ON A PRICED RESOURCE-BOUNDED ALTERNATING

µ-CALCULUS

Dario Della Monica and Giacomo Lenzi

University of Salerno, Salerno, Italy

Keywords:

µ-calculus, Multi-agent systems, Coalition logics, Bounded resources, Model checking.

Abstract:

Much attention has been devoted in artiﬁcial intelligence to the veriﬁcation of multi-agent systems and dif-

ferent logical formalisms have been proposed, such as Alternating-time Temporal Logic (ATL), Alternating

µ-calculus (AMC), and Coalition Logic (CL). Recently, logics able to express bounds on resources have been

introduced, such as RB-ATL and PRB-ATL, both of them based on ATL. The main contribution of this paper

is the introduction and the study of a new formalism for dealing with bounded resources, based on µ-calculus.

Such a formalism, called Priced Resource-Bounded Alternating µ-calculus (PRB-AMC), is an extension of

both PRB-ATL and AMC. In analogy with PRB-ATL, we introduce a price for each resource. By considering

that the resources have each a price (which may vary during the game) and that agents can buy them only if

they have enough money, several real world scenarios can be adequately described. First, we show that the

model checking problem for PRB-AMC is in EXPTIME and has a PSPACE lower bound. Then, we solve the

problem of determining the minimal cost coalition of agents. Finally, we show that the satisﬁability problem

of PRB-AMC is undecidable, when the game is played on arenas with only one state.

1 INTRODUCTION

Much attention has been devoted in the artiﬁcial in-

telligence ﬁeld to the veriﬁcation of multi-agent sys-

tems. In that regard, different logical formalisms

have been proposed, such as Alternating-time Tem-

poral Logic (ATL) (Alur et al., 2002), Alternating

µ-calculus (AMC) (Alur et al., 2002), and Coalition

Logic (CL) (Pauly, 2002). Such logics allow one to

predicate about the abilities of teams of agents with

respect to speciﬁc tasks. Recently, some efforts have

been done towards the deﬁnition of more powerful

formalisms, which are able to capture also quantita-

tive aspects related to the task to be performed. In par-

ticular, we mention RB-ATL (Alechina et al., 2009;

Alechina et al., 2010) and RAL (Bulling and Farwer,

2010). By means of formulae of these logics it is

possible to assign an endowment of resources to each

agent of a team and express the property that the team

is able to perform a given task with the available re-

sources. In (Della Monica et al., 2011), a further

variation of ATL, called Priced Resource-Bounded

Alternating-time Temporal Logic (PRB-ATL), has

been considered; in this logic a price for each re-

source is introduced and team operators are accord-

ingly extended. By means of these features, several

real world scenarios can be adequately described. All

the formalisms introduced so far are based on ATL

or CL. The main contribution of this paper is the

introduction and the study of a new formalism for

dealing with bounded resources, based on µ-calculus.

Recall that the µ-calculus is an extension of modal

logic with least and greatest ﬁxpoints of monotone

operators on sets. Intuitively, least ﬁxpoints corre-

spond to inductive deﬁnitions (e.g. liveness proper-

ties), and greatest ﬁxpoints correspond to coinductive

deﬁnitions (e.g. safety properties). Nesting ﬁxpoints

give further power to the µ-calculus so that it sub-

sumes many temporal, dynamic, and game-theoretic

logics used in system veriﬁcation, artiﬁcial intelli-

gence, game theory, etc.

The formalism we propose is called Priced

Resource-Bounded Alternating µ-calculus

(PRB-AMC). It is an extension of both AMC

and PRB-ATL.

We study the model checking problem for

PRB-AMC, which turns out to be decidable in EX-

PTIME and PSPACE-hard, analogously to what hap-

pens for PRB-ATL (Della Monica et al., 2011). We

remark that in our logic, agents can both consume

and produce resources. Note that, when production

is allowed, the model checking problem can be unde-

cidable (see, e.g, (Bulling and Farwer, 2010)). Our

222

Della Monica D. and Lenzi G..

