Alternating-time temporal logic with resource bounds

Many problems in AI and multi-agent systems research are most naturally formulated in terms of the abilities of a coalition of agents. There exist several excellent logical tools for reasoning about coalitional ability. However, coalitional ability can be affected by the availability of resources, and there is no straightforward way of reasoning about resource requirements in logics such as Coalition Logic (CL) and Alternating-time Temporal Logic (ATL). In this article, we describe a logic for reasoning about coalitional ability under resource constraints.We extendATLwith costs of actions and hence of strategies.We give a complete and sound axiomatization of the resulting logic, Resource-Bounded ATL (RB-ATL) and a model-checking algorithm for it.


Introduction
In many situations, a group of agents can cooperate to achieve an outcome which cannot be achieved by any agent in the group acting individually.For example, in the prisoners dilemma, a single prisoner cannot ensure the optimal outcome, while a coalition of two prisoners can.Similarly, it may be possible for a set of cooperating agents to solve a difficult computational problem by distributing it, while a single agent may not have sufficient memory or processor power to solve it.In the latter case, there is an interaction between the amount of resources available to the agents (or the amount of resources which they are willing to contribute), and their ability to jointly achieve a goal.
In this article we describe a logic, Resource-Bounded Alternating-time Temporal Logic (RB-ATL), for reasoning about coalitional ability under resource constraints.RB-ATL allows us to express and verify properties such as: (1) 'a coalition of agents A has a strategy to achieve a property ϕ provided they have resources b, but they cannot enforce ϕ under a tighter resource bound b 1 '; (2) 'A has a strategy to maintain the property ϕ, provided they have resources b'; and (3) 'A has a strategy to maintain ϕ until ψ becomes true, provided A has resources b'.
In Section 2.4, we illustrate the expressive power of RB-ATL on a simple example of a sensor network, where the agents (sensor nodes) require two resources: energy and memory.
In previous work, we studied a version of Coalition Logic (CL) with resource bounds, RBCL [5].RBCL can express properties of the form (1) above, but not of the form (2) and (3).Other work on temporal logics and logics of coalitional ability with resource constraints includes [2,7,8,10,11].However, this work concentrates on model-checking complexity, rather than axiomatization, which is the focus of this article.
This article is a revised and extended version of [4].In [4], we gave a sound and complete axiomatization and a model-checking algorithm for a version of RB-ATL without infinite resource bounds.The main differences from [4] are the addition of an infinite resource bound (to make the logic a conservative extension of ATL), and the addition of complete proofs and an illustrative example. 1  The remainder of this article is organized as follows.In Section 2, we present the syntax and semantics of RB-ATL and show how RB-ATL can be used to express properties of a simple sensor network.In Section 3 we provide a sound and complete axiomatization of RB-ATL.In Section 4, we give a model-checking algorithm for RB-ATL.Finally, we survey related work in Section 5 and conclude in Section 6.

Syntax and semantics of RB-ATL
Consider a system of agents that can perform actions to change the state (we assume concurrent execution of actions by all agents).We denote the set of agents by N. To reason about resources, we assume that actions have costs.Let R be a set of resources (such as money, energy, or anything else which may be required by an agent for performing an action).We assume that a cost of an action, for each of the resources, is a non-negative integer.The set of resource bounds B over R is defined as B = (N∪{∞}) r , where r =|R|.We denote by 0 the smallest resource bound (0,...,0) and ∞ the greatest resource bound (∞,...,∞).

Syntax of RB-ATL
The syntax of RB-ATL is defined as follows, where A is a non-empty subset of N and b ∈ B. ϕ means that A has a strategy to make sure that ϕ is always true, and the cost of this strategy is at most b.Similarly, A b ϕ Uψ means that A has a strategy to enforce ψ while maintaining the truth of ϕ, and the cost of this strategy is at most b.Notice the meaning of these operators when b = ∞ is the same as their counterparts in ATL; in other words, the ATL operator A corresponds to A ∞ for A =∅ and ∅ to the dual of N ∞ .

Semantics of RB-ATL
To interpret this language, we extend the definition of concurrent game structures [6] with resource requirements for executing actions.For consistency with [6], in what follows we refer to agents as 'players' and actions as 'moves'.

Definition 1
A Resource-bounded Concurrent Game Structure (RB-CGS) is a tuple S = (n,r,Q, ,π,d,c,δ) where: • n ≥ 1 is the number of players (agents), we denote the set of players {1,...,n} by N; 1 A preliminary version of RB-ATL with infinite bounds was introduced in [14].
• r is the number of resources; • Q is a non-empty set of states; • is a finite set of propositional variables; • π : → ℘(Q) is a function which assigns to each variable in a subset of Q; • d : Q×N → N is a function which indicates the number of available moves (actions) for each player a ∈ N at a state q ∈ Q such that d(q,a) ≥ 1.At each state q ∈ Q, we denote the set of joint moves available for all players in N by D(q).That is D(q) ={1,...,d(q,1)}×...×{1,...,d(q,n)} • c : Q×N ×N → B is a partial function which indicates the minimal amount of resources required by each move available to each agent at a specific state; • δ : Q×N |N| → Q is a partial function where δ(q,m) is the next state from q if the players execute the move m ∈ D(q).
We assume that each agent in each state has an available action with 0 cost (intuitively, it has the option of doing nothing).
Given a RB-CGS S, we denote by Q * the set of finite sequences of states or finite computations and by Q ω the set of infinite sequences of states or infinite computations.For a finite or infinite computation λ = q 1 q 2 ... ∈ Q * ∪Q ω , we use the notation λ[i]=q i and λ[i,j]=q i ...q j .We denote the set of finite non-empty sequences of states by Q + .

