently. A Boolean formula can represent a state in a model. As an example, consider a model in which \(S = \{s_0,s_1,\ldots ,s_7\}\). The Boolean formula \(f_0(x_1,x_2,x_3) = x_1 \wedge x_2 \wedge x_3\) can be used to encode \(s_0\), the Boolean formula \(f_1(x_1,x_2,x_3) = \lnot x_1 \wedge x_2 \wedge x_3\) to encode \(s_1\), etc. The number of Boolean variables required to encode a set \(S\) grows as \(O(log_2(S))\).
3 Modelling multiagent systems in ISPL
In this section, we describe Interpreted Systems Programming Language (ISPL), the language used to model MAS within MCMAS. ISPL is strongly based on Interpreted Systems as defined in Sect. 2. In this section, we present ISPL’s constructs and give its semantics.
An ISPL program describes a multiagent system as composed of a number of agents and an environment. Agents’ definitions in ISPL closely follow those of agents in Definition 1. To describe an agent in ISPL we declare the following components. Local states are private, internal states of the agents, declared by means of variables, and cannot be observed by the other agents. Agents interact with each other and the environment by means of publicly observable local actions. Actions are performed in accordance with a local protocol representing the agent’s decisionmaking process. Local states change value over time following a local evolution function, which returns the next local state on the basis of the current local state and the joint actions performed by all the other agents at a given instant. ISPL’s structure with actions and protocols is deliberately based on interpreted systems’ semantics which constitute a widely used framework for describing MAS.
We describe ISPL’s syntax using a simple example, the bit transmission protocol [24]. In this protocol, a Sender agent intends to deliver a message (the value of a bit) to a Receiver over an unreliable communication channel. The channel may randomly drop messages in either direction, but it does not modify the content of messages. To guarantee communication, the Sender keeps sending the same bit to the Receiver until it receives an acknowledgement; at this point the Sender stops sending the bit. The Receiver performs no action until a bit is received, and keeps sending acknowledgements thereafter.
The environment is described using the keyword Environment. One of its features is that some of its local states can be observed by other agents. As an example, the section Obsvars in Fig. 6 lists the variables that are observable by all the agents in the system. The variables in the Vars section of the environment, instead, can be observed by an agent only if they appear in the Lobsvars definition for that agent. For instance, agent A1 in Fig. 6 can observe variable v1 (but not v2), and agent A2 can observe variable v2 (but not v1). Both agents can observe variable v3. This feature enables faster communication and synchronisation between the agents and the environment.
Following the agents’ declarations, an ISPL model contains the section Evaluation declaring the atomic variables for the model. Figure 7 reports the definition of five atomic variables recbit, recack, bit0, bit1, envworks. The Boolean condition appearing on the righthand side of each line denotes the set of global states, where the propositions hold. The specifications to be verified (see the Formulae section in Fig. 8) are built on the propositions defined here.
The description of a system of agents is completed by providing a set of initial states, an optional set of fairness conditions, and the set of formulae to be verified. As shown in Fig. 8, the set of initial states is declared in the section InitStates by means of a Boolean function imposing conditions on local states. The Fairness section reports a list of Büchi fairness constraints, expressed as Boolean formulae. In the example of Fig. 8, it is required that the proposition envworks, which captures the fact the Environment is transmitting messages in both directions, must be true infinitely often: this means that the channel cannot block messages indefinitely. Finally, the section Formulae contains the specifications to be checked. These are formulas in the logic ATLK; since ATL subsumes CTL, formulas in the logics CTL or CTLK [59] are also supported. As an example, the first formula in Fig. 8 states that it is always true that, when recack is true and the value of the bit is 0, then the agent Sender knows that the agent Receiver knows that the value of the bit is 0. The second specification is stronger and states that when recack is true and the value of the bit is 0, it is common knowledge in the group g1 that the value of the bit is 0. The group g1 is defined above the specifications by the keyword g1, listing the groups of agents to be considered in the epistemic specifications for the model.

The set of agents in \(IS_P\) is the set of agents declared in \(P\), where the environment in \(P\) is mapped to \(Ag_0\) in \(IS_P\).

