The Thirty-Third AAAI Conference on Artificial Intelligence (AAAI-19)
Probabilistic Alternating-Time
µ
-Calculus
∗Fu Song,
1Yedi Zhang,
1Taolue Chen,
2,5Yu Tang,
3Zhiwu Xu
4 1ShanghaiTech University,2University of London3East China Normal University,4Shenzhen University
5Nanjing University
Abstract
Reasoning about strategic abilities is key to an AI system con-sisting of multiple agents with random behaviors. We propose a probabilistic extension of Alternatingµ-Calculus (AMC), named PAMC, for reasoning about strategic abilities of agents in stochastic multi-agent systems. PAMC subsumes existing logics AMC and PµTL. The usefulness of PAMC is exempli-fied by applications in genetic regulatory networks. We show that, for PAMC, the model checking problem is inUP∩co-UP, and the satisfiability problem isEXPTIME-complete, both of which are the same as those for AMC. Moreover, PAMC ad-mits the small model property. We implement the satisfiability checking procedure in a toolPAMCSolver.
Introduction
Temporal logics play a key role in specification and veri-fication of ICT systems. There are two fundamental deci-sion problems of temporal logics: satisfiability checking and model checking. Given a temporal logic formula, the former asks whether a system satisfying the formula exists, while the latter asks whether a further given system satisfies the formula. Temporal logics for reasoning about strategic abil-ities in Multi-Agent Systems (MAS) have been proposed, typically in the form of alternating-time temporal logics (e.g., ATL, ATL∗and AMC) (Alur, Henzinger, and Kupfer-man 2002), and strategic logics (Chatterjee, Henzinger, and Piterman 2010; Mogavero et al. 2014; 2017). Both model checking and satisfiability checking for these logics have been extensively investigated (Schewe and Finkbeiner 2006; Walther et al. 2006; Schewe 2008).
In practice, agents or the environment may exhibit ran-dom behaviors because of unpredictable physical conditions. Hence, it is vital to reason about strategic abilities of agents in astochasticsetting. From the modeling perspective, this gives rise to stochastic MAS, which consist of a set of agents oper-ating concurrently in a stochastic environment. Probabilistic variants of ATL and ATL∗, for instance PATL, PATL∗(Chen and Lu 2007; Chen et al. 2013) and SGL (Baier et al. 2007),
∗
This work is partially supported by NSFC (No. 61532019, 61761136011, and 61502308), EPSRC (EP/P00430X/1) and ARC (DP160101652, DP180100691).
Copyright c2019, Association for the Advancement of Artificial Intelligence (www.aaai.org). All rights reserved.
have been proposed, for which the model checking prob-lem was also studied. In contrast, the satisfiability probprob-lem for these logics turns out to be much more difficult. Indeed, the satisfiability problem for PCTL, a (proper) fragment of these logics, is a long-standing open problem in the formal verification community (Chakraborty and Katoen 2016).
This paper aims to propose a temporal logic withdecidable satisfiability checking for reasoning about stochastic MAS. Apart from being theoretical appealing, admitting decidabil-ity for satisfiabildecidabil-ity is useful in practice. Examples include debugging specifications (as writing formal specifications is often error-prone (Rozier and Vardi 2010)), social proce-dure or mechanism design (Pauly 2011), and assertion-based design (i.e., verifying consistency of all requirements by for-malizing designer’s intention by assertions, and checking the full set of assertions to be satisfied before the model is ready for examination (Foster, Krolnik, and Lacey 2004)). To this end, we propose a probabilistic variant of AMC, prob-abilistic alternating-timeµ-calculus(PAMC), which is ac-quired by equipping the coalition modalities with ( qualita-tive) probability quantifiershhAii./kin AMC. For instance,
νZ(a∧ hh{1,2}ii≥0.9Z) states that the agent group{1,2}has a coalition strategy at each step such that the opponent coali-tion can escape thea-region in one step with probability less than 0.1. PAMC subsumes both AMC and PµTL (Liu et al. 2015), but is incomparable with PATL and PATL∗mentioned before. Remark that we do not extend AMC in aquantitative way such as (Mio 2012), in order to retain decidability.
com-plexities of the model checking and satisfiability problems of PAMC lie in the same complexity classes of AMC. We imple-ment a satisfiability solver for PAMC, which is, to our best knowledge, the first satisfiability checker for (probabilistic) alternating-time temporal logics.
Due to space restriction, proofs and experimental results are given in the full version, which, together with our tool PAMC and benchmarks, is available at http://faculty.sist. shanghaitech.edu.cn/faculty/songfu/Projects/PAMCSolver.
Probabilistic Concurrent Game Structures
For a natural numberk∈N, let [k] denote the set{1, ...,k}. Given a probability distributionPr:X→[0,1], letsupp(Pr) denote the set{x∈X |Pr(x)>0}. We denote byD(X) the set of probability distributions onX.
Probabilistic concurrent game structuresare a model of concurrent stochastic multi-agent systems, in which the tran-sition probability function gives a probability distribution over the successor states after considering a joint action from the agents. In this work, we consider stochastic multi-agent systems with perfect information only.
Definition 1. Fix a finite set AP of atomic propositions
(i.e., observations). Aprobabilistic concurrent game struc-ture(PCGS) is a tupleM=(Ag,Act,Q,Γ, δ, λ,q0), where:
Ag=[n]is a finite set of agents.Act=S
i∈AgActiandActi
is a non-empty set of actions of agent i ∈ Ag. Adecision d = ha1, ...,aniis a joint action of all the agents in which
ai, denoted byd(i), is the action chosen by agent i. We
de-note byDthe set of decisions. Q is a finite set of statesand q0 ∈Q is the initial state.Γ = (Γi)i∈AgwithΓi : Q→2Acti
assigning to agent i at a state q a finite setΓi(q)⊆Actiof
actions available at the state q. We denote byD(q)the set Q
i∈AgΓi(q)of available decisions at q.δ:Q×D→ D(Q)is
a (partial)probability transition functionwhich associates each pair(q,d)∈Q×Dsuch thatd∈D(q)with a probability distribution over Q.λ : Q → 2AP is thelabeling function assigning to each state q∈Q a set of atomic propositions.
Given a stateq ∈Qand a decisiond∈D(q),Mmoves to a next stateq0 ∈ Qwith probability δ(q,d)(q0), which is also denoted asδ(q,d,q0). We indicate bysupp(q,d) the setsupp(δ(q,d)). TheoutdegreeofMis the maximum of
|supp(q,d)|forq∈Qandd∈D(q).