Definition 2
Given a RB-CGS S and a state q ∈ Q, a move (or a joint action) for a coalition A ⊆ N is a tuple σ A = (σ a ) a∈A such that 1 ≤ σ a ≤ d(q,a).
By D A (q) we denote the set of all moves for A at state q.Given a move m ∈ D(q), we denote by m A the actions executed by A, m A = (m a ) a∈A .We define the set of all possible outcomes of a move σ A ∈ D A (q) at state q as follows: For convenience, we define the projection of ∞ components of a resource bound b on another bound d as d ∞ ← b where for all i ∈{1,...,r}: The cost of a move σ A ∈ D A (q) is defined as cost(q,σ A ) = a∈A c(q,a,σ a ).(Note that we use c for the cost of single actions and cost for the cost of joint actions.)

Definition 3
Given a RB-CGS S, a strategy for a subset of players A ⊆ N is a mapping F A which associates each sequence λq ∈ Q + to a move in D A (q). [1,i])).We denote by out(q,F A ) the set of all such sequences λ starting from q, i.e. λ [1]=q.Given a non-empty finite prefix λ of a computation which is consistent with a strategy F A , we define the cost of In other words, all executions of a b-strategy cost at most b resources.Note that this means that each computation of such a strategy starts with a finite prefix where some non-( 0 ∞ ← b) cost actions are executed, and continues with an infinite sequence of ( 0 ∞ ← b)-cost actions.
Notice that the truth definition of A ∞ is the same as that of A in ATL.

Example
To conclude this section, we describe a concrete scenario to illustrate the notions introduced above, and give some examples of the expressive power of RB-ATL.Consider a sensor network consisting of two agents (sensor nodes), 1 and 2. The agents monitor for movement.If they detect movement, they can inform their neighbour.If an agent receives a communication from its neighbour, it can save it.If an agent has more than one record of movement, the agent can report this to the base station.We assume that 2 is closer to the base station than 1.We consider two resources, energy and memory.Sending a message requires energy (depending on the distance to the receiver) and saving a communication requires memory.Sending from 1 to 2 (send12) and from 2 to 1 (send21) both require 2 units of energy and 0 memory.Saving a record requires 0 units of energy and 1 unit of memory.Sending from 1 to the base station (send1b) requires 3 units of energy, and sending from 2 to the base (send2b) requires 1 unit of energy.The option of doing nothing (idle) is always available and costs nothing.In the initial state q 0 , each agent has a record of having itself seen movement.The system is shown in Figure 1, where transitions between states are annotated with tuples of actions (the first element is an action by agent 1, and the second is an action by agent 2).We omit self-loops in each state by the joint action (idle,idle) for readability.
In this scenario, both agents together can enforce the outcome q 6 , where we assume that a proposition p (which means that the base station has been informed) holds.Moreover, they can achieve this by spending 3 units of energy and 1 unit of memory by choosing the following actions: (send12,idle) in q 0 , (idle,save) in q 1 and (idle,send2b) in q 4 .This can be expressed in RB-ATL as {1,2} (3,1)  Up.It is also the case that the agents cannot achieve this without using some memory, even if they use unlimited energy: ¬ {1,2} (∞,0)  Up.Clearly, neither of the agents can single-handedly enforce q 6 ; however, once the system is in q 6 , either agent can trivially maintain p forever, since the only choice of action available to each agent there is idle.This can be expressed as {1,2} (3,1)  U {1} 0,0 p.

Axiomatization
In this section we present the axiomatic system for RB-ATL.To make the formulas below more readable, we define the following abbreviations: The axiomatic system consists of the following axiom schemas and rules of inference, where A, A 1 and A 2 are non-empty subsets of N, and b, d ∈ B. Axioms: Inference rules: A b ϕ Uψ → θ Before proving soundness and completeness, i.e. every formula derived by the above system is valid and every valid formula can be derived by the above system, we give an intuitive explanation of the axioms and compare them with the axiomatic system for ATL given in [13].
First of all, observe that with the resource bounds removed, the axioms (⊥), ( ), (S), and the inference rules ( A b -Monotonicity) and ( N b -Necessitation) are identical to their ATL counterparts.Unlike ATL, we need several versions of (S) since we do not have the ∅ b modality and, as a result, (S N ), (S N+ ) and (S N− ) are not derivable from (S).The axiom (B) says that if A can enforce ϕ under a resource bound b, then it can also enforce ϕ if it has more than b resources.The axiom (FP 2 ) is similar to its ATL counterpart.However, unlike in ATL, there are two ways to 'unwind' A b ϕ in RB-ATL: one way is to make a move which costs a non-trivial amount of resources d, and then maintain ϕ with b−d resources; the second way is to make a trivial ( 0 ∞ ← b)-cost move, and then maintain ϕ with b resources.Similarly for (FP U ). Finally, the rules ( A b -Induction) and ( A b U-Induction) correspond to the ATL axioms (GFP 2 ) and (LFP U ); the first one says that 2 corresponds to the greatest fixed point and the second that U corresponds to the least fixed point.This will be made more precise after we give fixed point characterizations of the temporal operators.