For each agent \(i\), the set of possible local states \(L_i\) in \(IS_P\) is defined by taking the Cartesian product of the corresponding sets defined for the local variables for agent \(i\) in \(P\) (Section Vars).

For each agent \(i\), the set \(Act_i\) is the corresponding set of actions agent \(i\) in the programme \(P\) (Section Actions).

For each agent \(i\), the protocol \(P_i\) is defined by the list of Boolean conditions in the Section Protocol for the agent \(i\) in the programme \(P\).

For each agent \(i\), the function \(\tau _i\) is defined by the list of Boolean conditions in the Section Evolution for the agent \(i\) in the programme \(P\).

The set of global initial states \(I\) is defined by evaluating the Boolean conditions in InitStates in \(P\).
Given an ISPL program \(P\) and the interpreted system \(IS_P\) denoted by \(P\), we construct the induced model \(M_{IS_P}\) by applying Definition 2 to \(IS_P\) and by taking the evaluation \(h\) defined by the Boolean conditions in \(P\) (Section Evaluation). Formally, an ISPL program \(P\) satisfies a specification \(\phi \) (given in section Formulae) if \(M_{IS_P} \models \phi \). Note that since the semantics of ISPL programs is defined in terms of their corresponding interpreted system, their evolution is deterministic. Also observe that the composition of the different agents is synchronous via joint actions as in interpreted systems.
As described in Sect. 2.1, under uniformity, an interpreted system induces not just one but a set of uniform models. In this case, we say that an ISPL program \(P\) satisfies \(\phi \) under uniform semantics if there exists an induced uniform model \(M_{IS_P}\) such that \(M_{IS_P} \models \phi \).

Global states are represented by rectangles; only the local states of the Sender and Receiver are reported, for readability the Environment is not. For instance, the global state ((b0,false),empty) in the topleft corner encodes a state in which variable bit for the Sender has value r0, variable ack is false, and variable state for the Receiver is empty.

The temporal transitions are represented by solid arrows and for brevity are labelled with the Environment action only (the Sender’s and Receiver’s actions are derived deterministically from the protocol).