Paths and tracks. Apath π ∈ Qω (a.k.a. play) is an
in-finite sequence of states q0q1... such that for all j ≥ 1,
qj ∈ supp(qj−1,dj−1) for somedj−1 ∈ D(qj−1). Atrackis
a finite prefix of a path. Given a trackπ=q0q1...qm(resp. a
pathπ=q0q1...) and 0 ≤i ≤mfor track, letπidenote the
stateqiandπ≤idenoteq0...qi.
Strategies.A(mixed) strategyof agenti∈Agis a function
θi:Q+→ D(Acti) that assigns to each trackq0. . .qm,
repre-senting the past history of the game, a probability distribution on its available actionsΓi(qm)⊆Acti. Therefore, the choice
of the next action can behistory-dependentandmixed. A strategyθiispureif for allπ ∈ Q+,θi(π) is aDirac
distri-bution, i.e.,|supp(θi(π))| =1. A strategyθi ismemoryless
(i.e., positional or imperfect recall) if for allπ, π0∈Q+and q ∈ Q,θi(πq) = θi(π0q), namely, the agent takes an action
only depending on the last state. We denote byΘithe set of
all strategies for agentiand letΘ =S
i∈AgΘi.
Acoalitionis a set of agentsA⊆Ag. Acoalition strategy forAis a functionυA:A→Θassigning to each agenti∈A
a strategyυA(i)∈Θi. LetΥAdenote the set of all coalition
strategies for the coalitionA. We denote byAthe setAg\A.
Outcomes.Given a trackπ∈Q+such that the last state of
πisqand a decisiond ∈ D(q), we denote byPrυA,υA(π,d)
the probabilityQ
i∈AυA(i)(π)(d(i))·Qi∈AυA(i)(π)(d(i)),that is,
the probability of the decisiondchosen by the agentsAgwith respect toπ,υAandυA. We say that a pathπiscompatible
with respect toυAandυA, if for all j≥0, there is a decision
dj∈D(πj) such thatδ(πj,dj, πj+1)>0 andPrυA,υA(π≤j,dj)>
0.We denote byOυA,υA
q the set of paths starting fromqthat
are compatible with respect toυAandυA, which is the set of
paths that can be followed by the game when agents enforce strategiesυAandυA.
Probability Space.Aσ-algebraover a setΩis a setE ⊆2Ω
such thatEcontains∅and is closed under countable union and complement. An element ofEis called anevent. A prob-ability spaceis a triple (Ω,E,Pr), whereΩis asample space,
E is aσ-algebra overΩ,Pr : E → [0,1] is aprobability measuresuch thatPr(Ω) = 1 and the countable additivity property is satisfied, i.e.,Pr(U∪V)=Pr(U)+Pr(V) whenever U∩V =∅.
Given a coalitionA⊆Ag, a stateq,twocoalition strate-giesυA andυA, one can construct a probability space over
the set of pathsOυA,υA
q with the probability measurePr υA,υA q
defined in the standard way (Vardi 1985), where an event is a measurable set of paths.
CGS, MDP and MC.Aconcurrent game structure(CGS) is
a PCGS such that all the probability distributions involved in the PCGS (i.e., probability transition function and strategies) are Dirac distributions. A Markov decision process (MDP) is a PCGS such that|Ag|=1. AMarkov chain(MC) is an MDP such that|Act|=1.
Probabilistic Alternating-Time
µ
-Calculus
Probabilistic alternating-timeµ-calculus is a simple and suc-cinct, but natural, probabilistic extension of AMC (Alur, Henzinger, and Kupferman 2002). In this logic, coalition modalitieshhAiiφfrom AMC are replaced withprobabilistic coalition modalitieshhAii./kφ, which probabilistically
quan-tify over the strategic choices of a groupAof agents.
Definition 2. Let Zbe a finite set of propositional
vari-ables. The syntax of probabilistic alternating-timeµ-calculus (PAMC for short) is defined as follows:
φ::=p| ¬φ|Z|φ∧φ|φ∨φ|µZ.φ|νZ.φ| hhAii./kφ|[[A]]./kφ
where p ∈ AP, Z ∈ Z,./∈ {≥, >, <,≤}, k ∈ [0,1]is a ra-tional constant, A⊆Ag, and for eachµZ.φandνZ.φ, each occurrence of Z inφis under the scope of an even number of negations inφ.
Letφbe a PAMC formula.Z∈ Zis afree variableinφif an occurrence ofZis not in the scope of a fixed-point operator
hereafter that each variableZis quantified by eitherµorνat most once in each PAMC sentence. We denote by⊥ ≡ p∧ ¬p and> ≡ p∨ ¬p. For simplifying the presentation, formulae likehh{i1, ...,im}ii./kφmay be written ashhi1, ...,imii./kφ.
Given two PAMC sentencesηZ.φforη∈ {µ, ν}andϕ, let
φ[ϕ/Z] be the sentence obtained fromφby replacing every occurrence ofZwithϕ. The Fisher-Ladner closureFL(φ) of a PAMC sentenceφcontains all the sub-sentences ofφwith the rule: ifηZ.ϕ∈FL(φ) forη∈ {µ, ν}, thenϕ[ηZ.ϕ/Z]∈FL(φ). The size ofFL(φ) is at most double that ofφ. We denote by FL∃(φ)⊆FL(φ) andFL∀(φ)⊆FL(φ) the set of sentences of the formhhAii./kϕand [[A]]./kϕ, respectively.
Example 1. Let φ = νZ1.(hh1,2ii≥
1
2Z1∧φ0)where φ0 = µZ2.(p∨ hh1,3ii≥
1
3Z2), we haveFL(φ) = {φ,hh1,2ii≥ 1 2φ∧ φ0,hh1,2ii≥1
2φ, φ0,p∨ hh1,3ii≥ 1
3φ0,p,hh1,3ii≥ 1 3φ0}.
The semantics of PAMC is defined w.r.t. PCGSs and a valuationξ:Z →2Q. Letξ[S/Z] denote the valuation such
thatξ[S/Z](Z)=S andξ[S/Z](Z0)=ξ(Z0) forZ,Z0.