Consider an operation [ A b
], which, given a set of states X, returns the set of states from where A can enforce an outcome to be in X under resource bound b (this is the same as Pre(A,X,b) defined in Section 4, which is in turn similar to Pre from [6]): By the Knaster-Tarski theorem, f has the least and the greatest fixed point.The least fixed point of f is denoted by μX.f (X) and the greatest fixed point by νX.f (X).We are going to show that the meanings of 2 and U correspond to the greatest and the least fixed points of certain operations on sets of states.

Lemma 1
For all q ∈ Q, the following fixed point characterizations hold: iff there is a b-strategy F A for A such that for all λ ∈ out(q,F A ), there exists i ≥ 1 such that λ[i]∈ ψ and λ[j]∈ ϕ for all j < i Proof.We will only provide the proof for the first case as the second can be done in a similar way.For convenience, let us denote ](X 2 ); hence q ∈ f (X 2 ).Therefore, f (X) is monotone and there is the greatest fixed point νX.f (X).We now show that , by the semantics definition, we have that there is a b-strategy F A such that for any λ ∈ out(q,F A ), λ[i]∈ ϕ for all i ≥ 1.Then, For each q ∈ out(q,F A (q)), we define a b -strategy F q as the remainder of F A from q , i.e.F q ,A (q κ) = F A (qq κ) for all κ ∈ Q * .Then, for all q κ ∈ out(q ,F q ,A ), we have that qq κ ∈ out(q,F A ).It is straightforward that any computation in out(q ,F q ,A ) costs at most b .Then, q ∈ A b ϕ .
ϕ is in fact the greatest fixed point of f (X), we show that, for every post-fixed point Z, Z ⊆ Y .
We have Assume q ∈ Z, we have: We define a b-strategy F A which will maintain ϕ by induction on the length of inputs for F A .Let i denote the set of inputs of length i ≥ 1 for F A .Initially, 1 ={q}.We will define F A for input of length i and i+1 inductively on i ≥ 1 such that, for all λ ∈ i+1 , either cost(λ,F A ) ≤ d and ϕ ; then, we have: Then, we define F A (q) = σ A and 2 ={qq | q ∈ out(q,σ A )}. Obviously, we have: ](Z), we have: Then, we have: • Case i > 1, let us assume that F A for inputs of length i−1 and i have been defined.By the induction hypothesis, we have, for all λ ∈ i , either cost(λ,F A ) ≤ d and λ ϕ , we have: Furthermore, by considering the (d −d )-strategy F A,λq where F A,λq (λ ) = F A,λ (q λ ) for all λ ∈ Q + , we have: ϕ ; then, we have: Then, we define for every λ Obviously, we have: ](Z), we have: Obviously, we have: Given the above construction of F A , we have that In other words, q ∈ A b ϕ , i.e., q ∈ Y .
Therefore, Z ⊆ Y ; hence, Y is the greatest post-fixed point of f (X), hence also the greatest fixed point of f (X).

Soundness of RB-ATL
First, we prove that the axioms of RB-ATL are valid.
then the same F A is also a d-strategy which has the same property.(S) is valid because if there exists a strategy F A 1 to enforce ϕ and a strategy F A 2 to enforce ψ, then there exists a joint strategy F A 1 ∪A 2 (with the same moves for A 1 and A 2 as F A 1 and F A 2 , respectively) to enforce both ϕ and ψ. (S N ) is valid because if there exists a b-strategy F A to enforce ϕ, and, for all strategies of N, ψ is inevitable, then ϕ ∧ψ can be enforced in by F A .(S N+ ) is valid because if there exists a b-strategy F N to enforce ϕ, and, for all strategies of N which cost at most b, ψ is inevitable, then ϕ ∧ψ can be enforced in by F N .(S N− ) is valid because if, for all strategies of N, ϕ and ψ are inevitable, then so is ϕ ∧ψ.(FP 2 ) is valid by Lemma 1(1) and (FP U ) by Lemma 1 (2).
Then, we prove that the inference rules preserve validity (the proof for (MP) is standard, hence it is omitted):

Completeness of RB-ATL
The proof of completeness is based on [13].We construct a satisfying model for a formula ϕ 0 which is consistent with the axiomatic system for RB-ATL.
In the proof, we assume when convenient that all formulas are in negation normal form of RB-ATL.The syntax of negation normal form RB-ATL is as follows: where A is a non-empty coalition and b ∈ B. Given a normal form formula ϕ of RB-ATL, we denote by ∼ϕ the normal form negation of ϕ.Given an RB-CGS S and a state q, the semantics of normal form RB-ATL is the same as RB-ATL, except for formulas ¬ A b ϕ, ¬ A b ϕ and ¬ A b ϕ Uψ which are defined as follows: • S,q |= ¬ A b ϕ iff for every b-strategy F A , there exists λ ∈ out(q,F A ) such that S,λ [1]|=∼ϕ The model is constructed in a way very similar to the construction in [13].It is assembled from finite trees where nodes are labelled by sets of formulas.First, we define the set of formulas used in the labelling.