The epistemic relations for the Sender are represented by dotted lines; dashed lines represent the epistemic relations for the Receiver. All reflexive relations are omitted.
4 MCMAS: implementation and usage
MCMAS is implemented in C++ and can be compiled from its source code on most platforms (including Windows, Linux, Mac, Raspberry Pi and various other UNIX systems). The build recognises most architectures automatically. Precompiled versions of the tool are also available from the support pages [68]. In what follows, we describe MCMAS ver 1.2.2, released in March 2015.
4.1 Implementation details
To illustrate the tool, we discuss a number of implementation choices that affect the overall performance of MCMAS. In particular, we consider the following issues: (1) variable ordering in OBDDs; (2) computation of the set of reachable states; (3) construction of the temporal and epistemic transition relations; (4) consistency checks of the input model.
4.1.1 Variable ordering in OBDDs
The size of an OBDD is very sensitive to the choice of the ordering for its Boolean variables. A good ordering can use less memory and speed up OBDD operations by orders of magnitude with respect to an alternative one. Finding a static ordering that generates a compact OBDD representation for representing the state space and the transition relation is challenging. Dynamic reordering is a useful technique aimed at finding a good compromise between continuous variable reordering and efficiency with the aim of reducing memory consumption during model checking.
 1.
Boolean variables for current and successor states are interleaved. The resulting ordering is: \( (v_0,v_0')\cdots (v_n,\) \( v_n') X_0,\ldots X_n\), where \(v_0,\ldots v_{i_0}\) are variables encoding the local states for agent 0 and the other agents, and primed variables encode successor states. The variables \(X_0 \ldots X_n\) encode actions and are grouped at the end.
 2.
A variation of the above whereby variables for actions are interleaved with states: \((v_0,v_0') X_0 \cdots \) \((v_n,v_n') X_n\).
 3.
For each agent, all the variables for the current state are grouped at the beginning, followed by the variables used to encode actions, followed by primed variables to encode successor states: \((v_0,\ldots ,v_{i_0}) X_0 (v_0',\) \(\ldots ,v_{i_0}')\ldots (v_{i_nN+1},\ldots , v_n) X_n(v_{i_N+1},\ldots , v_n)\),
 4.
A variation of the case above whereby variables for the actions follow the variables for the states: \((v_0,\ldots ,\, v_{i_0})\) \( (v_0',\ldots ,v_{i_0}')\, X_0 \cdots (v_{i_N+1},\ldots , v_n)\,(v_{i_N+1},\ldots ,v_n)\, X_n\).
4.1.2 Computing the set of reachable states
4.1.3 Building the temporal and epistemic relations
 1.We first compute an OBDD \(X'\) by removing the OBDD variables in \(V\backslash V_i\), which are not part of agent \(i\)’s encoding. \(X'\) is characterised by the following set:where \(s_i\) and \(s_i'\) are the local states of agent \(i\) in \(s\) and \(s'\), respectively.$$\begin{aligned} X'=\{s\in L_0\times \cdots \times L_n\mid \exists s'\in X \text{ such } \text{ that } s_i'=s_i \},\end{aligned}$$
 2.
The OBDD \(Y\) is computed as \(Y=X'\cap S\), where \(S\) is the set of reachable states.
For efficiency purposes, MCMAS implements optimised algorithms for the verification of CTL operators, rather than employing the procedures for the ATL modalities.
4.1.4 Fairness for ATLK
In a number of circumstances, it is desirable to remove certain unwanted behaviours from the possible executions of a system. For instance, consider the code for the Environment of the bit transmission protocol reported in Fig. 5. The protocol for this agent allows the environment to block messages forever; yet, the designer is likely to want to describe a situation in which the communication channel randomly drops messages, but it is not continuously faulty.
The removal of unwanted behaviours is often achieved by imposing fairness conditions. In the case of branching logics such as ATLK, this requires the definition of constraints outside the model and the use of purposebuilt verification algorithms which extend the standard labelling algorithm presented in Sect. 2.3.
Fairness conditions are declared in MCMAS using an optional set of Boolean formulae constructed using atomic propositions from \(AP\). In line with the standard literature, an infinite path in a model is said to be fair if all the Boolean formulae from the set of fairness conditions are true infinitely often along the path.
MCMAS implements the standard algorithms for the verification of temporal operators under fairness [15]. When the fairness options are enabled all operators are evaluated on the set of fair paths as discussed in [9].
4.1.5 Witnesses, counterexamples and strategy synthesis
 1.
by generating fair counterexamples and witnesses upon request. This is done by integrating the algorithm described in [14] for the temporal operators with the generation of the counterexample tree;
 2.
by including additional cases for the additional operators. Algorithm 11 generates a counterexample for the formula \(K_i \phi \) by picking a state not satisfying \(\phi \) (the remaining epistemic operators are treated in a similar way). A similar algorithm can be used to compute a counterexample for \(K_i \phi \) under fairness by replacing \(SAT_{\phi }\) is replaced by \(SAT^F_{\phi }\) and \(S\) by \(S^F\) in Algorithm 11. Algorithm 12 builds a witness for a formula of the form \(\langle \!\langle \varGamma \rangle \!\rangle X \phi \) by computing the set \(Z\) of states in which agents not in \(\varGamma \) can move to a state satisfying \(\lnot \phi \), and returns an element from its complement. Algorithm 13, instead, returns a witness for a formula of the form \(\langle \!\langle \varGamma \rangle \!\rangle \phi _1 U \phi _2\). This algorithm and a similar one for formulae of the form \(\langle \!\langle \varGamma \rangle \!\rangle G \phi \) (not reported here) follow the standard procedure for Until and Globally operators of CTL (see [15] for additional details).
Notice that, in line with [1], MCMAS currently generates ATL witnesses and counterexamples without fairness, as this would require doubleexponential algorithms and cause visualisation problems for the traces obtained.

MCMAS generates a counterexamples for universal formulas containing an existential formula. For instance, if the formula \(AF EG \phi \) is false, MCMAS generates a counterexample consisting a loop reachable from the initial state in which \(EG \phi \) does not hold.

