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40Логические исследования 2018. Т. 24. № 2. С. 59-69 УДК 16+510.64
Logical Investigations 2018, Vol. 24, No. 2, pp. 59-69 DOI: 10.21146/2074-1472-2018-24-2-59-69
Неклассическая логика
Non-classical Logic
Natasha Alechina
Model checking for coalition announcement logic
Natasha Alechina
University of Nottingham,
Jubilee Campus, Nottingham, NG8 1BB, UK.
E-mail: natasha.alechina@nottingham.ac.uk
Abstract: This talk is based on joint work with Rustam Galimullin and Hans van Ditmarsh, published in the German Conference on Artificial Intelligence (KI 2018). First I will introduce background and motivation for the work. I will introduce multi-agent Epistemic Logic (EL) for representing knowledge of (idealised) agents, Public Announcement Logic (PAL) for modelling knowledge change after truthful announcements, Group Announcement Logic (GAL) for modelling what kinds of changes in other agents' knowledge a group of agents can effect, and Coalition Announcement Logic (CAL) which is the main subject of the talk. CAL studies how a group of agents can enforce a certain outcome by making a joint announcement, regardless of any announcements made simultaneously by the opponents. The logic is useful to model imperfect information games with simultaneous moves. It is also useful for devising protocols of announcements that will increase some knowledge of some agents, but also preserve other agents' ignorance with respect to some information (in other words, preserve privacy of the announcers). The main new technical result in the talk is a model checking algorithm for CAL, that is, an algorithm for evaluating a CAL formula in a given finite model. The model-checking problem for CAL is PSPACE-complete, and the protocol requires polynomial space (but exponential time).
Keywords: epistemic logic, public announcement logic, coalition announcement logic, model checking algorithm
For citation: Alechina N. "Model checking for coalition announcement logic", Logicheskie Issledovaniya / Logicallnvestigations, 2018, Vol. 24, No. 2, pp. 59-69. DOI: 10.21146/20741472-2018-24-2-59-69
1. What this talk is about
Logics for describing announcements by (groups of) agents, and how announcements affect agents' knowledge. More precisely, model checking algorithm for Coalition Announcement Logic.
The report based on the paper just accepted for the German Conference on Artificial Intelligence (KI 2018).
© Alechina N.
First, we will introduce some background on logic of knowledge, logic of public announcements, and logics of group and coalition announcements.
1.1. Multi-agent epistemic logic: example
For more detailed exposition of epistemic logic see [Hintikka, 1962].
Consider the following example: there are three agents, a, b and c.
Suppose that, a and b are households that either consume or not consume power (p1 is true if a's power is on, and p2 is true if b's power is on). c is an electricity substation that needs to know how many households consume power, but not whether individual households consume power or not.
We describe above the situation of c not knowing anything about power consumption (and a and b knowing their own and each other's status).
Possible worlds (or states) are w0, w1, w2, w3 with different truth values of p1 and p2. In wo, a and b know that they are in wo and —p1 Ap2 is true c does not know whether p1 and p2 are true (see Fig. 1):
Fig. 1. Example 1.
2. Multi-agent epistemic logic
Fix a non-empty finite set of agents A and a set of propositional variables P. Definition 1. Kripke model A Kripke model is a triple M = (W, V), where
• W is a non-empty set of states,
• A ^ P(W x W) assigns an equivalence relation to each agent, and
• V : P ^ P(W) assigns a set of states to each propositional variable.
• M is called finite, if W is finite.
• A pair (M, w) with w e W is called a pointed model, where w e W is an actual world.
Consider the following epistemic language Lel:
p,^ ::= p | -p | (p A | Kap
where p e P, a e A, and all the usual abbreviations of propositional logic and conventions for omitting parentheses hold. Kap stands for 'a knows that p'. The dual operator 'a considers p possible' Kap is defined as -Ka-ip. Forcing relation is defined as follows:
(M, w) |= p iff w e V(p)
(m, w) = -p iff (M, w) = p
(m, w) = p A ^ iff (m, w) = p and (M, w) = ^
(M, w) = Kap iff Vv e W : w v implies (M, v) = p
Let us consider the following example:
(M, wo) = Ka(-p1 A p2) (M, wo) = -Kc(-p1 A P2) (M, wo) = -Kc-p1
See the Example 1 on Fig. 1.
3. Public Announcement Logic (PAL)
Public announcement logic was initally proposed by Plaza [Plaza, 2007]. Let us consider the following examples:
Suppose c hears that (-p1 A p2) V (p1 A -p2) is true. Some worlds become impossible (see Fig. 2).
w3
Fig. 2. Example 2.
Suppose c hears that exactly one of the households consumes power: (—p1 A p2) V (pi A —p2). After this announcement, c only considers w0 and w\ possible
(see Fig. 3).