Definition 3. Given a PCGSM = (Ag,Act,Q,Γ, δ, λ,q0),
the semantics of PAMC is defined via the denotation function
~◦ξM, which is defined as follows:
• ~pξM={q∈Q|p∈λ(q)};
• Boolean connectors are defined in a standard way;
• ~µZ.φξM =
T{Q0⊆Q|
~φξ[Q 0/Z]
M ⊆Q
0};
• ~νZ.φξM =
S{Q0⊆Q|
~φξ[Q 0/Z]
M ⊇Q
0};
• ~hhAii./kφ
ξM=
(
q∈Q| ∃υA∈ΥA,∀υA∈ΥA: PrυA,υA
q ({π∈ O υA,υA
q |π1∈~φξM})./k )
;
• ~[[A]]./kφ
ξM= (
q∈Q| ∀υA∈ΥA, ∃υA∈ΥA: PrυA,υA
q ({π∈ O υA,υA
q |π1∈~φξM})./k )
.
We sometimes drop the superscriptξfrom~φξMifφis a PAMC sentence. When it is clear from context, the subscript
Mis also dropped from~φξMand~φM.
AMC (resp. PµTL (Liu et al. 2015)) is an extension ofµ -calculus in which the next-modalities are replaced byhhAiiφ
and [[A]]φ(resp. [Xφ]./k) and the semantics is defined over CGSs (resp. MCs). PATL/PATL∗ (Chen and Lu 2007) are probabilistic variants of ATL/ATL∗ (Alur, Henzinger, and Kupferman 2002) and the semantics is defined over PCGSs.
Theorem 1. PAMC subsumes AMC and PµTL. PAMC and
PATL/PATL∗are incomparable.
Proof sketch. Each AMC (resp. PµTL) formula can be
encoded as a PAMC formula, which can be shown eas-ily by structural induction. For instance, hhAiiφ in AMC (resp. [Xφ]./kin PµTL) can be encoded ashhAii≥1φ0(resp.
hh∅ii./kφ0), whereφ0denotes the encoding ofφin respective cases. Some PAMC formulae (e.g., hhAii>0.5φ) cannot be
expressed in AMC or PµTL.
Since PµTL and PCTL are incomparable on Markov chains (Liu et al. 2015), there exists a PCTL formula (and thus a PATL formula) which is inexpressible in PAMC. More-over, we note that AMC is more expressive than ATL∗, by which a formula in PAMC can be constructed which is
inex-pressible in PATL∗.
Proposition 1 is directly from the PAMC semantics. To simplify the notation, we write>for<,<for>,≤for≥and
≥for≤, and writeb≥for<,b>for≤,b≤for>andb<for≥.
Proposition 1. Given a PAMC formulahhAii./kφor[[A]]./kφ
for k∈[0,1], ./∈ {≥, >,≤, <}, a PCGSMand a valuation ξ, we have the following deduction rules.
1. ~hhAii./kφξ=~hhAii./1−k¬φξ. 2. ~[[A]]./kφξ=~[[A]]./1−k¬φξ. 3. ~¬hhAii./kφξ=~[[A]]b./1−kφξ.
4. ~¬[[A]]./kφξ=~hhAiib./1−kφξ.
5. ~hhAgii./kφξ=~[[∅]]./kφξ. 6. ~[[Ag]]./kφξ=~hh∅ii./kφξ.
A PAMC sentence is innegation normal form(NNF) if
¬only appear in front of atomic propositions. Using Propo-sition 1 and the rule¬µZ.φ ≡ νZ.¬φ[¬Z/Z], every PAMC sentence can be equivalently transformed into NNF. Here-after, we assume that all PAMC sentences are in NNF.
In this work, for a given PAMC sentenceφ, we consider the following two fundamental problems.Model checking is to determine whetherq0 ∈ ~φM for the initial stateq0 of a given PCGSM.Satisfiability checkingis to determine whether there exists a PCGSMwith the initial stateq0such thatq0∈~φM.
In PAMC, only probabilistic next-time coalition modalities
hhAii./kφand [[A]]./kφare allowed, hence it suffices to
con-sider memoryless strategies for the coalitionA. By leveraging the model checking algorithm for AMC (Alur, Henzinger, and Kupferman 2002) that computes~φrecursively, we can get a model checking algorithm for PAMC. The key diff er-ence lies in the handling ofhhAii./kφand [[A]]./kφ, which
can be done in polynomial time by linear programming. The model checking problem for AMC lies inUP∩co-UP, and can be solved in exponential time in the alternation depth of the formula. We have that
Theorem 2. The model checking problem for PAMC is in
UP∩co-UPand can be decided inO((|φ| · |M|)c·d)time for
some constant c, where d denotes the alternation depth ofφ.
An application: Genetic Regulatory Networks
To demonstrate the usage of PCGSs and PAMC, we present an example from precision medicine based on the probabilis-tic Boolean network (PBN) model for geneprobabilis-tic regulatory net-works (Shmulevich and Dougherty 2010). Consider a PBN withngenes, each of which has two local states{0,1}where 0 and 1 indicate that the corresponding gene is not expressed and expressed, respectively. A (global) state of the PBN is a Boolean vector of local states with widthn. Each geneihas a finite set of predictor functionsFi={f1i, ...,fkii}denoting the
inter-gene relationship, where each function fi
jdetermines
the local state of geneiat the next step when the j-th element ofFi(i.e., fij) is chosen, andc
i
jis the probability that f i j is
selected (soPki j=1c
i
j=1). The probability of the PBN
evolv-ing from a state~q=[q1, ...,qn] to the stateq~0=[q01, ...,q0n] is
P
f1∈F1:f1(~q)=q01· · · P
fn∈Fn:fn(~q)=q0n
Qn
i=1c
i, whereciis the
For intervention purposes, control inputs are introduced into PBN to reason about treatment strategies. For instance, in cancer therapy, control inputs (e.g. radiation, chemotherapy) may be employed to move the state probability distribution vector away from one associated with uncontrolled cell pro-liferation or markedly reduced apoptosis (Shmulevich and Dougherty 2010, Section 5.2). Suppose there arex1, ...,xm
control inputs ranging over binary values{0,1}, which con-trol the probabilities of predictor functions. The concon-trol input being 1 indicates that a corresponding predictor function (intuitively a treatment) is applied; 0 otherwise. Under the values~v=v1, ...,vmof control inputs, the probability of the
predictor function fi
j ∈ Fibecomescij,~vwithPkji=1c i j,~v =1.