Definition 6
The closure cl(ϕ 0 ) is the smallest set of formulas satisfying the following closure conditions: • all sub-formulas of ϕ 0 including ϕ 0 itself are in cl(ϕ 0 ); , then so is ∼ϕ; and • cl(ϕ 0 ) is also closed under finite positive Boolean operators (∨ and ∧) up to tautology equivalence.
Note that cl(ϕ 0 ) is finite.Let be the set of maximal consistent subsets of cl(ϕ 0 ).We define trees (T ,V ,C) over in a similar way as [13] where is a labelling function that assigns to each node a consistent set; and • C : T ×N ×N → N r is a (partial) cost function that assigns a cost to each action available at a node.
Intuitively, nodes in a tree are identified with finite words corresponding to the sequence of joint actions by the grand coalition which leads to that node.The root is the empty word and each node t corresponds to a finite computation the last state of which is t.An interior node of the tree is a node but not a leaf.A formula is in V (t) intuitively means that the formula is true in t.Finally, the cost of an action j of an agent i at a node t is given by C(t,i,j).As in [13], the construction proceeds in three stages.The first stage is producing locally consistent trees, namely trees where the labelling satisfies conditions on successor nodes which makes it possible to prove a truth lemma for the next step modalities.The second stage is proving the existence of trees which realize eventualities (essentially, make the labelling consistent with the truth conditions for the 2 and U modalities).Finally, the finite trees realizing eventualities are combined into one infinite tree model.

Definition 7
A tree (T ,V ,C) is locally consistent iff for any interior node t ∈ T : (1) If A b ϕ in V (t), then there is a move σ A such that cost(t,σ A ) ≤ b and for any t ∈ out(t,σ A ) we have ϕ ∈ V (t ); and (2) , then for any move σ A with cost(t,σ A ) ≤ b, there exists t ∈ out(t,σ A ) where ¬ϕ ∈ V (t ).
Two following lemmas are used as a crucial step in the local consistency proof.
Proof.When k = 0 (or m = 0), we can always add the axiom A 0 (or ¬ N ∞ ⊥) into .Hence, it is sufficient to prove this lemma with k > 0 and m > 0.
Let A = i A i , b = i b i and ϕ = i ϕ i and χ = j ∼χ j .Assume to the contrary that is inconsistent, we have: by ( 1), ( 2) and (3) by ( 4) and ( 6) Similarly, we have the following lemma: χ m } be a consistent set of formulas in which: • The A i 's are both non-empty and pair-wise disjoint • i b i ≤ d j for all j Then, ={ϕ 1 ,...,ϕ k ,∼χ 1 ,...,∼χ m } is also consistent.
Proof.When k = 0 (or m = 0), we can always add the axiom N 0 (or ¬ N ∞ ⊥) into .Hence, it is sufficient to prove this lemma with k > 0 and m > 0.
Let A = i A i , b = i b i and ϕ = i ϕ i and χ = j ∼χ j .Assume to the contrary that is inconsistent, we have: ( by ( 1), ( 2), ( 3) and ( 4) by ( 5) and ( 7) Let be a finite consistent set of formulas.Let be the subset of which contains all formulas of the form A b ϕ or their negations.Let k ∈ N be such that | | < k, then there is a locally consistent tree (T ,V ,C) of height one where T ={ }∪{1,...,k} n and V ( ) = .
Proof.Denote T ={1,...,k} n ; hence T ={ } ∪T where we denote by ∪ the disjoint union operator.Furthermore, we assume that Let us construct a tree with a root labelled by and k n children denoted by t = (a 1 ,...,a n ) ∈ {1,...,k} n .Intuitively, we allow each agent to perform k different actions where the special action k always costs 0. For convenience, we denote the action of agent i in t by t i = a i and the joint move by a coalition A in t by t A = (t i ) i∈A .In the following, we define the labelling function V (t) for each leaf t and the cost function C( ,i,a) for each agent i ∈ N and action a ∈{1,...,k}: (a) For each A b p p ϕ p ∈ + where A p =∅, ϕ p is added to V (t) for all t such that ∀i ∈ A p : t i = p.Let min A p be the smallest number in A p , we assign the cost of action p performed by min A p to be deinf(b p ), i.e.C( ,min A p ,p) = deinf(b p ).For other agents i in A p \{min A p }, we assign C( ,i,p) = 0.For other unassigned-cost actions, their costs are assigned as follows: We define C(t,A) = i∈A C( ,i,t i ) as the cost of the joint action by the coalition A and C(t) = C(t,N) as the cost of the joint action by the grand coalition.(b) For each ¬ N e p χ p ∈ − N , ∼χ p is added to V (t) whenever C(t) ≤ e p .(c) Finally, we will add at most one formula from − to V (t).Let We now prove that the constructed tree (T ,V ,C) is locally consistent.First, we show that all labels are consistent.It is obvious that V ( ) = is consistent.For each child t ∈ T , since at most one formula ∼ψ p such that ¬ B d p p ψ p ∈ − is added into V (t), we consider the following two cases: Case ∀p ∈{1,...,l}:∼ψ p / ∈ V (t) : Let us assume that Therefore, by Lemma 3, V (t) is consistent.Case ∃!q ∈{1,...,l}:∼ψ q ∈ V (t) : Let us assume that Recall that: hence, q ∈{q 1 ,...,q l t }, and also Similar to the previous case, we have: Then, we have: ⇒∀j ∈{1,...,h}:e j = ∞→C(t) ≤ e j as f +1 ≤ e j ⇒∀j ∈{j 1 ,...,j h t }:e j = ∞ by (b) and Therefore, by Lemma 2, V (t) is consistent.
Let us now prove that (T ,V ,C) satisfies the two local consistency conditions of Definition 7.
( ψ p ∈ V ( ) where B p = N, let σ be an arbitrary joint move for the coalition B p such that cost( ,σ ) ≤ d p .We will determine an outcome t ∈ out( ,σ ), such that ∼ψ p ∈ V (t).As t ∈ out( ,σ ), t i = σ i for all i ∈ B p ; it remains to determine t i for i / ∈ B p .Let t ∈ T such that ψ p ∈ − (σ ).Let p = i j * for some j * ∈{1,...,l σ }.
Let ι be an arbitrary agent in N \B p =∅.We define: Let us prove that ∼ψ p ∈ V (t).
We have: Recall that i j * = p.Then, we have: Therefore, according to (c), ∼ψ p ∈ V (t).
The next stage of the proof is to consider what conditions on tree labelling we need to be able to prove the truth lemma for other temporal modalities.The definition of what it means to 'realize' formulas of the form A b ϕ Uψ, ¬ A b ϕ, A b ϕ, ¬ A b ϕ Uψ is similar to the one in [13] (essentially the truth conditions for the formulas with 'satisfied' replaced by 'in the labelling of').
The following lemma and its proof are similar to the correspondings in [13], but for formulas of RB-ATL.