MCMAS generates a witness for existential formulas containing a universal formula. For instance, if the formula \(EF AG \phi \) is true, MCMAS generates a witness consisting of a path leading from the initial state to a state in which \(AG \phi \) holds.

MCMAS generates witnesses or counterexamples for Boolean combinations of either universal or existential formulae (including the cases mentioned above).
4.1.6 Consistency checking
It is known that bounded integer types may generate overflows in a reachable state. Due to the encoding strategy adopted in MCMAS, the value of a variable exceeding its upper bound is truncated to a value within its domain. This might lead to unexpected behaviours. To avoid this MCMAS allows users to verify the presence of overflows as an optional check. This is carried out by encoding the transition relation \(T_{ overflow}\) in such a way that an expression on the righthand side of an assignment is assumed to be beyond the bounds of the lefthand side variable, and then by constructing the conjunction of \(T_{ overflow}\) and the reachable states \(R\). A nonempty conjunction indicates the existence of an overflow.
The ISPL semantics requires that each state must have a successor state. So, checking deadlock states that do not have successor states is necessary to guarantee correct results of model checking, as deadlock states violate the premises of the model checking algorithm for \(EG\). Checking the presence of a deadlock state is implemented by model checking the formula \(EG\; true\) on the model. If the formula does not hold in all states, then there exists a deadlock state.
4.2 MCMAS usage

The options o, g, and d are used, respectively, to select the algorithm to be used to order OBDD variables, to group OBDD variables, and to disable dynamic OBDD reordering.

The option e is used to select the algorithm to be used to generate the reachable state space (see Algorithm 10).

The options k and a are used to check for deadlocks and arithmetic overflows in the model.

The option c is used to select the way in which counterexamples and witnesses are displayed. If this option is selected, the user can also tune the generation of counterexamples and witnesses using additional parameters provided using the options p, f, l, and w (we refer to the online documentation for a detailed description of these).

The option uniform is used to force the generation of uniform models as described in Sect. 2.2.

ISPL program editing This helps users create MCMAS projects and ISPL program skeletons; it implements syntax highlighting; it performs dynamic syntax checking (a separate ISPL compiler was implemented in Java + ANTLR [58] for this); it provides an outline view and the synchronisation between the outline view and the ISPL editor; it also supports text formatting and content assist.

Interactive execution mode This mode implements the s option described above to perform interactive simulations. It allows users to execute their model step by step by selecting an initial state first, and subsequently by choosing a successor state among those reachable via the enabled transitions. Users can backtrack an execution to the beginning at any step. Simulations can be performed either explicitly, i.e. without OBDD encoding, or symbolically. The explicit simulation is performed by the Eclipse plugin and does not require interaction with MCMAS. The symbolic simulation is more appropriate for large models and requires the installation of MCMAS.

Verification In this modality the GUI invokes MCMAS to execute the model checking procedures. Counterexamples and witness traces are displayed in a graphic way using the dot utility from the Graphviz package [27]; states are shown as nodes and transitions as edges. When the mouse is rolled over a node in the graph, the corresponding state is highlighted. The executions can be projected onto a subset of agents to mask unwanted information of other agents.
5 Scenarios and applications
In the past 10 years, MCMAS has been used to verify a wide number of systems ranging from simple multiagent systems protocols to industrial scenarios [21, 22, 23, 28, 29, 44, 48, 49, 50]. In the following, we summarise a few instructive examples, but refer to the references above for more details.
The ISPL encodings for the scenarios below were either manually given, or automatically generated by means of dedicated compilers. While other traditional model checkers could be used to model the scenarios, as we will see below, the specifications checked are based on epistemic or ATL formulas, hence normally not verifiable with traditional checkers.
5.1 Example scenarios
We begin with simple scenarios from the artificial intelligence and MAS literature and then move on to describe more complex use cases.
5.1.1 The muddy children puzzle
The muddy children puzzle [24] concerns a group of \(n\) children out playing in the field; \(k\) of them get mud on their forehead. They sit in a circle and see only the foreheads of other children, but not their own. An adult arrives and announces “At least one of you is muddy!”; he pauses and then asks: “Does anyone of you know whether you are muddy?”. The adult repeats this question over and over. After each question, the children, assumed to be perfect reasoners, answer “I do not know”. By reasoning about what children know and what they learn from each announcement it is possible to show that at round \(k\) the muddy children say that they know they are muddy.
Given the small size of the state space per child, the analysis can be performed on a very large number of children. Keeping track of the number of round of questions is not problematic but leads to a larger state space.
Experimental results for muddy children
Number of children  Number of OBDD vars  Time (s)  