ON A PRICED RESOURCE-BOUNDED ALTERNATING µ-CALCULUS.

DOI: 10.5220/0003750102220227

In Proceedings of the 4th International Conference on Agents and Artiﬁcial Intelligence (ICAART-2012), pages 222-227

ISBN: 978-989-8425-96-6

 2012 SCITEPRESS (Science and Technology Publications, Lda.)

decidability property is due to the fact that although

agents can produce resources, the production should

not exceed the initial availability of the resources.

Such a restriction to the notion of production makes

sense as, in practical terms, it allows one to model sig-

niﬁcant real-world scenarios, such as, acquiring mem-

ory by a program, leasing a car during a travel, and,

in general, any scenario in which an agent is releasing

resources previously acquired.

We also tackle the problem of coalition forma-

tion. How and why agents should aggregate is not

a new issue and has been deeply investigated, in

past and recent years, in various frameworks, as for

example in algorithmic game theory, argumentation

settings, and logic-based knowledge representation,

see (Wooldridge and Dunne, 2006; Dunne et al.,

2010; Bulling and Dix, 2010). Analogously to what

has been done in (Della Monica et al., 2011) for

PRB-ATL, here we face this problem in the setting of

priced resource-bounded agents with the goal speci-

ﬁed by an PRB-AMC formula. In particular we study

the problem of determining the minimal cost coali-

tions of agents acting in accordance to rules expressed

by a priced game arena and satisfying a given for-

mula. We show that also the optimal coalition prob-

lem is in EXPTIME and has a PSPACE lower bound.

Finally, we show that the satisﬁability problem of

PRB-AMC is undecidable, when the game is played

on a one-point arena, that is, the underlying graph is

constituted by a single vertex. (Notice that such an

undecidability result does not immediately extend to

generic graphs.) While the result seems to be weak

per se, we conjecture that the problem is undecidable

in the general setting and we hope to use the present

result as a preliminary step towards the proof of the

general case.

2 SYNTAX AND SEMANTICS

The scenario is the same as PRB-ATL. So, we have

a set A G of n agents, a set RES of r resources, the

set M = (N∪{∞})

of resource availabilities, the set

N = (N ∪ {∞})

of money availabilities, where N is

the set of all natural numbers 0,1,2,.... We let

b,~m

range over M and

$ range over N . Moreover, given a

vector

$, we will refer to the component correspond-

ing to the agent a as

$[a].

On the logical side, we use a set of atomic propo-

sitions Π and a set of ﬁxpoint variables VAR, to be

used in µ-calculus formulas. The syntax of formulas

is as follows:

φ ::= p | X | ¬φ | φ∧ φ | hhA

ii  φ | µX.φ(X) | ∼

where p ∈ Π, X ∈ VAR, A ⊆ A G ,

$ ∈ N ,

b ∈ M

and ∼∈ {<, >, =, ≤, ≥}. Moreover, µX.φ(X) is de-

ﬁned only when X occurs in an even number of nega-

tions in φ, so that formulas deﬁne monotonic opera-

tors on sets and we can apply Knaster-Tarski Fixpoint

Theorem (Tarski, 1955). Recall that the greatest ﬁx-

point operator νX.φ(X) can be deﬁned as usual, that

is, νX.φ(X) = ¬µX.¬φ(¬X).

The semantics is based on priced game structures

with environment, i.e., tuples G = (Q,π,ENV, d,qty,

δ,ρ). They are analogous to the priced game struc-

tures used in (Della Monica et al., 2011), the only

new ingredient being the environment ENV : VAR →

Q×M

, with which we can evaluate formulas contain-

ing ﬁxpoint variables. Recall that:

The semantics is based on priced game struc-

tures with environment analogous to the ones used

in (Della Monica et al., 2011), i.e. tuples G =

(Q,π,ENV,d,qty, δ, ρ); here there is one extra fea-

ture, that is an environment ENV : VAR → 2

Q×M

with which we can evaluate formulas containing ﬁx-

point variables. Recall that:

• Q is a ﬁnite set of locations, usually denoted

q,q

,....