(The specific definitions ofci
j,~vare irrelevant here.) The main
problem is to synthesize values for the control inputs at each state to ensure that the network behaves as desired.
The control of PBN can be naturally modelled as a PCGS
M=(Ag,Act,Q,Γ, δ, λ,q0), whereAg=[n+m] withi∈[n]
denotes the genei, andn+i∈ {n+1, ...n+m}denotes the control inputi;Q = {0,1}n corresponding to states of the
network;Acti =Fifori ∈[n] denoting the predictor
func-tions andActn+i={0,1}forn+i∈ {n+1, ...n+m}denoting
treatment choices;Γiis the function such thatΓi(~q)=Acti
for every agentiand state~q;q0is the initial state of the net-work which is determined by the patient’s physiology;δis the probability transition function such that for every~q∈Q,
fi
ji ∈Fiand~v=v1, ...,vm∈ {0,1} m:
δ(~q,hf1 j1, ...,f
n
jn,v1, ...,vmi,[f 1 j1(~q), ...,f
n jn(~q)]) :=
Qn
i=1ciji,~v.
Note that we do not specifyλexplicitly as this is usually application-specific.
With PCGS defined above, we can formulate the achieve-ment propertyas: µZ.(p∨ hhAii≥0.8Z) for some A ⊆ {n+
1, ...,n+m}. This formula expresses that some desired-region
is reachable, and there is treatment strategy using at most treatments inAat each step such that it has at least probability 0.8 to go on with the right direction.
Themaintenance propertycan be formulated asνZ.(p∧ hhAii>0.9Z) stating that there is a treatment strategy using at most treatments inAat each step such that the network has less than 0.1 probability to escape from the desired-region.
Deciding Satisfiability
In this section, we present a reduction from the satisfiability problem of a PAMC sentenceφto solving a (turned-based, two-player) parity gameGφsuch thatφis satisfiable iff Player-0 has a winning strategy inGφ. Intuitively, as in the classic decision procedure ofµ-calculus, the fixpoint is handled by parity games. In particular, the two players are model “con-structor” and “spoiler”. The constructor intends to show that there a model witnessing the satisfiability ofφ, while the spoiler tries to defeat the constructor by demonstrating a path in the game to show that a model cannot exist. Each state controlled by the constructor consists of a set of sub-sentences ofφfor which the constructor strives to show that it is satisfiable. In addition, the coalition strategies for the set of sub-sentences are tackled by the notions of intersec-tion graph and maximally independent set (cf. Definiintersec-tion 5),
which are used to make sure that the strategies of an agent in different coalitions are consistent. To handle probabili-ties, weighted covers are used which would guarantee that there exist distributions over covers satisfying constraints
./ k. To illustrate the reduction, the PAMC sentenceΨ =
hh1,2ii>0.5p
1∧ hh3ii>0.5(¬p1∨p2)∧[[1,3]]>0.4(¬p1∧ ¬p2)
will be used as an example. We start with some concepts.
Parity automata. Aparity automaton(PA) is a tupleP=
(S,Σ, δ,S0,F), whereS is a finite set of states,Σis the input
alphabet,δ : S ×Σ→ 2S is a transition function,S0 ⊆S is the set of initial states and F : S → {0, ...,k}is arank function. ArunρofPover anω-wordα0α1... ∈Σω is an
infinite sequence of states s0s1· · · such that s0 ∈ S0, and
for everyi≥0,si+1 ∈δ(si, αi). Letinf(ρ) be the set of states
visited infinitely often inρ.ρisacceptingiffmins∈inf(ρ)F(s) is
even. An infinite word isacceptedbyPiffPhas an accepting run over this word. We denote byL(P)⊆Σωthe set of all infinite words accepted byP. AB¨uchi automaton(BA) is a special PA (S,Σ, δ,S0,F) in whichF : S → {0,1}. A PA
P = (S,Σ, δ,S0,F) isdeterministicif|S0| = 1 and for all
(s, α)∈S ×Σ,|δ(s, α)| ≤1.δandS0in a deterministic parity
automaton (DPA for short) are simplified as the function
δ:S ×Σ→S and a states0.
Turned-based two-player parity games. A(turned-based
two-player) parity gameGis a tuple (V=V0]V1,E,vinit,Ξ),
whereV0is a finite set of states (i.e., vertices) of Player-0,V1
is a finite set of states (i.e., vertices) of Player-1,E⊆V×V is a finite the set of edges,vinit ∈V0is the initial state, and
Ξ : V → {0, ...,k}is arankfunction. Aninfinite playρof
Gis an infinite sequence of statesv0v1...such thatv0=vinit,
and for everyi ≥0, (vi,vi+1)∈ E. Afinite playρofGis a
finite sequence of statesv0v1...vnsuch thatv0=vinit, and for
everyi∈[n], (vi−1,vi)∈E. Astrategyof Player-iis a partial
functionθ:V∗Vi→Vsuch that for everyρ∈V∗andv∈Vi,
ifθ(ρ·v) is defined, then (v, θ(ρ·v))∈E. Given a strategyθ0
for Player-0 and a strategyθ1for Player-1, letGθ0,θ1 be the play such that Player-0 and Player-1 enforce their strategies
θ0andθ1during the play. Player-0winson a finite playρiff
the last state ofρis controlled by Player-1 and it cannot move to the next state anymore. Player-0winson an infinite playρ iffmins∈inf(ρ)F(s) iseven. Player-0 has a winning strategy iff
Player-0 has a strategyθ0such that Player-0 wins on the play Gθ0,θ1, for each strategyθ1Player-1 chooses.
Covers and weighted covers. Given a set of sentencesΦ,
a cover cof Φis a set c ⊆ 2Φ such that S
v∈cv = Φ. A
weighted coverofΦis a pair (c,w) such thatcis a cover ofΦandw: c→ (0,1] is a probability function such that P
v∈cw(v) =1. Thewidthof a covercis the cardinality of
c. Given a weighted cover (c,w) ofΦ = {φ1,· · ·, φm}, let
Φ(φi) :={v∈c|φi∈v}andw(φi) :=Pv∈Φ(φi)w(v). We write Cb(Φ) for the set of covers ofΦwith width at mostb. Given
band a setΦ,|Cb(Φ)|is at most 2|Φ|·2(b|Φ+1)|−−12|Φ| (Chakraborty and Katoen 2016). Intuitively, a weight cover (c,w) ofΦcan be seen as a distributionwover the coverc.