Lemma 5
For any subset Y ⊆ , there is a formula χ Y ∈ cl(ϕ 0 ), called the characteristic formula of Y , such that for every y ∈ , χ Y ∈ y iff y ∈ Y .
Proof.For any maximally consistent subset y of cl(ϕ 0 ), we define: In what follows, is the set of formulas of the form A b ϕ or ¬ A b ϕ from cl(ϕ 0 ).

Lemma 6
Given A b ϕ Uψ and x ∈ , there is finite tree (T ,V ,C) over such that: • every interior node of (T ,V ,C) has k n children where Proof.The proof is similar to the corresponding proof in [13], but also uses induction on the bound b.
Let Z ⊆ such that, for any x ∈ Z, there is a finite tree obeying all conditions of Lemma 6.We shall prove the lemma by showing that Z = .In the following, assume that x ∈ . Therefore, However, it remains to prove that η is a theorem.This is done by showing that η belongs to any maximal consistent set q of RB-ATL in three cases: ∈ q, let us construct a simple tree (T ,V ,C) where T ={ } and V ( ) = q∩cl(ϕ 0 ).Since (T ,V ,C) satisfies all conditions of the lemma, q∩cl(ϕ 0 ) ∈ Z.Then, we have: Let us consider the following two sub-cases: if ψ ∈ q, let us construct a simple tree (T ,V ,C) where T ={ } and V ( ) = q∩cl(ϕ 0 ) ψ.
Then, (T ,V ,C) satisfies all conditions of Lemma 5; hence, q∩cl(ϕ 0 ) ∈ Z; therefore, similar to the above argument of the case Then, by Lemma 4, there exists a locally consistent tree (T 0 ,V 0 ,C 0 ) of height one where T 0 ={ } ∪{1,...,k} n and V 0 ( ) = q∩cl(ϕ 0 ).For each c ∈{1,...,k} n , let c be an arbitrary set from such that c ⊇ V 0 (c).Let (T 1 ,V 1 ,C 1 ) be a finite tree such that For every child c ∈{1,...,k} n such that χ Z ∈ V 1 (c), we have: which satisfies all conditions of Lemma 6 Let us consider a finite tree (T ,V ,C) where for all t ∈ T : for all t ∈ T , i ∈ N and j ∈{1,...,k}: It is straightforward that (T ,V ,C) is also locally consistent and all of its interior nodes have is a joint move which costs at most 0 ∞ ← b,we have: ←b ϕ Uψ from the root of (T ,V ,C).Hence, as T ( ) = q∩cl(ϕ 0 ), we have: us consider the following three sub-cases: if ψ ∈ q, the proof is the repetition of that for the base case.
if ϕ and A 0 ∞ ←b χ Z ∈ q, the proof is the repetition of that for the base case.
; therefore, < k.By Lemma 4, there exists a locally consistent tree (T 0 ,V 0 ,C 0 ) of height one where T 0 ={ } ∪{1,...,k} n and V 0 ( ) = .For each c ∈{1,...,k} n , let c be an arbitrary set from such that c ⊇ V 0 (c).Let (T 1 ,V 1 ,C 1 ) be a finite tree such that For each c ∈{1,...,k} n such that A b 2 ϕ Uψ ∈ V (c), as b 2 < b, we have: Let us consider a finite tree (T ,V ,C) where for all t ∈ T : for all t ∈ T , i ∈ N and j ∈{1,...,k}: It is straightforward that (T ,V ,C) is also locally consistent and all of its interior nodes have k n children.
Let us show that (T ,V ,C) realizes A b ϕ Uψ at .Let c ∈{1,...,k} n such that C(c,A) ≤ b 1 , i.e., c A is a joint move which costs at most b 1 ,we have: Let us consider a b-strategy F A where It is straightforward that F A realizes A b ϕ Uψ from the root of (T ,V ,C).Hence, as T ( ) = q∩cl(ϕ 0 ), we have: Similarly, we have the following result:

Lemma 7
Given ¬ A b ϕ and x ∈ , there is finite tree (T ,V ,C) over such that: • every interior node of (T ,V ,C) Proof.The proof is similar to that of the previous lemma.Let Z ⊆ such that, for any x ∈ Z, there is a finite tree obeying all conditions of Lemma 7. We shall prove the lemma by showing that Z = .
In the following, assume that x ∈ .
However, it remains to prove that η is a theorem.Again, this is done by showing that η belongs to any maximal consistent set q of RB-ATL in three cases: ϕ / ∈ q, let us construct a simple tree (T ,V ,C) where T ={ } and V ( ) = q∩cl(ϕ 0 ).Since (T ,V ,C) satisfies all conditions of the lemma, q∩cl(ϕ 0 ) ∈ Z.Then, we have: ¬χ Z ∈ q, we have either ¬ϕ ∈ q or ¬ A 0 ∞ ←b ¬χ Z ∈ q.Let us consider the following two sub-cases: if ¬ϕ ∈ q, let us construct a simple tree (T ,V ,C) where T ={ } and V ( ) = q∩cl(ϕ 0 ) ¬ϕ.
Then, (T ,V ,C) satisfies all conditions of Lemma 5; hence, q∩cl(ϕ 0 ) ∈ Z; therefore, similar to the above argument of the case Then, by Lemma 4, there exists a locally consistent tree (T 0 ,V 0 ,C 0 ) of height one where T 0 = { } ∪{1,...,k} n and V 0 ( ) = q∩cl(ϕ 0 ).For each c ∈{1,...,k} n , let c be an arbitrary set from . Then, for every c ∈{1,...,k} n such that C(c,A) ≤ 0 ∞ ← b, i.e., c A is a joint move which costs at most 0 ∞ ← b, we have: which satisfies all conditions of Lemma 7 Let us consider a finite tree (T ,V ,C) where for all t ∈ T : for all t ∈ T , i ∈ N and j ∈{1,...,k}: It is straightforward that (T ,V ,C) is also locally consistent and all of its interior nodes have k n children.
Let us show that (T ,V ,C) realizes A 0 ∞ ←b ϕ at .Let c ∈{1,...,k} n such that C(c,A) ≤ 0 ∞ ← b, i.e. a joint move that costs at most 0 ∞ ← b, we have: Hence, as T ( ) = q∩cl(ϕ 0 ), we have: us consider the following two sub-cases: if ¬ϕ ∈ q, the proof is the repetition of that for the base case.
ϕ ∈ q for all (b 1 ,b 2 ) ∈ split(b).Let = q∩cl(ϕ 0 ).Obviously, ⊆ ; therefore, < k.By Lemma 4, there exists a local consistent tree (T 0 ,V 0 ,C 0 ) of height one where T 0 ={ } ∪{1,...,k} n and V 0 ( ) = .For each c ∈{1,...,k} n , let c be an arbitrary set from such that c ⊇ V 0 (c).Let (T 1 ,V 1 ,C 1 ) be a finite tree such that . Then, for every c ∈{1,...,k} n such that C(c,A) ≤ b, i.e., c A is a joint move which costs at most b, we have: which satisfies all conditions of Lemma 7 Let us consider a finite tree (T ,V ,C) where for all t ∈ T : and for all t ∈ T , i ∈ N and j ∈{1,...,k}: It is straightforward that (T ,V ,C) is also locally consistent and all of its interior nodes have k n children.
Let us show that (T ,V ,C) realizes A b ϕ at .Let c ∈{1,...,k} n such that C(c,A) ≤ b, i.e. a joint move which costs at most b, we have: ϕ is realized at the root of (T ,V ,C) Hence, as T ( ) = q∩cl(ϕ 0 ), we have: Now we have almost all the ingredients for constructing the model for ϕ 0 .For each consistent set x in and an eventuality ϕ of cl(ϕ 0 ), we have a finite tree (T x,ϕ ,V x,ϕ ,C x,ϕ ) with the root having label x which realizes ϕ.Let the eventualities in cl(ϕ 0 ) be listed as ϕ e 1 ,...,ϕ e m .Next, we define the final tree.
• Initially, select an arbitrary x ∈ such that ϕ 0 ∈ x.As that formula is consistent, such a set exists.
Let (T x,ϕ e 1 ,V x,ϕ e 1 ,C x,ϕ e 1 ) be the initial tree.• Given the tree constructed so far and the last used eventuality ϕ e i .We replace every leaf labelled by y ∈ of the currently constructed tree with the tree (T y,ϕ e j ,V y,ϕ e j ,C y,ϕ e j ) where j = i mod m+1.
Let S ϕ 0 be the model which is based on (T ϕ 0 ,V ϕ 0 ,C ϕ 0 ).(It is easy to define the assignment π using V .) Proof.Let us consider the first case when ϕ e i = A b ϕ Uψ ∈ V (t) where t is a node of (T ϕ 0 ,V ϕ 0 ,C ϕ 0 ).The proof for the case of ϕ e i =¬ A b ϕ is also done similarly.
• If t happens to be the root of the sub-tree (T t,ϕ e i ,V t,ϕ e i ,C t,ϕ e i ), then the proof is done as ϕ e i is realized within this sub-tree at t, hence also in the final tree.
• Otherwise, we define inductively on b a b-strategy as follows: Base case: ←b ϕ Uψ ∈ V (tc ).Let F A (t) = c A .Then, we can continue with the same argument to define the strategy F A until a node t labelled by y in (T ϕ 0 ,V ϕ 0 ,C ϕ 0 ) is reached where t is the root of some sub-tree (T y,ϕ e i ,V y,ϕ e i ,C y,ϕ e i ).Such a node must exist because, according to the construction of the (T ϕ 0 ,V ϕ 0 ,C ϕ 0 ), eventual formulas in cl(ϕ 0 ) are cycled through.As (T y,ϕ e i ,V y,ϕ e i ,C y,ϕ e i ) realizes ϕ e i , we can extend F A to a b-strategy to realize ϕ e i .