• π : Q → 2

is a labeling function assigning to each

location the set of all atomic propositions which

are true on it.

• d(q,a) is the number of actions available for the

agent a on state q. We code actions with num-

bers from 1 to d(q,a). We assume that d(q,a) ≥ 1

(there is always at least one action available) and

the action 1 means “doing nothing”.

For each location q ∈ Q and team A =

,... ,a

} ⊆ A G , we denote by D

(q) the set

of action proﬁles available to the team A at the

location q, deﬁned as D

(q) = {1,..., d(q,a

)}×

... × {1,..., d(q, a

)}. For the sake of readability,

we denote D

A G

(q) by D(q). Given a team A, an

agent a ∈ A, and an action proﬁle

, we will re-

fer to the component of the vector

correspond-

ing to the agent a as

[a]. Actions (resp., action

proﬁles) are usually denoted by α,α

,... (resp.,

α,

,...).

• qty(q,a,α) is an element of Z

representing the

quantity of resources consumed or produced by

the agent a while performing the action α ∈

d(q,a) on the location q (Z is the set of integers).

Positive components represent resource produc-

tions, negative ones represent resource consump-

tions. qty(q, a, 1) is the zero vector, for all q ∈ Q,

a ∈ A G . With an abuse of notation we also de-

note by qty the function deﬁning the amount of re-

sources required by an action proﬁle

∈ D

(q),

that is qty(q,

) =

∑

a∈A

qty(q,a,

(a)).

ON A PRICED RESOURCE-BOUNDED ALTERNATING μ-CALCULUS

223

• δ(q,hα

,... ,α

i) is the transition function giving

the state reached from q when the n agents per-

form the action proﬁle hα

,... ,α

i ∈ D(q).

• ρ(~m,q,a) is the price of the r resources depending

on resource availability ~m ∈ M , the location q ∈

Q, and the agent a.

In order to deﬁne the semantics of PRB-AMC,

we must introduce the notion of strategy. Unlike

(Della Monica et al., 2011), here it is enough to con-

sider only one-step strategies.

Let us ﬁx the initial global availability of resources

and let A be a set of agents. A one-step strategy

for A is a function giving for each (q,~m) ∈ Q× M

an action proﬁle

containing a move

α[a] for each

a ∈ A. The outcome of a one-step strategy on (q,~m)

is the set of all conﬁgurations (q

′

,~m

′

) ∈ Q × M such

that there is an extension

A G

to A G such that:

• q

′

= δ(q,

A G

• ~m

′

= ~m + qty(q,

A G

• 0 ≤ ~m+ qty(q,

A G \A

) ≤ ~m

where

A G \A

is the restriction of

A G

to A G \ A.

A one-step (

$,~m

)-strategy F

is a strategy such

that for every (q

′

,~m

′

) in the outcome of F

on (q,~m),

we have:

• 0 ≤ ~m

′

≤ ~m

;

• ρ(~m,q,a)·consumed(q, a, F

(q)[a]) ≤ $[a], for all

a ∈ A,

where consumed(q,a,α) is obtained from qty(q,a,α)

by replacing the positive components, representing

resource productions, with zeros, and the negative

ones, representing resource consumptions, with their

absolute values.

We deﬁne the semantics of our logic in two steps.

As a ﬁrst step, we deﬁne a preliminary pre-modelhood

relation, and as a second step, we deﬁne the proper

modelhood relation, that makes use of the former one.

The pre-modelhood relation is a quinary relation, de-

noted by:

G,~m

,q,~m |=

φ,

where G is a priced game structure with environment,

is the initial availability, q is a location, ~m is the

current availability and φ is a formula.