Definition 4. A set v of sentences istransitiveif the following conditions hold: (1) Ifφ1∧φ2 ∈ v, thenφ1, φ2 ∈ v; (2) If
φ1 ∨φ2 ∈ v, then φ1 ∈ v orφ2 ∈ v; (3) IfηZ.ϕ ∈ v for
one sentence of the formhhAii./kϕor[[A]]./kϕin v.
LetVtdenote the set of transitive sets of sentences. For
everyv∈Vt, we denote byhhviiand [[v]] the sets{hhAii./kϕ∈
v}and{[[A]]./kϕ∈v}, respectively. Intuitively, consider a set
vthat is controlled by the constructor in the parity gameGφ. Ifvis transitive, then the constructor should give successor states to satisfy sentences inhhviiand [[v]].
Example 2. The set v={hh3ii>0.5(¬p
1∨p2),hh1,2ii>0.5p1,
[[1,3]]>0.4(¬p1 ∧ ¬p2)} is a transitive vertex. We have: hhvii = {hh1,2ii>0.5p
1,hh3iii>0.5(¬p1 ∨ p2)}, and [[v]] = {[[1,3]]>0.4(¬p1∧ ¬p2)}.
Definition 5. An intersection graph Gv = (hhvii,Ev)
formed from v ∈ Vt is an undirected graph such that
(hhA1ii./1k1ϕ1,hhA2ii./2k2ϕ2)∈EviffA1∩A2,∅.
Amaximal independent set(MIS)Mof Gvis a maximal
set of vertices in Gvsuch that there are no edges between
each pairs of vertices fromM.
Given a set B⊆Ag, a B-dominant MIS(B-MIS)Mof Gv
is a maximal set of vertices in Gvsuch that (1) there is no
edge between each pairs of vertices fromMand (2) for all
hhAii./kϕ∈M, A⊆B.
Let misv (resp. misvB) denote the set of MIS (resp. B
-MIS) in Gv. Note that misv and misvB are sets whose
members are sets of sub-sentences. LetMISv :=
misv\
S
[[B]]./kϕ∈[[v]]misBv
∪nM∪ {[[B]]./kϕ} |M∈misB
v,[[B]]./kϕ∈
[[v]]o.As mentioned before, MIS andB-dominant MIS are used to ensure the consistency of the strategies of an agent in different coalitions. Intuitively, consider a transitive setv con-trolled by the constructor. To satisfy sentences inhhviiand [[v]], probabilistic constraints in each setMofMISvshould
be ensured simultaneously by one distribution, but this distri-bution need not satisfy the probabilistic constraints outside ofMby choosing proper actions.
Example 3. Consider the vertex v defined in Example 2,
we have:misv= n
hh1,2ii>0.5p
1,hh3ii>0.5(¬p1∨p2) o
,mis{v1,3}= n
hh3ii>0.5(¬p 1∨p2)
o ,MISv=
n
hh1,2ii>0.5p
1,hh3ii>0.5(¬p1∨p2), hh3ii>0.5(¬p
1∨p2), [[1,3]]>0.4(¬p1∧ ¬p2) o
.
Proposition 2. (Moon and Moser 1965) Given a set v of
sentences, the size ofmisv(resp.misvB) is at most3
|v|
3 and
can be computed in timeO(3|v3|).
Definition 6. Given a setM∈MISv, theobjectiveofMis
the setOM:={ϕ| hhAii./kϕ∈Mor[[A]]./kϕ∈M}.
A weighted cover (cM,wM) of OM such that cM ∈
Cb(O
M)satisfies M, denoted by (cM,wM) |= M, if for all
hhAii./kϕ,[[A]]./kϕ∈M, wM(ϕ)./k, where b=|M|+1.
Example 4. Consider the vertex v from Example 2. We have
MISv={M,M0}, whereM={hh1,2ii>0.5p
1,hh3ii>0.5(¬p1∨
p2)}andM0={hh3ii>0.5(¬p1∨p2),[[1,3]]>0.4(¬p1∧ ¬p2)}.
Furthermore, OM = {p1,¬p1 ∨ p2}, and OM0 = {¬p1 ∨ p2,¬p1∧ ¬p2}.OMhas one cover c1={p1,¬p1∨p2} such
that(c1,w1) |= Mfor some weight w1, e.g., w1({p1,¬p1∨
p2})=1.M0have two covers c2={¬p1∨p2,¬p1∧ ¬p2}
and c02 = {¬p
1∨p2}, {¬p1 ∧ ¬p2} such that(c2,w2) |=
M0 and (c02,w02) |= M0 for some weights w2 and w02, e.g.
w2({¬p1∨p2,¬p1∧ ¬p2}=1), w02({¬p1∨p2})=0.55and
w02({¬p1∧ ¬p2})=0.45).
Definition 7. Given a PAMC sentence φ, the two-player
game Gφ is a triple(V = V0 ]V1,E,vinit), where V0 and
V1 form a partition of V, Vifor i ∈ {0,1}is the finite set of
vertices for Player-i, vinit ={φ} ∈V0is the initial vertex of
the game and E⊆V×V is a finite set of edges.
V0and V1are defined below. E⊆V×V is the least set of
edges satisfying the following conditions.
1. (v,v∪ {φi})∈E ifφ1∨φ2∈v andφi<v, for i∈ {1,2}. 2. (v,v∪ {ϕ[ηZ.ϕ/Z]})∈ E ifηZ.ϕ ∈ v andϕ[ηZ.ϕ/Z] <v,
forη∈ {µ, ν}.
3. (v,v∪ {φ1, φ2})∈E ifφ1∧φ2∈v and{φ1, φ2}*v. 4. (v,⊥)∈E if both p∈v and¬p∈v for some p∈AP. 5. (v,>)∈E if v<Vtand Items 1-4 cannot be applied to v.
6. For each v∈ Vt, letMISv ={M1v,· · ·,M kv
v}, Cj :={cM |
(cM,wM)|=Mvj}, and C =Sj∈[kv]Cj. If Cj =∅for some
j∈[kv], then(v,⊥)∈E; otherwise(v,C)∈E,(C,Cj)∈E
for j ∈[kv],(Cj,c)∈ E for each c∈Cj, and(c,v0)∈E
for each v0∈c.