OBDD ordrng 1  OBDD ordrng 2  OBDD ordrng 3  OBDD ordrng 4  
20  111  0.61  0.50  0.76  0.76 
40  213  11.46  6.57  8.76  10.68 
60  313  40.29  39.58  54.16  97.67 
80  415  165.62  144.44  190.46  192.49 
100  515  736.04  431.85  466.05  385.31 
120  615  920.02  1036.89  784.44  787.35 
Experimental results for 100 prisoners
Number of prisoners  Number of OBDD vars  Time (s)  

OBDD ordrng 1  OBDD ordrng 2  OBDD ordrng 3  OBDD ordrng 4  
5  40  0.40  0.34  0.41  0.43 
10  88  13.79  13.00  11.50  11.48 
15  123  118.23  122.57  126.75  126.93 
20  161  1573.96  1692.87  1636.44  913.02 
25  196  6541  8682  6083  7107 
Table 1 shows that MCMAS is able to handle a large number of children. The default OBDD ordering works well up to 100 children; ordering 3 and 4 are more efficient for the larger models.
5.1.2 One hundred prisoners
This puzzle concerns 100 prisoners kept in solitary confinement [19]. One day the warden gathers all the prisoners in the dining hall for dinner and announces that from the following day he will randomly choose one prisoner every day for questioning. The interrogation room has only a light governed by a toggle switch. Prisoners can observe whether the light is on when they enter the room. During their visit they are allowed to switch the light on or off as they please. While in the interrogation room, a prisoner may announce that he believes that all prisoners have already visited the interrogation room. If a prisoner makes this announcement and this corresponds to the truth, then all prisoners are set free; if the announcement is not correct, all prisoners are executed. The prisoners are granted one meeting to coordinate their actions before the interrogations begin. It is assumed that prisoners can count days and that the light is initially off.
5.1.3 Tian Ji racing horses
Experimental results for racing horses
Number of children  Number of OBDD vars  Time (s)  

