Definition 5.3.1. Given two propositional valuationsV, V′ :
P→(A→ {w,l}), we say thatV′ extends V (notation V ⊆V′) if for all p∈
P, i∈A
V(p)(i) =w impliesV′(p)(i) =w.
Similarly, given a frame (W, R)and two valuations over that frame V, V′ :
W →(P→(A→ {w,l})),V′ extendsV if for allw∈W,p∈
Pandi∈A
V(w)(p)(i) =w impliesV′(w)(p)(i) =w.
Intuitively, in case ofMPL, the valuationV′ extendsV if for every proposi-
tion letterp, the valuationV′makesptrue for all the agents for whichV makes
ptrue and possibly more. Or, in other words, for everyp, the set of winners of pgivenV is a subset of the set of winners forpgivenV′. In the modal case,V′ extendsV if for every proposition letterpand every statew, the set of players that winpgivenV atwis a subset of the set of winners forpgivenV′ atw.
Definition 5.3.2. Consider the smallest and largest valuations Vs and Vl,
respectively, defined as follows. In the propositional case,
Vs(p)(i) =l for allp∈
Pandi∈A, and,
Vl(p)(i) =w for allp∈
Pandi∈A. Given any Kripke frame (W, R)define,
Vs(w)(p)(i) =l for allw∈W,p∈
Pandi∈A, and
Vl(w)(p)(i) =w for allw∈W,p∈
Pandi∈A.
Because usually no confusion will arise we use the same notation for both valuations defined over the set of propositional lettersPand valuations defined over frames. It follows immediately from the above definition that for any valuation V: Vs ⊆ V ⊆ Vl. It will be convenient to extend valuations to
functions over arbitrary formulas.
Definition 5.3.3. For any MPL formula φdefine:
V(φ)(i) =w⇔playerihas a winning strategy for the game G(φ, V)@(φ, Id), and in the case of a MML formula φ, given a multi-player model M = (W, R, V)andw∈W,
V(w)(φ)(i) =w⇔ playeri has a winning strategy for the game
G(φ, M)@(φ, w, Id), that isM, wiφ.
Proposition 5.3.4. Let φ be a MPL formula and V, V′ two valuations such
thatV′ extendsV. If for somei∈
A,V(φ)(i) =w, thenV′(φ)(i) =w.
Proof. Let V and V′ be as in the statement. The proposition can be proved
by an easy induction on the complexity ofφ. We will only discuss the case of φ=¬jkψ. AssumeV(¬jkψ)(i) =w. Distinguish cases: i∈ {j, k}andi /∈ {j, k}.
In the first case, assume wlog thati=j. By assumptionV(¬jkψ)(j) =w. That
is, playerjhas a winning strategy for the gameG(¬jkψ, V). From this we may
conlcude thatk has a winning strategy for the gameG(ψ, V) by mimickingj’s strategy during the game. By induction hypothesis V′(ψ)(k) = w and hence
V′(¬
jkψ)(j) = w. Also, in the second case i /∈ {j, k}, the result is easily
obtained: V(¬jkψ)(i) =w implies thatV(ψ)(i) =w. By induction hypothesis
V′(ψ)(i) =w and henceV′(¬
jkψ)(i) =w.
Remark 5.3.5. From the above proposition and the observation thatVlextends
all valuations, it follows that if aMPLformulaφisi-satisfiable thenVl(φ)(i) =
Theorem 5.3.6. i-satisfiability of MPL is in P.
Proof. By the above remark 5.3.5 it follows that in order to determine whether a givenMPL-formulaφisi-satisfiable it suffices to evaluateVl(φ)(i). This can
be computed in very little (read: polynomial) time, since it involves only one check per connective.
Theorem 5.3.7. i-validity of MPL is in P .
Proof. We apply similar reasoning as in the above theorem. In order to deter- mine whether a givenMPL-formulaφisi-valid it suffices to evaluateVs(φ)(i).