As a result, V0 := 2FL(φ) ∪ {⊥} ∪ {Cj | (C,Cj) ∈ E}and
V1 :=V1,0∪V1,1∪ {>}, where V1,0 ={C|(v,C)∈E},V1,1 = {c|(c,v0)∈E}, C,C
jand c are defined as above.
The intuition of Items 1-5 are straightforward. To see the intuition behind Item 6. Let us begin with v =
{hhA1ii./1k1φ1,· · · ,hhAnii./nknφn}. If Ai ∩Aj = ∅ for some
i,j∈[n], then for each pair of possible strategies ofAiandAj,
there always exist some common decisions whenAiandAj
enforce their strategies, hence their probabilistic constraints should besimultaneouslyensured by one distribution. Sup-posemisv ={M1,· · ·,Mm}(cf. Definition 5), then, (1) for
each pair ofhhAii./kφandhhA0ii./0k0
φ0inM
i,A∩A0=∅, (2)
for each sentencehhAii./kφfromM
i, for every setMj,Mi,
there exists a sentencehhA0ii./0k0
φ0inM
jwithA∩A0,∅, and (3) each sentence invmust occur at least in one setMi. This
allows us to consider each setMiindividuallyby assigning
the actionajto players inAj. For each setMj, we compute
its weighted covers, which check existence of distributions that satisfying the probabilistic constraints inMj.
For the general casev={hhA1ii./1k1ϕ1,· · ·,hhAmii./mkmϕm,
[[B1]]∼1h1ψ1,· · ·,[[Bn]]∼nhnψn}, we assume there exists a
player j<Sni=1Bi(cf. Remark 1). By assigning the action
bito the player jfor each sentence [[Bi]]∼ihiψi, it suffices to
checkindividuallyeach of{[[B1]]∼1h1ψ1,· · ·,[[Bn]]∼nhnψn}.
For eachBi, letmisBvi={Mi1,· · · ,M i
mi}, then (1) for each
sen-tencehhAii./kϕfrommisBi
v ,A⊆Bi, and (2)Mi`for 1≤`≤mi
must be a subset of a set inmisv. By assigning proper actions
to players (cf. previous paragraph), for each [[Bi]]∼ihiψi, it
suffices to check existence of a distribution for sub-sentences inMi`and [[Bi]]∼ihiψisimultaneously, which is done by
com-puting weighted covers. Moveover, each MISM∈misvsuch
thatM⊆M0for someM0∈misBi
v is omitted to avoid double
checking.
Overall, for eachv ∈ VtwithMISv = {M1v,· · ·,M kv v}, if
v1={hh1,2ii>0.5p1∧ hh3ii>0.5(¬p1∨p2)∧[[1,3]]>0.4(¬p1∧ ¬p2)}
v2={hh1,2ii>0.5p1∧ hh3ii>0.5(¬p1∨p2),[[1,3]]>0.4(¬p1∧ ¬p2)}
v3={hh1,2ii>0.5p1,hh3ii>0.5(¬p1∨p2),[[1,3]]>0.4(¬p1∧ ¬p2)}
C=n{p1,¬p1∨p2} ,{¬p1∨p2,¬p1∧ ¬p2} ,{¬p1∨p2},{¬p1∧ ¬p2} o
{p1,¬p1} v7={p1, p2}
{¬p1,¬p1∧ ¬p2}
{p2,¬p1∧ ¬p2} {¬p1}v8={p2}v9={¬p1,¬p2}
⊥
C1=n{p1,¬p1∨p2} oC2=n{¬p1∨p2,¬p1∧ ¬p2},{¬p1∨p2},{¬p1∧ ¬p2} o
c1={p1,¬p1∨p2} c2={¬p1∨p2,¬p1∧ ¬p2} c3=
{¬p1∨p2},{¬p1∧¬p2}
v4={p1,¬p1∨p2} {¬p1∨p2,¬p1∧ ¬p2} v5={¬p1∨p2} v6={¬p1∧ ¬p2}
>
Figure 1: The gameGΨforΨ, whereC,>,c1,c2,andc3are
Player-1 vertices, others are Player-0 vertices.
probabilistic constraints ofMvj, we added (v,⊥) intoE. i.e., Player-0 loses the game. Otherwise, the play goes toCand let Player-1 to choose oneMvjto dissatisfy, i.e., (C,Cj)∈E. At
Cj, Player-0 chooses one coverc∈Cjsuch that (c,w)|=M j v,
i.e., the distributionwsatisfies the probabilistic constraints of Mvj. Next, Player-1 chooses one set of sub-sentencesv0∈c to dissatisfy.
Example 5. Recalling the sentence Ψ = hh1,2ii>0.5p 1 ∧ hh3ii>0.5(¬p
1∨p2)∧[[1,3]]>0.4(¬p1∧ ¬p2), the
correspond-ing game is shown in Figure 1, in which c1,c2,c3,>and C are
Player-1vertices, others are Player-0vertices.hh1,2ii>0.5p1
andhh3ii>0.5(¬p
1 ∨p2) have some common decisions, as
well ashh3ii>0.5(¬p1∨p2)and[[1,3]]>0.4(¬p1∧ ¬p2). While, hh1,2ii>0.5p
1and[[1,3]]>0.4(¬p1∧ ¬p2)can have disjoint
decisions by assigning proper actions to Player-1. We have:
MISv={M,M0}, whereM={hh1,2ii>0.5p1,hh3ii>0.5(¬p1∨
p2)}andM0={hh3ii>0.5(¬p1∨p2),[[1,3]]>0.4(¬p1∧ ¬p2)}.
If Player-1chooses C1(resp. C2) to dissatisfy, then
Player-0can choose c1 (resp. c3) as the witness ofM(resp.M0).
It is easy to verify that Player-0has a winning strategy in the game GΨ by choosing c1,c3,v7,v8 and v9. The model
construction from a winning strategy is given in Example 6.
Finally, in order to prevent from regeneration sequences for eachµ-sentence ψ (i.e., derivation sequences that de-riveψinfinitely often), we construct a DPA which accepts all terminating regeneration sequences. By constructing the cross-product of the two-player gameGφand the DPA, we get the resulting parity gameGφsuch thatφis satisfiable iff Player-0 has a winning strategy in the parity gameGφ.