We always suppose that ~m ≤ ~m

and ~m

has the

same inﬁnite components as ~m.

The deﬁnition of |=

is by induction on φ, and the

clauses are:

• G,~m

,q,~m |=

p iff p ∈ π(q);

• G,~m

,q,~m |=

X iff (q,~m) ∈ ENV(X);

• G,~m

,q,~m |=

¬φ iff not G,~m

,q,~m |=

φ;

• G,~m

,q,~m |=

φ ∧ φ

′

iff G,~m

,q,~m |=

φ and

G,~m

,q,~m |=

′

;

• G,~m

,q,~m |=

hhA

ii  φ iff there exists a

(

$,~m

)-strategy F

such that, for all conﬁgura-

tions (q

′

,~m

′

) in the output of F

, it holds that

G,~m

′

,~m

′

|= φ;

• G,~m

,q,~m |=

µX.φ(X) iff (q,~m) belongs to the

smallest set E such that E = {(q

′

,~m

′

)|G[X :=

E],~m

′

,~m

′

φ}, where G[X := E] is the same

priced structure with environment as G, except

that ENV(X) = E;

• G,~m

,q,~m |=

∼

b iff ~m ∼

Finally, the proper modelhood relation is deﬁned:

G,q,~m |= φ ↔ G,~m,q,~m |=

φ.

3 EXPRESSIVENESS

Recall from (Della Monica et al., 2011) that

PRB-ATL has the following syntax:

φ ::= p | ¬φ | φ∧ φ | hhA

ii  φ | hhA

iiφU φ

| hhA

iiφ | ∼

where p ∈ Π, A ⊆ A G ,

$ ∈ N ,

b ∈ M and

∼∈ {<,>,=,≤,≥}.

Intuitively hhA

iiφU φ

′

means that A can ensure φ

until φ

′

holds, and hhA

iiφ means that A can ensure

that φ holds forever.

So, PRB-ATL extends ATL, hence also the tem-

poral logic CTL. Moreover, it is well known that

CTL (resp., ATL) can be efﬁciently translated into

µ-calculus (resp., the alternation-free fragment of

AMC), but not conversely, and that CTL

∗

(resp.,

ATL

∗

) can be translated into the µ-calculus (resp.,

AMC), but not conversely.

In our more general setting, we extend the previ-

ous results as follows:

Theorem 3.1. PRB-ATL can be translated in

PRB-AMC.

Proof. The proof hinges on the model checking algo-

rithm for PRB-ATL. In fact, in order to make it clear

that these operators are ﬁxpoint deﬁnable, it sufﬁces

to rewrite the subroutines of the model checking al-

gorithm 1 of (Della Monica et al., 2011) for the oper-

ators hhA

iiφ

U φ

and hhA

iiφ.

We intend that the vector

$ can contain ﬁnite and

inﬁnite components. The rewriting process goes by

induction on the sum of the ﬁnite components of

We say that

$ is zero-inﬁnite if it consists only of

zeros and inﬁnites, and, for every

$, we denote by

the least vector with the same inﬁnite components as

(which is necessarily zero-inﬁnite). In the algorithms

we assume the convention ∞ − ∞ = ∞.

Rather than distinguishing two subroutines

for zero and nonzero money assignments as in

ICAART 2012 - International Conference on Agents and Artificial Intelligence

224

(Della Monica et al., 2011), we distinguish two

subroutines for zero-inﬁnite and non-zero-inﬁnite

money assignments.

In all our subroutines we replace the Pre operators

with next operators hhA

iiφ, which are available in

PRB-AMC.

We ﬁx a priced arena with environment G and an

initial availability ~m. Given a formula φ, we use the

notation [φ] to denote the set {(q

′

,~m

′

) | G,~m,q

′

,~m

′

φ}, where |=

is the auxiliary pre-modelhood relation

deﬁned in the previous section. By deﬁnition of the

proper modelhood relation |=, we have (q,~m) ∈ [φ] if

and only if G,q,~m |= φ, for each q ∈ Q.