Again, this can be done in polynomial time, since it involves only one logical operation per connective.
At this point we can hear the reader wonder: what abouti-satisfiability (ori- validity) ofMML? Can it be decided in polynomial time as well? Unfortunately, at present we do not have any decicive answer to this question. The reason being that we cannot prove (or disprove) that every i-satisfiable formulaφ of MML
with arbitrary valuations is satisfiable on a model that is polynomial in the size ofφ. We will come back to this so-calledpolysize model property in section 5.4.
Restricted Valuations
We have just discussed the complexity of the i-satisfiability for MPL in case of arbitrary valuations. It seems natural however, to consider the case of re- stricted valuations as well. Remember that restricted valuations were defined as follows: for every proposition letterp there have to be two playersi and j such thatV(p)(i) =w andV(p)(j) = l. Similarly, in case of MML, for every proposition letterpand every statewthere have to be two playersiandjsuch that V(p)(w)(i) = w andV(p)(w)(j) =l. In chapter 2 we already briefly dis- cussed some of the significant implications of moving from arbitrary to restricted valuations. In this part, we will see that if we confine ourselves to restricted valuations we lose the nice computational properties of MPL as discussed in above. That is, we will show thatMPLwith arbitrary valuations is in NP.
Lemma 5.3.8. The satisfiability problem of PL is polytime reducible to the
i-satisfiability problem of MPL with restricted valuations.
Proof. Letφbe a formula ofPL. Fix a player 0∈A. In order to check whether φis satisfiable we can translate it into a formulaφ′ ofMPLand check whether
φ′is 0-satisfiable. The translation will be as follows: Let 16= 0∈
A, then in the following order
- distribute¬over∨and∧and apply double negation elimination whenever possible (that is, rewriteφinto negation normal form),
- replace proposition lettersW pthat are not under the scope of¬with
- replace⊥with⊥0,⊤with⊤0,
- replace¬ψ with¬01ψ(note thatψcan only be a proposition letterp,⊥0
or⊤0),
- replace∨with∨0,
- replace∧with∨1.
This algorithm, translating a PLformulaφ into a MPLformula φ′ can be
executed in polynomial time. What remains to be shown is that the above translation really reduces satisfiability ofPLto 0-satisfiabilility ofMPL.
Claim 5.3.9. A formula φ of PL is satisfiable iff the MPL formula φ′ is 0-
satisfiable.
Proof. Without loss of generality we may assume that φis in negation normal form.
For the direction from left to right, assumeφis satisfiable inPL. LetV be the valuation evaluatingφto 1. Define aMPLvaluationV′ as follows:
V′(p)(k) =wifV(p) =1, for all playersk6= 1,
V′(p)(1) =wifV(p) =0.
First of all, note that since V is a standard propositional valuation, it follows that V′ is a valuation in the restricted sense. By induction on φwe will show
that player 0 has a winning strategy for the game G(φ′, V′). The inductive
argument can be found in the appendix 7.4.
In order to show the direction from right to left assume φ′ is 0-satisfiable
using restricted valuations. Let V′ be the valuation such that player 0 has a
winning strategy for the gameG(φ′, V′). Define a propositional valuationV as follows:
V(p) =1if there is a k6= 1 such thatV′(p)(k) =w,
V(p) =0ifV′(p)(1) =w.
By induction on φ we show that V(φ) = 1. The proof can be found in the appendix (7.4). We may conclude that the satisfiability problem of PLis polytime reducible to thei-satisfiability problem ofMPL.
Theorem 5.3.10. i-Satisfiability of MPL with restricted valuations is NP- complete.
Proof. By the above lemma it follows that i-satisfiability ofMPL is NP-hard. What remains to be shown is that it is in NP. This boils down to showing that if we are given a (restricted) valuation by a nondeterministic Turing machine we can deterministically compute in polynomial time whether the given valuation i-satisfiesφor not. Just like in case of classical propositional logic this can easily be done since we have to do only one logical operation per connective.