Definition 8. Given a PAMC sentenceφ, we define a B¨uchi
automatonPµ =(Sµ,Σ, δµ,Sµ0,Fµ)where Sµ =S0µ={ψ ∈ FL(φ)| ψcontains someµ-sentence},Σ =2FL(φ),δ
µ(q, α)= q0, if q0is derived from q and q0∈α, and Fµ(s)=0for all s∈Sµ.
Pµ precisely accepts all the sequences in which some
µ-sub-sentence of φ is derived infinitely often. LetPφ =
(Sφ,Σ, δφ,s0
φ,Fφ) be a DPA such thatL(Pφ) = Σω\L(Pµ).
Therefore,Pφ accepts all the sequences for which all the
µ-sentences inFL(φ) are regenerated finitely often.
Definition 9. Based on Definition 7 and Definition 8, we
define a turn-based two-player parity game Gφ = (V0 = V00]V10,E0,v0init,Ξ)where
• V00=V0×(Sφ∪ {♦}), V10=V1×(Sφ∪ {♦}), v0init=(vinit,s0φ),
• for every v∈V such that v <Vt, s∈Sφand(v,v0)∈ E:
then (v,s), (v0, δ
φ(s,v))∈E0; • for every v∈Vtand s∈Sφ:
– (v,s),(⊥,♦)∈E0if(v,⊥)∈E; – (v,s),(C,♦)∈E0if(v,C)∈E;
– (C,♦),(Cj,♦)∈E0for every(C,Cj)∈E;
– (Cj,♦),(c,♦)∈E0for every c∈Cj;
– (c,♦),(v0, δφ(s,v))∈E0for every v0∈c.
• Ξ : V0 → {0,· · ·,k} such thatΞ(v,s) = Fφ(s)for all s∈Sφ,
where♦is a placeholder, C,Cjand c are the vertices as in
Definition 7.
Lemma 1. Player-0has a winning strategy in the gameGφ
for every satisfiable sentenceφ.
To show the other direction of Lemma 1, we shall show how to construct a model from a winning strategy of Player-0. W.l.o.g., we assume that{1,· · · ,g}is the set of all players appeared in φ. It is well-known that, for parity games, if Player-0 has some winning strategies, then Player-0 has a pure memoryless winning strategy (Gurevich and Harrington 1982). Letζ0: V0
0 → V
0be a winning strategy of Player-0
inGφ. Since the DPAPφis deterministic, we can directly extract fromζ0:V0
0→V
0a winning strategyζ:V
0→Vfor
Player-0 by projecting fromV0ontoV.
The winning strategyζ and the gameGφtogether yield the digraph Gζφ in which only the edges specified by the strategyζ are retained. LetΠbe the set of all finite paths
~v=v1· · ·vr∈V0+(V1,0∪ {>}) inGζφsuch thatv1is either the
initial vertexvinitor there is an edge (v,v1) inG ζ
φsuch that
v∈V1,1. Note thatΠis a finite set. We denote byfst(~v) and
lst(~v) the vertexv1andvr, respectively. Given a vertexv∈V0,
letΠv⊆Πbe the set of finite paths starting fromv.
Let AP be the set of atomic propositions appearing in
φ,Mφ = (Ag,Act,Q,Γ, δ, λ,q0) be a PCGS obtained from
Gζφ, whereAg :={1,· · ·,g+1},Act:=S
i∈AgActi,Acti :=
S
q∈QΓi(q),Q := {q~v |~v ∈ Π} ∪ {q>},q0 := q~vsuch that
fst(~v)=vinit,λ(q~v) :=(Si∈[r−1]vi)∩AP, for~v=v1· · ·vr∈Π
andλ(q>) :=∅.Γ:=(Γi)i∈Agandδare defined as follows. Consider a path~v=v1· · ·vr ∈Πsuch thatvr ,>, then
vr−1∈Vt, we assume that
hhvr−1ii:={hhA1ii./1k1ϕ1,· · ·,hhAmii./mkmϕm}
[[vr−1]] :={[[B1]]∼1h1ψ1,· · ·,[[Bn]]∼nhnψn}.
Letι(hhAjii./jkjϕk) (resp.ι([[Bj]]∼jhjψj)) denote the index j.
Case 1: If [[vr−1]]=Si∈[m]Ai=∅, then we use one actiona
and letΓi(q~v) :={a}for everyi∈Ag.
Case 2: Otherwise, we consider m + n distinct actions
a1, ...,am,b1, ...,bn. LetΓg+1(q~v) := {b1,· · · ,bn}andΓi(q~v)
1. for every j∈[m] andi∈Aj,aj∈Γi(q~v)
2. for every j∈[n] andi∈Bj,bj∈Γi(q~v).
In the sequel, we defineδ. We first extract essential infor-mation from~v. SupposemisBj
vr−1 :={M
j 1,· · ·,M
j
`j}, for j ∈
[n] andmisvr−1\ S
j∈[n]mis Bj
vr−1 :={M
0 1,· · ·,M
0
`0}.Note that inCase 1, there is only one MIS that ishhvr−1ii.
By the construction ofGφ, for everyM∈MISvr−1, there is a weighted cover (c,w) such that|c| ≤ |M|+1, (c,w)|=M. LetwMbe the distribution such thatwM(q~v0)=w(fst(v~0)) for allv~0∈S
v∈cΠv,w>be the distribution such thatwM(q>)=1.
Moreover,vr≡ {c|(c,w)|=M, M∈MISvr−1}. To defineδ, we introduce some auxiliary notations.
• For a sentenceψ=hhAii./kϕ∈ hhv r−1ii, let
D[ψ]∃:={d∈D(q~v)| ∀i∈A, d(i)=aι(ψ)}.
• For an MISM∈ {M0 1,· · ·,M
0 `0}, let
D[M]∃:=Tψ∈MD[ψ]∃andD∃:=SM∈{M0 1,···,M0`0}
D[M]∃.
• For a sentenceϕ=[[B]]∼hψ∈[[vr−1]], let
D[ϕ]∀:={d∈D(q~v)| ∀i∈B, d(i)=bι(ϕ)}.
• For a sentenceϕ=[[B]]∼hψ∈[[v
r−1]] and aB-MISM∈ misvB
r−1, let
D[M, ϕ]∀:=D[M]∃∩D[ϕ]∀and
D∀:=S[[B]]∼hψ∈[[v
r−1]],M∈misBvr−1
D[M,[[B]]∼hψ]
∀.
The following proposition reveals the property of action assignments of agents.