Induction
Step: Assume that b > 0 ∞ ← b, since A b ϕ Uψ ∈ V (t), and V (t) is a maximally consistent set, we have that ψ ∨(ϕ ∧( -If ψ ∈ V (t), the proof is done as A b ϕ Uψ is immediately realized at t.
Then, we can continue with the same argument to define the strategy F A until a node t labelled by y in (T ϕ 0 ,V ϕ 0 ,C ϕ 0 ) is reached where t is the root of some sub-tree (T y,ϕ e i ,V y,ϕ e i ,C y,ϕ e i ).Such a node must exist because, according to the construction of the (T ϕ 0 ,V ϕ 0 ,C ϕ 0 ), eventualities in cl(ϕ 0 ) are cycled through.As (T y,ϕ e i ,V y,ϕ e i ,C y,ϕ e i ) realizes ϕ e i , we can extend F A to a b-strategy to realize ϕ e i .

Lemma 9
Proof.Let us consider the case A b ϕ ∈ V (t).The proof for ¬ A b ϕ Uψ is similar.In the following, let T ={1,...,k} n .We shall define a b-strategy F A which realizes A b ϕ by induction on the length of inputs.Let i denote the set of inputs of length i.Initially, 1 ={t}.We define F(A) for inputs of length i and i+1 of inputs of length i+1 inductively on i ≥ 1 such that for all λ ∈ i+1 , cost(λ,F A ) ≤ b 1 and ϕ ∈ V (tc ) for all tc ∈ 2 .Induction step: Assume that i > 1 and we have defined F A for inputs of length i and i+1 such that for all λ Given the constructed strategy F A , we have that out(t,F Finally, we show the following truth lemma.
Proof.The proof is done by induction on the structure of ϕ.
• For the cases of propositions, negative proposition and disjunction, the proofs are trivial.
• Assume ϕ = A b ψ, Lemma 4 ensures that there is a move c A for some c ∈{1,...,k} n where C(c,A) ≤ b such that for all c ∈{1,...,k} n , we have ψ ∈ V (tc ).Then by the induction hypothesis, we have that Finally, we have the following theorem.

Theorem 1
The axiom system for RB-ATL is sound and complete.

Model-checking RB-ATL
In this section, we describe a model-checking algorithm for RB-ATL which runs in time polynomial in the size of the formula (if resource bounds are encoded in unary) and the structure, and is exponential in the number of resources.The algorithm is similar to the model-checking algorithm for ATL given in [6].The main differences from the algorithm for ATL are that we need to take the costs of strategies into account, and, instead of working with a straightforward set of subformulas Sub(ϕ) of a given formula ϕ, we work with an extended set of subformulas Sub + (ϕ).Sub + (ϕ) includes Sub(ϕ), and in addition: We assume that Sub + (ϕ) is ordered in the increasing order of complexity and of resource bounds (so e.g. if b ≤ b , then

Related work
Recent work on ATL and CL (e.g.[1,6,12,13,15,16]) has allowed the expression of many interesting properties of coalitions and strategies.However, there is no natural way of expressing resource requirements in these logics.The only work in this tradition that considered resources is [17], which introduced Coalitional Resource Games and studied complexity of decision problems for these games.A logic for describing Coalitional Resource Games and a model-checking procedure for the logic were proposed in [3]; however, the only modality that logic has is A b (only one-step games were considered).
More recently, several extensions of temporal logics and logics of coalitional ability that are capable of expressing resource bounds have been proposed in the literature, e.g.[2,7,8,10,11].All of these papers consider only the model-checking problem, and some of the logics allow both consumption and production of resources by actions.There are many different proposals for the syntax and semantics of resource logics.In [8] several versions are given, e.g.considering resource bounds both on the coalition A and the rest of the agents in the system, considering a fixed resource endowment of A in the initial state which affects their endowment after executing some actions, etc.In [10,11], a different syntax and semantics are considered, also involving resource endowment of the whole system when evaluating a statement concerning a group of agents A. As observed in [8], subtle differences in truth conditions for resource logics result in the difference between decidability and undecidabiliity of the model-checking problem.In [8], undecidability of the model-checking problem for several versions of the logics is proved.The only decidable cases considered in [8] are an extension of Computation Tree Logic (CTL) [9] with resources (essentially one-agent ATL) and the version where on every path only a fixed finite amount of resources can be produced.Similarly, [10] gives a logic PRB-ATL (Priced Resource-Bounded ATL) with a decidable model-checking problem where the total amount of resources in the system has a fixed bound.The model-checking algorithm for PRB-ATL runs in time polynomial in the size of the model and exponential in the number of resources and the resource bound on the system.In [11], an EXPTIME lower bound in the number resources is shown.Recently, it has also been shown that if a zero-cost action is always available, the model-checking problem for RB-ATL with both production and consumption of resources is decidable; however, it is EXPSPACE-hard [2].