Let us begin with the subroutine for φ =

hhA

iiψ

U ψ

when

$ is zero-inﬁnite.

1: τ ← [ f alse]

2: σ ← [ψ

]

3: while τ 6= σ do

4: τ ← σ

5: σ ← τ∪ ([hhA

ii  τ] ∩ [ψ

])

6: end while

7: [φ] ← σ

Now we observe that the while loop (line 3) cal-

culates ﬁxpoints. More precisely, it is equivalent to

a simultaneous assignment σ,τ := µX.ψ

∨(hhA

ii

X ∧ ψ

). By replacing the while loop with a ﬁxpoint

assignment we obtain the algorithm:

1: τ ← [ f alse]

2: σ ← [ψ

]

3: σ,τ ← µX.ψ

∨ (hhA

ii  X ∧ ψ

)

4: [φ] ← σ

where it is clear that the semantics of φ is deﬁnable in

PRB-AMC.

Likewise, if

$ is not zero-inﬁnite then we have:

1: τ ← [hhA

iiψ

U ψ

]

2: for all

′

$ with the same inﬁnites as

$ do

3: σ ← τ∪ ([hhA

$−

′

ii  hhA

′

iiψ

U ψ

] ∩ [ψ

])

4: while τ 6= σ do

5: τ ← σ

6: σ ← τ∪ ([hhA

ii  τ] ∩ [ψ

])

7: end while

8: end for

9: [φ] ← σ

The ﬁrst line of the algorithm is a ﬁxpoint def-

inition by the zero-inﬁnite case. Moreover, in each

iteration of the for loop (line 2), the ﬁrst assigment is

a ﬁxpoint deﬁnition by induction, and the while loop

(line 4) is equivalent to a ﬁxpoint assignment on the

variables σ and τ. By replacing the while loop with

this ﬁxpoint assignment, we have a ﬁxpoint deﬁnition

of σ and τ at the end of every iteration of the for loop.

So, at the end of the algorithm we have a ﬁxpoint def-

inition of the semantics of φ.

The situation is analogous for φ = hhA

iiψ. Let

us begin with the subroutine for hhA

iiψ when

$ is

zero-inﬁnite.

1: τ ← [true]

2: σ ← [ψ]

3: while τ 6= σ do

4: τ ← σ

5: σ ← [hhA

ii  τ] ∩ [ψ]

6: end while

7: [φ] ← σ

In this case, the while loop (line 3) calculates a

greatest ﬁxpoint, i.e., σ,τ := νX.hhA

ii  X ∧ ψ. By

replacing the while loop with a ﬁxpoint assignment,

we have a ﬁxpoint deﬁnition of the semantics of φ.

Finally, if

$ is not zero-inﬁnite then we have:

1: τ ← [hhA

iiψ]

2: for all

′

$ with the same inﬁnites as

$ do

3: σ ← τ ∪ ([hhA

$−

′

ii  hhA

′

iiψ] ∩ [ψ])

4: while τ 6= σ do

5: τ ← σ

6: σ ← τ ∪ ([hhA

ii  τ] ∩ [ψ])

7: end while

8: end for

9: [φ] ← σ

The ﬁrst line of the algorithm is a ﬁxpoint deﬁni-

tion by the zero-inﬁnite case. Moreover, in each iter-

ation of the for loop (line 2), the ﬁrst line is a ﬁxpoint

deﬁnition by induction, and the while loop (line 4)

calculates a least ﬁxpoint. So by replacing the while

loop with a ﬁxpoint assignment on σ and τ, we have a

ﬁxpoint deﬁnition of σ and τ at the end of every iter-

ation of the for loop. At the end of the algorithm, we

have a ﬁxpoint deﬁnition of the semantics of φ.