Fig. 3. Example 3.
3.1. Definitions
The language of PAL is Lpal:
f, 0 ::= p | ->f | (f A 0) | Kaf | [0]f,
where as before, p e P, a e A.
[0]f stands for 'after 0 is truthfully announced, f holds'.
The dual operator (0)f is defined as -[0]-f and means '0 is true, and after it is announced, f is true'.
Given (M, w) and f e LEL, an updated model (M, w)^ is a restriction of the original model to the states where f holds.
(M, w) = [f ]0 iff (M, w) |= f implies (M, w)^ = 0 (m, w) = (f )0 iff (m, w) |= f and (M, w)^ = 0
4. Group and Coalition Announcement Logics (GAL and CAL)
Intuition: announcements by (group of) agents:
• announcements are made by agents
• agents can only announce what they know
• for example, a can announce Ka((-p1 A p2) V (p1 A -p2)), but c can not
• the only thing c can announce in (M, wo) is KcT
• a group of agents can announce a conjunction of formulas, each formula known by an agent in the group.
The language Lgal of group announcement logic [Agotnes, van Ditmarsch, 2008] is Lpal extended with (G)f, where G C A, which stands for 'there is a truthful announcement by G, after which f holds'. Let Lgl denote the set of formulas of the type /\aeG Kafa, where for every a e G it holds that fa e LEL.
(M,w) = (G)f iff 30gLgl : (M,w) = (0)f
The language Lcal [Agotnes et all, 2010] of coalition announcement logic is Lpal extended with ([G])f, where G C A, which stands for 'there is an announcement by G such that whatever agents in A \ G announce simultaneously, afterwards f holds':
(M, w) |= ([G])f iff 30glgl Vxg4lg : (M, w) = 0 A [0 A x]f
• a and b together can make an announcement after which Kc(((—p1 Ap2) V (P1A-P2)) A—Kc-P1) holds: (M,wo) = (a, b)Kc(((—p1 AP2) V(p1 A-P2)) A —Kc—P1); also,
(M,wo) = (a,b])Kc(((—p1 A P2) V (p1 A —P2)) A —Kc—p^
• a can make an announcement after which Kc(((—p1 A p2) V (p1 A —p2)) A —Kc—P1) holds: (M, wo) = (a)Kc(((—P1 A P2) V (p1 A —P2)) A — Kc— P1)
• a cannot make an announcement after which Kc(((—p1 Ap2) V(p1 A—p2)) A —Kc—p1) holds, no matter what other agents announce simultaneously (because b can announce Kbp2):
(M, wo) = (a])Kc(((—p1 A P2) V (p1 A —P2)) A —Kc—P1)
5. Model checking CAL
Model checking problem for a logic L: given a (finite) model M of L and a formula 0 of L, does it hold that M = 0?
Model-checking problem for CAL: given a pointed Kripke model (M, w) and a formula 0 of CAL, does it hold that M = 0?
The model checking problem for CAL is interesting because we can use it to plan epistemic actions. What can we tell other agents so that we are guaranteed to get just the right information to them without revealing too much. For example, epistemic planning and verification of distributed protocols.
Definition 2. Bisimulation Let two models M = (W, ~ V) and M' = (W', V') be given. A non-empty binary relation Z C W x W' is called a bisimulation if and only if for all w e W and w' e W' with (w, w') e Z:
• w and w' satisfy the same propositional variables;
• for all a e A and all v e W: if w v, then there is a v' such that w' v' and (v, v') e Z;
• for all a e A and all v' e W': if w' v', then there is a v such that w v and (v, v') e Z.
Let (M, w), (M', w') are pointed models and Z C W x W' is a bisimulation, then (M, w) and (M , w ) are bisimilar.
If Z1,Z2 are bisimulations, then Z1 U Z2 is a bisimulation. Union of all bisimulations is a maximal bisimulation.
Definition 3. The quotient model of M with respect to some relation R is MR = (WR, ~R, VR), where WR = {[w] | w e W} and [w] = {v | wRv}, [ w] [v] iff 3w' e [ w], 3v' e [ v] such that w' v' in M, and [ w] e VR(p) iff Vw' e [w] : w' e V (p).
Definition 4. Bisimulation contraction of M (written |M|) is the quotient model of M with respect to the maximal bisimulation of M with itself, i.e. bisimulation contraction is the minimal representation of M.
Definition 5. A model M is bisimulation contracted if M is isomorphic to |M |.
Lemma 1. (| M |, w) = <p iff (M,w) = <p for all G Lcal ■
Every pointed model (M, w) is distinguished from all other non-bisimilar pointed models (M, v) by some distinguishing formula ¿w G Lel.