OBDD ordrng 1  OBDD ordrng 2  OBDD ordrng 3  OBDD ordrng 4  
5  39  0.09  0.03  0.03  0.02 
10  65  0.47  0.34  0.35  0.47 
15  85  5.66  3.17  2.86  5.62 
20  111  50.22  259.95  37.78  37.67 
25  131  179.48  589.59  148.47  145.36 
30  151  1000.06  1612.75  1011.45  1001.54 
35  177  2449.73  2307.21  525.21  2078.08 
40  197  224.21  329.09  Timeout  3278.78 
The experimental results for various number of horses are reported in Table 3. On this scenario, the OBDD ordering 1 offers the best performance on large models. It is worth pointing out that the running time for the model with 40 horses is shorter than that with 35 horses under the first OBDD ordering. This is because the former model has better structural regularity, which makes the OBDD operations more efficient.
5.2 Applications
We now turn to larger applications and use cases outside the domain of multiagent systems.
5.2.1 Verification of authentication protocols
Authentication protocols are a class of security protocols whereby two or more agents need to acquire knowledge of their identity, typically to initiate secure communication. Authentication protocols are notoriously difficult to analyse, due to the possible existence of subtle bugs such as maninthemiddle attacks and impersonation. Formal models have been employed to analyse authentication protocols; however, they are often limited to reachability analysis only, thereby imposing rather severe limitations on the class of specifications that can be verified. Specifications concerning authentication are amenable to be expressed in a temporalepistemic logic language as they concern states of knowledge of the principals in a system.
Authentication protocols expressed in CAPSL, a mainstream language for the description of security protocols, were automatically verified with MCMAS in [5]. Specifically, a compiler was built to translate protocols from the Clark–Jacobs and SPORE repository [62] into MCMAS readable input. This involved devising a translation from the SPORE protocol description into ISPL and a translation from the SPORE protocol specifications (“goals” in CAPSL) into appropriate temporalepistemic formulas. The methodology is completely automatic and was evaluated on wellknown key establishment protocols. The tool confirmed bugs already known in some key establishment protocols and verified the correctness of others. Since MCMAS also provides counterexamples when a specification is false, an attack could easily be derived by inspecting the output of the checker.
The performance of the methodology was in line with stateoftheart model checkers for security protocols. It has been argued, see, e.g. [8, 31], that epistemic specifications are considerably more intuitive for the security analyst as they refer precisely to the states of knowledge of the principals, which are the basic primitive in security analysis. This increase in expressiveness allows also the validation of other specifications, including the distributed detection of attacks at runtime [6]. Lazy intruder models are known to increase the effectiveness of model checking approaches for security protocols [3]; they can also be applied in the context of security specifications [46].
5.2.2 Verification of anonymity protocols
Anonymity protocols are a class of protocols aimed at establishing the privacy of principals during an exchange. For example, the onion routing protocol can guarantee the communication between two parties without their identities being revealed to any third party.
Onion routing and a number of other anonymity protocols have been analysed with MCMAS. One wellknown example in this class is the dining cryptographers protocol [12], which can be described as follows. A group of \(n\) cryptographers shares a meal around a circular table. Either one of them paid for the meal or their employer did. They would like to discover whether one of them paid without revealing the identity of the payer (in case one of them did pay). To this end, every cryptographer tosses a coin and shows the outcome to his righthand neighbour. Comparing his own coin to the coin shown to him, each cryptographer who did not pay for dinner announces whether the two coins agree or not. However, if a cryptographer paid for the meal, he announces the opposite of what he sees. By parity considerations, one can show that an even number of cryptographers claiming that the two coins are different entails that their employer paid for the dinner, while an odd number of “different” utterances signifies that one of the cryptographers paid for the dinner. The protocol can naturally be formalised in ISPL. The specification of the protocol can very easily be captured in an epistemic setting: after the announcements the cryptographers acquire common knowledge of the payer; but if this is one of them no one knows who this is, other than the payer itself. This analysis cannot be replicated on traditional checkers. We refer to Sect. 6.2 for more details on this scenario including the epistemic specifications checked.
5.2.3 Automatic verification of WSBPEL services
Web services consist of distributed, networked applications exchanging messages to perform a given function. A key problem in serviceoriented computing is to design and manage the service composition, whereby two or more services collaborate to achieve a certain task. Model checking has been used in this context to verify whether or not services are composed correctly according to some temporal specifications. A common approach involves modelling the services as finitestate machines and verify their composition by means of a traditional checker.
In this context, to avoid the difficulties and the errorprone task of manually encoding several services, a compiler from WSBPEL into ISPL was built [51]. WSBPEL [57] is the leading language and de facto industrial standard for service composition. The compiler parses the input and constructs an internal automatabased representation for the finitestate machines defined in the WSBPEL code and encodes these in ISPL. The resulting code can then be passed to MCMAS for verification.
This methodology was evaluated in the context of a large use case of software procurement developed within a collaborative EU project. In the use case a client company places an order for software and hardware to be deployed by a number of parties. Several providers propose the equipment to the company which is given the opportunity of changing the design a number of times before deployment. Penalties are applied to the parties should they deviate from the procurement process. The interaction continues into the integration and testing phases of the hardware with additional contracts regulating the extent to which modifications can be requested by the client, compensation claims, reports from technical experts and insurance providers. The reachable state space of the scenario consists approximately of \(10^6\) states.
In the scenario, a number of specifications pertaining to the knowledge of the parties in the exchange can be verified. For example, it can be shown that as long as the client is not in breach of contract, he knows that either the system is installed correctly or some penalty to third parties will apply. This and other specifications can be verified in a few seconds. We refer to [51] and the source code for additional details.
Other scenarios from services and business process modelling were similarly investigated by means of MCMAS. We found that scenarios generating state spaces up to \(10^{12}\) could be analysed with no difficulty.
6 Related work
The first version of MCMAS was first made available as opensource in 2003. In the past 10 years, the checker underwent a number of extensions and revisions that lead to a first documented release in [53] and a second in [47]. Unsupported, experimental releases continue and include a module to perform parameterised verification [41, 42], and dedicated for the verification of strategy logic specifications [10, 11]. This area is fast evolving; in the past few years a number of checkers have appeared that offer functionalities related to those offered by MCMAS. We compare the various functionalities and, when possible, the performance of the most prominent ones below.
6.1 Verics
Experimental results comparing the performance of MCMAS, MCK, and MCTK on the dining cryptographer protocol
Number of cryptographers  MCMAS  MCK  MCTK  