Proposition 3. The following statements hold:
1. For everyM∈ {M0 1,· · · ,M
0 `0},[[B]]
∼hψandM0∈misB vr−1:
D[M]∃∩D[M0,[[B]]∼hψ]∀=∅.
2. For everyM1,M2∈ {M01,· · ·,M0`
0}, ifM1 ,M2, then
D[M1]∃∩D[M2]∃=∅.
3. For every[[B]]∼hψ,[[B0]]∼0h0ψ0,M ∈ misvB
r−1 andM
0 ∈
misvB0
r−1, ifM,M
0or[[B]]∼hψ
,[[B0]]∼ 0h0
ψ0, then
D[M,[[B]]∼hψ]
∀∩D[M0,[[B0]]∼ 0h0
ψ0] ∀=∅. 4. For everyϕ, ϕ0∈ hhv
r−1ii, ifϕ,ϕ0, then
(D[ϕ]∃\(D∃∪D∀))∩(D[ϕ0]∃\(D∃∪D∀))=∅.
5. For everyϕ, ϕ0∈[[vr−1]], ifϕ,ϕ0, then
(D[ϕ]∀\(D∃∪D∀))∩(D[ϕ0]∀\(D∃∪D∀))=∅. 6. For everyϕ∈ hhvr−1iiandϕ0∈[[vr−1]],
(D[ϕ]∃\(D∃∪D∀))∩(D[ϕ0]∀\(D∃∪D∀))=∅.
We now defineδas follows.
1. ForM∈ {M01,· · ·,M0`
0}andd∈D[M]∃,δ(q~v,d) :=wM. 2. For all ϕ = [[B]]∼hψ ∈ [[v
r−1]], M ∈ misBvr−1 and d ∈
D[M, ϕ]∀,δ(q~v,d) :=wM.
3. For allϕ∈ hhvr−1iiandd∈D[ϕ]∃\(D∃∪D∀), for some M∈MISvr−1 such thatϕ∈M,δ(q~v,d) :=wM. Note that suchMalways exists.
4. For allϕ=[[B]]∼hψ∈[[v
r−1]] andd∈D[ϕ]∀\(D∃∪D∀), for someM ∈ MISvr−1 such thatϕ∈ M,δ(q~v,d) := wM.
Note that suchMalways exists as well.
5. For all the otherd∈D(q~v),δ(q~v,d)=w>.
Example 6. Consider the game inFigure 1and the strategy
ζwithζ(C2)=c3,ζ(v4)=v7andζ(v5)=v8. Then,ζis a
win-ning strategy for Player-0from which we can construct the di-graphGζ
Ψ.G
ζ
Ψis the subgraph of the game shown in Figure 1 consisting of the vertices {v1,· · · ,v9,C,C1,C2,c1,c3,>}
and related edges between them. Πconsists of four finite paths{v1v2v3C,v4v7>,v5v8>,v6v9>}(blue paths in Figure 1).
From Π, using the weights defined in Example 4, we get the model of φ as shown in Figure 2, where players are
{1,2,3,4} and actions are {a1,a2,b1} with Γ1(q0) = {a1},
Γ2(q0) = {a1,b1} and Γ3(q0) = {a2}, λ(q1) = {p1,p2},
λ(q2)={p2}andλ(q3)=∅.
q0 q1
q2
q3 h
a1, a1, a2, b1i,1
ha1, b1, a2, b1i,0.45
ha1, b1, a2, b1i,0.55
q0=qv1v2v3C q1=qv4v7>
q2=qv5v8> q3=qv6v9>
Figure 2: The model ofΨ.
By the construction (in particular, the functionδand finite path setΠ), we can verify that
Lemma 2. Mφsatisfiesφ.
Theorem 3. The satisfiability problem for PAMC is
EXPTIME-complete. Moreover, if φis satisfiable, we can construct a model ofφin exponential size of|φ|such that
• the number of players is bounded by k+1, where k is the number of the players occurring inφ,
• the number of actions is bounded by|FL∃(φ)|+|FL∀(φ)|+1,
• the out-degree is bounded by|FL∃(φ)|+2.
Remark 1. Without adding the player g+1, our reduction
still works ifAg,(B∪B0)for each distinct pair of[[B]]∼hψ and [[B0]]∼0h0
ψ0 in[[v]]. The caseAg = B∪B0could be solved by adapting the reduction by adding new MISsMinto
MISv, which takes into account the common decisions for [[B]]∼hψand[[B0]]∼0h0ψ0. We leave it as future work.
Related Work
Probabilistic temporal logics such as PCTL, PCTL∗, PLTLK,
µ-calculi and their variants (Vardi 1985; Morgan and McIver 1997; Huth and Kwiatkowska 1997; de Alfaro and Majumdar 2001; McIver and Morgan 2002; 2007; Mio 2012; Cleave-land, Iyer, and Narasimha 2005; Huang and Kwiatkowska 2016; Fu et al. 2018) can express probabilistic properties, rather than coalitions of players.
focus of the current paper, is a long-standing open problem for PATL and PATL∗, notwithstanding with some progress (Br´azdil et al. 2008; Chakraborty and Katoen 2016).
Satisfiability checking of (non-probabilistic) alternating-time temporal logics has been considered (Goranko and Vester 2014). Our decision procedure shares some similarity, specifically, MIS plays a similar role as the distributed con-trol therein. Modulo the probabilistic aspects and fixpoints, the main difference is that, to handle coalition strategies we are consideringmaximallyindependent sets rather than all subsets of distributed controls. As a result, our construction leads to potentially smaller models, which is important for implementation, but is at the cost of a considerably more complicated construction.
Much research has been dedicated to algorithms of solv-ing stochastic games, e.g., (Chatterjee 2007; Chatterjee and Henzinger 2012; Br´azdil et al. 2006) in which objectives are usually given byω-regular properties or PCTL. Stochastic games are useful for synthesis and verification of reactive systems, but hard to express cooperation of players.
Conclusion and Future Work
We proposed a temporal logic PAMC for reasoning about strategic abilities of agents in stochastic multi-agent systems. We showed that the complexities of its model checking and satisfiability checking problems lie in the same complexity classes of the respective problems of modalµ-calculus and AMC. We also implemented a satisfiability solver for PAMC. Several questions are left open for PAMC, such as (com-plete) axiomatization and extensions with epistemic opera-tors. We leave them as future work.
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