Conclusions
We have provided a complete and sound axiomatization of RB-ATL, a logic which extends ATL with resource bounds.The resulting logic can express interesting properties of coalitions of agents involving resource limitations.For example, it can express that a coalition can maintain the system in a ϕ-state indefinitely given a finite amount of resources (this essentially means that after a while ϕ can be maintained for free).We have also presented a model-checking algorithm for RB-ATL, which runs in time polynomial in the size of the model and the formula (assuming that resource bounds are encoded in unary) and exponential in the number of resources.
The semantics for RB-ATL presented in this article, in particular the assumption that actions only consume but never produce resources, is motivated by verifying resource requirements for systems of agents where resources of interest are time, memory, bandwidth, etc., which cannot be generated by agents.In future work, we plan to study axiomatizations of variants of RB-ATL where actions can have a negative cost, such as in [2,7].
and ( A b U-Monotonicity) clearly preserve validity, since if ϕ ⊆ ψ and an outcome in ϕ can be enforced, then an outcome in ψ can also be enforced by the same strategy.( N b -Necessitation) is valid since if ϕ is logically true, then it is inevitable in perpetuity.( A b -Induction) and ( A b U-Induction) preserve validity by Lemma 1.

-If ϕ and A b 1 A b 2 ϕ
Uψ ∈ V (t) for some (b 1 ,b 2 ) = b (hence, b 2 < b), we have that ϕ ∈ V (t) and by Lemma 4 and there exists c ∈{1,...,k} n such that C(tc,A) ≤ b 1 and for all c ∈{1,...,k} n with c A = c A , we have A b 2 ϕ Uψ ∈ V (tc ).Let F A (t) = c A .As b 2 < b, by the induction hypothesis, there is a strategy F A,c which realizes A b 2 ϕ Uψ from tc.Hence, we just need to define F A (tcλ) = F A,c (cλ).This simply gives us a b-strategy which realizes A b ϕ Uψ from t. -Otherwise, we have ϕ and A 0 ∞ ←b A b ϕ Uψ ∈ V (t).Let us repeat the argument in the base case where ϕ ∈ V (t) and by Lemma 4 and we have that there exists c ∈{1,...,k} n such that C(tc,A) ≤ 0 ∞ ← b and for all c ∈{1,...,k} n with c A = c A , we have A Lemma 4  ensures that for all c ∈{1,...,k} n such that C(c,A) ≤ b, there exists c ∈{1,...,k} n such that c A = c A and ∼ψ ∈ V (tc ).Then by the induction hypothesis, we have that S ϕ 0 ,tc |=∼ψ.Then, S ϕ 0 ,t |= ¬ A b ψ.•For the cases ofA b ψ 1 Uψ 2 , ¬ A b ψ, ¬ A b ψ 1 Uψ 2 and A b ψ, the proofs are trivial due to Lemmas 8 and 9.
all states where A has a 0 ∞ ← b-cost strategy to maintain ψ forever.Note that A has a b-cost strategy to maintain ψ forever if and only if it has a b-cost strategy to force the system into one of the [ A 0 ∞ ←b ψ] S states, while maintaining ψ.In other words, to compute A b ψ for b containing b i ∈{0,∞}, we need to compute A b ψ U A 0 ∞ ←b ψ.This explains the similarity between the cases of A b ψ and A b ψ 1 Uψ 2 for the case of b not consisting solely of 0 and ∞.In the case of A b ψ, in the first execution of the foreach d ∈{d |(d,d ) ∈ split(b)} loop, we have d = 0 ∞ ← b and τ = Pre(A,[ A 0 ∞ ←b ψ] S ,b)∩[ψ] S , which includes Pre(A,[ A 0 ∞ ←b ψ] S , 0 ∞ ← b)∩[ψ] S , hence it also includes [ A 0 ∞ ←b ψ] S .In the nested while loop, ρ accumulates the results and τ adds the ψ-states from where A has a 0 ∞ ← b strategy to enforce the outcome to be in ρ.In the outer loop, d bounds are used in some order consistent with <, namely satisfying the condition that if b i < b j then b i is used before b j .In the case for A b ψ 1 Uψ 2 where b does not consist only of 0 and ∞, after the first iteration of the foreachd ∈{d | (d,d ) ∈ split(b)} loop, τ is [ A 0 ∞ ←b ψ 1 Uψ 2 ] S which includes [ψ 2 ] S .The rest is very similar to the case for A b ψ where b does not consist solely of 0 and ∞.Note that |split(b)| is O(b r ).If ϕ contains operators with bounds other than 0 and ∞, |Sub + (ϕ)| is O(|ϕ|×|ϕ| r ), assuming resource bounds are written in unary.In the A b ψ and A b ψ 1 Uψ 2 cases, the outer loop is executed O(|ϕ| r ) times and the inner loop is executed in total at most |S| times.This gives us complexity O(|ϕ|×|ϕ| r ×|ϕ| r ×|S|), or O(|ϕ| 2r+1 ×|S|).Note that the lower bound for model-checking complexity is given by the model-checking complexity of ATL, which is polynomial time in the size of the model and the formula.