Notice that the existence of an efﬁcient translation

from PRB-ATL to PRB-AMC (like the one of CTL

into µ-calculus) is an open problem currently under

investigation.

4 MODEL CHECKING

In (Della Monica et al., 2011), the authors consider

the model checking problem for PRB-ATL, proving

ON A PRICED RESOURCE-BOUNDED ALTERNATING μ-CALCULUS

225

that it is in EXPTIME and it is PSPACE-hard. In this

paper, we extend these results to PRB-AMC.

Theorem 4.1. The model checking problem for

PRB-AMC is in EXPTIME and it is PSPACE-hard.

Proof. The PSPACE-hardness directly follows from

the one of the model checking problem for PRB-ATL.

To prove the EXPTIME upper bound, we pro-

vide an exponential time recursive algorithm, called

set (see Algorithm 1, where M

≤~m

′

denotes the set

{~m ∈ M | ~m ≤ ~m

′

}, for a resource availability ~m

′

∈

M ), which, given a priced game structure with envi-

ronment G, a formula φ, and a resource availability

′

, outputs the set of all conﬁgurations (q,~m), with

~m ≤ ~m

′

, which verify φ in G. The algorithm is a com-

bination of those in (Della Monica et al., 2011) and

(Emerson, 1996).

Note that the time complexity of the algorithm

is O((|G| × |M|

)

|φ|

), while the space complexity is

O(|G| × |M|

), where M is the maximum component

occurring in the initial resource availability vector ~m

′

Finally, in order to check whether a formula φ

is true over a game structure G and a conﬁguration

(q,~m) in G, the model checking algorithm simply

consists in verifying if (q,~m) belongs to the output

of set(φ,G,~m).

Algorithm 1: set(φ, G,~m

′

)

// computes the set of configurations

(q,~m)

such that

~m ≤ ~m

′

and

G,~m

′

,q, ~m |=

φ.

1: if φ =∼

b then

2: return {(q, ~m) | ~m ∼

b and ~m ≤ ~m

′

}

3: else if φ = p /* p ∈ Π */ then

4: return {(q, ~m) | p ∈ π(q),~m ≤ ~m

′

}

5: else if φ = X /* X ∈ VAR */ then

6: return {(q, ~m) | (q, ~m) ∈ ENV(X),~m ≤ ~m

′

}

7: else if φ = ¬ψ then

8: return (Q× M

≤~m

′

) \ set(ψ, G, ~m

′

)

9: else if φ = ψ

∧ ψ

then

10: return set(ψ

,G,~m

′

) ∩ set(ψ

,G, ~m

′

)

11: else if φ = hhA

ii  ψ then

12: return Pre(A,ψ,

$,G,~m

′

)

13: else if φ = µX.ψ(X) then

14: X

′

←

15: X ← set(ψ(X), G,~m

′

)

16: while X

′

6= X do

17: X

′

= X

18: X = set(ψ(X), G,~m

′

)

19: end while

20: return X

21: end if

Observe that the problem is PSPACE-complete

when the number of resources is constant.

5 THE OPTIMAL COALITION

PROBLEM

In (Della Monica et al., 2011), an optimality prob-

lem is introduced, called the Optimal Coalition prob-

lem (OC). This is the problem of ﬁnding the coali-

tions which achieve the given formulas with least

cost, if such coalitions exist. Formally, we intro-

duce team variables Y

,... ,Y

(we use Y to avoid

confusion with ﬁxpoint variables), and we admit for-

mulas φ(Y

,... ,Y

) containing the team variables

,... ,Y

(in place of some of the teams) with the cor-

responding money endowments

,... ,

. We denote

by φ[Y

,... ,Y

,... ,A

] the formula in which each

team variableY

is replaced by the team A

⊆ A G . We

ﬁx a priced game structure G, a location q of G and

an initial global availability ~m. The output is a triple

hres,A

∗

,costi where:

• res ∈ {true, false} and res = true iff there is a

vector of teams hA

,... ,A

i such that G, q,~m |=

φ[Y

,... ,Y

,... ,A

];

• if res = true, A

∗

is a vector which minimizes the

cost (otherwise A

∗

is undeﬁned);

• cost = Σ

i=1

·A

is the cost of the vector of teams,

where A

is the characteristic vector of A

seen as

a subset of A G , and · denotes scalar product be-

tween vectors.