OBDD vars (\(9n+3\))  Time (s)  OBDD vars (\(6n+2\))  Time (s)  OBDD vars (\(6n+2\))  Time (s)  
5  48  0.017  32  1.401  32  0.024 
10  93  0.091  62  74.655  62  0.128 
20  183  0.667  122  47937  122  34.790 
30  273  1.476  Timeout  182  2.946  
40  363  5.053  Timeout  242  20.786  
50  453  13.437  Timeout  302  72.444  
60  543  14.180  Timeout  Timeout 
6.2 MCK
MCK was the first OBDDbased model checker supporting temporalepistemic specification [26]; it has recently been rereleased with improved functionalities including a graphical interface [67]. Its current version supports CTL\(^*\) as the underlying temporal logic. MCK implements a variety of semantics including observational semantics, perfect recall and clock semantics. Given the high computational cost of these semantics, some are supported only in limited form; for example perfect recall is only supported for one agent. Some functionality for probabilistic reasoning was also recently added [33] and an extension to bounded model checking has also recently been explored [34]. While the original version of MCK used a different OBDDhandler, the current one uses CUDD as MCMAS.
In the experiments, we used encodings of the scenario from both the MCMAS and MCK packages and adapted them to ensure the same number of variables were used in each model. In the experiments, we used observational semantics since several agents are present and no model checker supports perfect recall under this setting.
Experimental results comparing the performance of MCMAS and NuSMV on the dining cryptographer problem
Number of cryptos  MCMAS  NuSMV  

Num of OBDD vars  1.2.2 (CUDD 2.5.0)  1.0 (CUDD 2.4.1)  Num of OBDD vars  Without dyn reorder  With dyn reorder  
Time (s)  Mem (KB)  Time (s)  Mem (KB)  Time (s)  Mem (KB)  Time (s)  Mem (KB)  
10  93  0.10  11,056  0.10  10,972  62  0.87  16,060  0.09  12,668 
20  183  0.65  13,572  0.58  13,504  122  3151.35  4,891,100  0.27  13,932 
30  273  1.45  15,332  2.01  21,764  182  Overflow  2.05  17,040  
50  453  12.98  16,076  10.06  40,708  302  Overflow  8.64  21,004  
100  903  185.72  60,800  284.22  49,780  602  Overflow  117.77  42,516  
150  1353  1916  72,172  1619  91,976  902  Overflow  397  68,176  
200  1803  841.4  58,872  3057  76,608  1202  Overflow  2560  102,840  
250  2253  3040  80,680  8705  221,404  1502  Overflow  Timeout 
Experimental results comparing the performance of MCMAS and NuSMV on the card game scenario from [17]
Number of cards  MCMAS  NuSMV  