Given a finite model (M,w), distinguishing formula ¿w is constructed recursively as follows:
5kw+1 ::= ¿W A A ( A Ka5kv A K \ ¿V),
a€A w~av w~av
where 0 < k < | W|, and 5°w is the conjunction of all literals that are true in
w, La ¿w ::= Awev(p) P A Aw^v(p) -P.
A distinguishing formula for a set of states S is
Ss ::= V Sw.
w€S
See Fig. 4, 5, 6, 7.
Fig. 4. Example 4 (Alternative model M': a and b only know "their" variable).
Fig. 5. Example 5 (a's equivalence in w0: can announce Ka—p\).
Fig. 6. Example 6 (b's equivalence in wo: can announce Kbp2). Let M/a = {[wi]a,..., [wn]a} be the set of a-equivalence classes in M.
Definition 6. A strategy Xa for an agent a in a finite model (M, w) is a union of equivalence classes of a including [w]a.
Definition 7. The set of all available strategies of a is S(a, w) = {[w]a U Xa : Xa CU M/a}.
Definition 8. Group strategy XG is defined as naeGXa for all a £ G. The set of available strategies for a group of agents G is S(G,w) = {naeGXa : Xa £ S(a,w)}.
Fig. 7. Example 7 (w0 is the intersection of a's and b's equivalence classes in w0: can announce Ka—p\ and Kbp2).
Given a finite and bisimulation contracted model (M,w) and strategy XG, a distinguishing formula 5Xg for XG is VwexG $w •
In a bisimulation contracted model we propose alternative truth definition for {G]},
(M, w) \= {G~}y iff 3XG e S(G,w) yXA\G e S(A\G,w) : (M,w)xg<ixa\g = y.
6. Model checking algorithm
Algorithm mc(M,w,y0), where (M,w) is a pointed model and yo is some formula:
case yo
p : if w e V(p) then return true else return false;
—y : if —mc(M,w,y) then return true else return false;
y A ^ : if mc(M, w, y) and mc(M, w, ^) then return true else return false;
Kay : for all v ~a w
if —mc(M,v,y) then return false; return true
{^}y if —mc(M,w,^>) then return false, else compute the ^-submodel of M and return mc(M^,w,y).
([G]}^>: compute (|M |,w) and sets of strategies S(G,w) and S(A \ G,w) for all Xg G S(g, w) . check = true; . for all XA\G G S (A \ G, w)
. if -mc( |M |XGnXA\G ,w,(p) then check = false . if check then return true return false.
Theorem 1. The model checking problem for CAL is PSPACE-complete.
7. Summary
We can use model checking to verify properties of announcements (for example, communication protocols, or data collection).
We can also use it to produce strategies (the right announcements to make) given the properties that should hold after the announcement
8. Questions
Question (V.I. Shalack): All these epistemic logic are S5, so we have got a problem with paradox of omniscience. Is there some another way to interpret knolwedge differently? For example, in jurisprudence and legal practice we are led by the principle that ignorance of law was no excuse, i.e. if a law was accepted, than everyone should know this laws and its consequences.
Answer: I prefer the syntactical interpretation of an epistemic modality, where every agent has his own 'knowledge set' of formulas, with their own rules how to infer consequences from these formulas. This way allows to model epistemic situations without logical omniscience. I agree that S5 is not always a good logic for modelling knowledge. On the other hand, if we work with a small number of formulas, then it is less paradoxical to assume omniscience limited to these formulas only.
Acknowledgements. Paper is RAS Institute of Philosophy Seminar Talk, 19 June 2018 (See video https://www.youtube.com/watch?v=Tfgg-ZO4JkM). It is a joint work with Rustam Galimullin and Hans van Ditmarsch. This report is based on KI 2018 paper: Rustam Galimullin, Natasha Alechina and Hans van Ditmarsch. Model Checking for Coalition Announcement Logic. To appear in Proc. The 41st German Conference on Artificial Intelligence (KI 2018).
References
Plaza, 2007 - Plaza, J. "Logics of public communications", Synthese, 2007, Vol. 158, No. 2, pp. 165-179.
Hintikka, 1962 - Hintikka, J. Knowledge and belief. An introduction to the logic of the two notions. Ithaca, New York: Cornell University Press, 1962. 179 pp.
Ägotnes et all, 2010 - Agotnes, T., Balbiani, P., van Ditmarsch, H., Seban, P. "Group announcement logic", Journal of Applied Logic, 2010, Vol. 8, No. 1, pp. 62-81.
Ägotnes, van Ditmarsch, 2008 - Agotnes, T., van Ditmarsch, H. "Coalitions and announcements", in: Padgham, L., Parkes, D.C., Müller, J.P., Parsons, S. (eds.), 7th International Joint Conference on Autonomous Agents and Multiagent Systems (AAMAS 2008). IFAAMAS, 2008, pp. 673-680.