We have the following result:

Theorem 5.1. In PRB-AMC, the OC problem is in

EXPTIME and it is PSPACE-hard.

Proof. We check the cost of all possible (2

)

vectors

of teams by calling each time the model checking al-

gorithm of the previous section. As we have seen, this

algorithm is in EXPTIME; so also the OC problem is.

The PSPACE-hardness follows from hardness of

the decisional version, and hardness of the latter fol-

lows from the proof of Theorem 3.2 of (Della Monica

et al., 2011) (again because the PRB-ATL formulas

used there actually belong to PRB-AMC).

6 AN UNDECIDABILITY RESULT

In this section we show the following result:

Theorem 6.1. It is undecidable whether a formula of

PRB-AMC is satisﬁable in a one point arena (i.e. an

arena where Q is a singleton).

To prove the theorem we reduce to our satisﬁabil-

ity problem a well-known undecidable problem, the

solvability of equations A(n) = B(n), where n is a

vector of variables ranging over N and A and B are

ICAART 2012 - International Conference on Agents and Artificial Intelligence

226

polynomials with coefﬁcients in N, see (Matiyase-

vich, 1993). In this section we let the letters m,n, p,...

range over N.

The ﬁrst step of the reduction, which is standard,

is to start from an equation A(n) = B(n) and to ex-

press solvability of the equation via solvability of a

ﬁnite system Σ(A,B) of equations of the form m = a

(with a ∈ N) , m = n+ p and m = n× p. The second

step is the following lemma:

Lemma 6.1. Let Σ be a ﬁnite system of relations of

the form m = a (a ∈ N), m = n + p and m = n × p,

with a set X of unknown variables. Then one can ﬁnd

effectively:

• a set R

of resources and a subset Q

of R

• a formula A

of PRB-AMC over R

satisﬁable in

a one point arena

• a formula ∆

of PRB-AMC

such that in every one point model M of A

, Σ holds

in M if and only if M veriﬁes ∃Q

∆

The proof of the lemma is omitted for lack of

space and will be provided in a future extended ver-

sion. Now, the theorem follows from the next Corol-

lary of Lemma 6.1.

Corollary 6.1. Let Σ be a ﬁnite system of equations

of the form m = a (a ∈ N), m = n+ p and m = n× p.

Then one can ﬁnd effectively:

• a set R

of resources

• a formula A

over R

• a formula ∆

of PRB-AMC

such that Σ is solvable if and only if A

∧∆

is satisﬁ-

able in a one point arena in PRB-AMC.

7 CONCLUSIONS

In this paper, we have presented an extension of µ-

calculus, called PRB-AMC, suitable for modeling

collective behavior of groups of agents acting in envi-

ronment where resource availability is limited.

The present work follows previous approaches in

that direction (Alechina et al., 2010; Bulling and

Farwer, 2010; Della Monica et al., 2011), the main

difference being the formalism underlying the logic,

namely, the µ-calculus instead of the Alternating-

time Temporal Logic. Even though our logic is

more expressive than logics introduced in previous

work, in particular PRB-ATL, the complexity of both

the model checking problem and the optimal coali-

tion problem is not harder than in PRB-ATL, i.e,

EXPTIME with PSPACE lower bound. The exact

complexity of both problems is conjectured to be

EXPTIME-complete. Additionally, we have explored

the satisﬁability problem for PRB-AMC, proving its

undecidability in the particular case when the game

structure is an arena with only one state. The satis-

ﬁability problem in the general case is an interesting

open problem currently under study.

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