Num of OBDD vars  1.2.2 (CUDD 2.5.0)  1.0 (CUDD 2.4.1)  Num of OBDD vars  Without dyn reorder  With dyn reorder  
Time (s)  Mem (KB)  Time (s)  Mem (KB)  Time (s)  Mem (KB)  Time (s)  Mem (KB)  
8  59  0.37  11,592  0.41  11,552  56  0.16  17,772  1.58  15,880 
10  97  7.61  38,712  12.47  38,376  94  507.63  2,441,688  1035.89  154,124 
12  113  515.4  94,100  273.8  88,992  110  Overflow  Timeout  
14  129  17,783  1,191,864  10,552  539,552  126  Overflow  Timeout 
In summary, both MCK and MCMAS offer functionalities to verify epistemic logic under different semantics. They differ in the modelling language employed as well as some advanced features supported. While MCMAS also supports ATL, MCK supports a probabilistic version of epistemic logic and a very limited form of perfect recall. They are both OBDD based. Our tests appear to suggest that MCMAS is more efficient in the treatment of large state spaces; this may be due to a more effective construction of the global state space.
6.3 MCTK
MCTK is a NuSMVbased [13] model checker for knowledge and time [63, 69]. In MCTK, epistemic formulas are encoded by exploiting locality of propositions and labelling of transitions. As such, it does not support interpreted systems semantics which is a feature of MCMAS. A model checker for knowledge based on NuSMV with similar characteristics was previously presented in [45]. Note that NuSMV is among the fastest and most mature OBDDbased model checker available. As above, we compared the performance of MCTK to that of MCK and MCMAS on the same example above by adapting the encoding of the dining cryptographers protocol to ensure the same state space is present. In our tests, we found MCTK to be considerably slower than MCMAS due to the efficient implementation of the model checking algorithm for epistemic logic in MCMAS. It should be also noted that MCTK’s input is given in SMV; this is adequate for modelling reactive systems, but may not be suitable for MAS where actions and protocols feature prominently. As a further consequence of this, no support for ATL is offered in MCTK. In addition, none of the debugging facilities present in MCMAS, including counterexample generations for epistemic specifications, are offered by MCTK.
6.4 NuSMV
To evaluate MCMAS in a broader context, we now report the results obtained by comparing MCMAS to NuSMV when checking the dining cryptographer benchmark and a gametheoretical scenario from [17] against plain CTL specifications. As in previous cases, to ensure the tests are robust, we inspected and compared the resulting models and state spaces.
Both MCMAS and NuSMV offer several features to finetune certain parameters in the model checking algorithms. We used the defaults for both of them and tested both with reordering enabled and disabled. Differently from NuSMV, which often offers a better performance without reordering, MCMAS’s statespace construction is heavily dependent on reordering; given this MCMAS’s results are reported with this feature enabled. Indeed, Table 6 shows that NuSMV is faster and uses more memory when reordering is disabled. NuSMV could not verify the model with 12 cards due to memory overflow irrespective of whether or not reordering was enabled. In contrast, MCMAS could handle the cases for 12 and 14 cards before timing out with 16 cards. To present a fair comparison, we also linked MCMAS version 1.0 to CUDD 2.4.1, which is the version used by NuSMV version 2.5.4^{1} [70]. Table 6 shows the results from both versions of MCMAS.
The tables above are not intended to give a comprehensive performance evaluation of the two checkers. They are purely meant to show that MCMAS’s performance is broadly in line with NuSMV, one of the most commonly used symbolic model checkers. We expect NuSMV to be faster than MCMAS on other models not tested here.
7 Conclusions
The continuous rise in the number of autonomous systems that are being deployed has made formal verification of multiagent systems a very active area of research. In this paper, we have presented the toolkit MCMAS, a model checker supporting specifications tailoring multiagent systems. We have discussed the details of the underlying semantics, its input language, the functionalities offered and evaluated it in the context of significant use cases and other checkers.
MCMAS is released as GNU GPL opensource software and is currently used in a number of projects worldwide [21, 28, 29, 44]. Several extensions are currently being developed by various groups. Many of these extensions already have inhouse prototypes featuring, for example, abstraction, symmetry detection, and combinations with bounded model checking. This paper does not address these new features but instead focuses on the core, underlying technology of the MCMAS checker.
Footnotes
 1.
MCMAS version 1.2.2 does not support CUDD 2.4.1, as the C++ interface between CUDD 2.5.0 and CUDD 2.4.1 is considerably different.
Notes
Acknowledgments
The authors would like to thank Jakub Michaliszyn and the anonymous reviewers for valuable feedback on earlier versions of this paper.
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