Dialogue games for Dishkant’s quantum modal logic Текст научной статьи по специальности «Математика»

Dialogue games for Dishkant's quantum modal logic

Vladimir L. Vasyukqy1

abstract. Recently some elaborations were made concerning the game theoretic semantic of Land its extension. In the paper this kind of semantics is developed for Dishkant's quantum modal logic LQ which is also, in fact, the specific extension of L. As a starting point some game theoretic interpretation for the SL system (extending both Lukasiewicz logic Land modal logic S5) was exploited which has been proposed in 2006 by C. Fermiiller and R. Kosik. They, in turn, based on ideas already introduced by Robin Giles in the 1970th to obtain a characterization of L in terms of a Lorenzen style dialogue game combined with bets on the results of binary experiments that may show dispersion.

Keywords: Lukasiewicz's logic, quantum loigic, dialogue games, risk value

1 Introduction

In [4],[5] Robin Giles determines a logic for reasoning about physical theories with dispersive experiments, meaning that repeated trials of the same experiment may yield different results. Giles refers to Lorenzen's dialogue games for intuitionistic and classical logic which systematically reduce arguments involving logically complex assertions to arguments about atomic assertions.

In the issue Robin Giles formally defined a characterization of infinite-valued Lukasiewicz logic in terms of a game that combines dialogue rules for logical connectives with a scheme for betting on results of dispersive experiments for evaluating atomic propositions.

xThis study comprises research findings from the 'Game-theoretical foundations of pragmatics' Project № 12-03-00528 carried out within The Russian Foundation for Humanities Academic Fund Program.

In this connection it is interesting that Herman Dishkant introduced the modal extension of Lukasiewicz's infinite-valued logic which allows to consider physical objects obeying to the rules of quantum mechanics. This suggests to extend Giles' method to Dishkant's logic for obtaining a characterization of that in terms of a dialogue game too. The starting position and conditions in this case would be as follows.

The main idea of H. Dishkant's quantum modal logic (LQ) [1] is to include Mackey's axioms for probabilities of quantum-mechanical experiments [6] into the calculus of Lukasiewicz's infinite-valued logic Lk0 treating probabilities as truth-values. It is done not directly and Mackey's construction plays the role of a leading idea only and resulting calculus is, in essence, a modal extension of Lukasiewicz logic where the last is enriched with the modal symbol Q and four modal inference rules. The proposition QA expresses such a property which can be observed and the presence of which confirms A ('A is confirmed by observation').

The system LQ contains four axioms and five rules of inference:

A1. A2. A3. A4

B5. B6. B7. B8.

A — (B — A)

(A — B) — ((B — C) — (A — C)) ((A — B) — B) — ((B — A) — A) (—A — —B ) — (B — A) A, A — B

B

A

QA

A —Q—A

A — B QA — QB

B9 QA—QB

' (QB — QA) o Q(QB D QA)

where A D B =def (-A — B) — B.

Semantically Dishkant' system LQ would be interpreted in the following way. Usually a quantum object is described by a wave function — by a unit vector of a complex Hilbert space HR. Let ^ be the set of all states of an object and besides these states we consider also questions which are described by closed subspaces of HR. Each such closed subspace p determines a probability p(^) of a positive answer to the question for any ^ € It is known that this probability is equal to the squared modulus of the projection of ^ on the subspace p, i.e. p(^) =

Since p1 = p2 ^ p1 = p2 then we do not identify the question with the corresponding function of HR but with the corresponding function p : ^ — [0,1] for which there exists such p that p(^) = \^p\2. Here [0,1] is the closed segment of real numbers.

Let P be the set of all questions and for any p € P let p be the corresponding subspace of HR. We call any function g : ^ — [0,1] a generalized question and the set of all generalized questions will be denoted by S. Obviously P C S. The set S is partially ordered by the relation < which is defined by

g < h =def < h(^)) for any g,h €S.

Now let us fix a function q : S — P satisfying the conditions

q1. g < h ^ q(g) < q(h) q2. q(p) = p

for any g,h € S; p €P. It is easy to see that there is at least such a function q (e.g. one may take q(g) equal to that p for which p is the minimal subspace containing all ^ € ^ for which q(^) = 1).

Any function ID : W0 — S is an interpretation of LQ if it satisfy the following conditions:

(I) ID(A - B) = min(1,1 - ID(A) + ID(B));

(II) ID(-A) = 1 - ID(A);

(III) ID(QA) = q(ID(A))

for any A,B £ W0, where W0 — a set of formulas of LQ. Here 1 : ^ ^ {1}, where ^ is a set of all states of an object. It is obvious that

ID

may be defined on V (the infinite list of propositional variables) arbitrarily and then extended uniquely on W0, if q is fixed.

It seems that under such definition q plays for modal formules the same role as Mackey's function r which assigns to every triple (A, a, E) (where A is an observable, a is a state and B is a Borel subset of the real line) the number r(A,a,E), 0 < r(A,a,E) < 1. So, we can treat W0 as the set of observables, dom(S) as the set of states and rng(S) as the set of all Borel subsets of the real line. The following result holds for such an interpretation ID [1, p. 152]:

Theorem 1. For any A £ Wif h A then ID (A) = 1 for any intepretation ID.

The weak completeness (semantic correctness) of LQ was proved just relative to the usual quantum propositional logic QPL (by embedding QPL in LQ). In view of this the problem was formulated to construct semantic model like those of Kripke-Grzegorczyk but for LQ. In [7] such Kripke-type model for L^0 was yielded where an accessibility relation is a ternary one and in [8, p. 67] such model was extended to LQ and the soundness and completeness of LQ in respect to those was proved.

According to [8] the ternary semantic of LQ would be described as follows. L-frame is a quadruple (O, K, R*) where K is nonempty set of observation points (states), O £ K, R is a ternary accessibility relation on K and * — a unary operation on K. The following conditions for R and * are satisfied:

(p1) ROaa (p2) Raaa

(p3) R2abcd ^ R2acbd (p4) R2Oabc ^ Rabc (p5) Rabc ^ Rac*b*

(p6) a** = a (p7) ROab ^ ROba (q1) Rabc ^ Rbac (d1) a < b =def ROab

(d2) R?abcd =def 3x(Rabx&Rxcd&x € K).

A valuation v is defined as a mapping assigning the truth value from truth-value matrix for LQ to propositional variables in every point of K accounting the binary relation < from (d1). An interpretation I is a natural extending of v on all formulas of L^0 under condition that in any point of K the usual explication of connectives takes place. The formal definition is as follows:

a) v is a valuation in L-frame, i.e. v is a function v : V x K —

M[o,i] (where M[0)i] is a logical matrix for Li.e. M[0)i] = ([0,1], —, —, {1}) where — x = 1 — x,x — y = min(1,1 — x + y). For any p € V and any a,b € K the following condition is satisfied:

(1) a < b&v(p, a) = 0 ^ (p, b) = 0;

b) I is an interpretation associated with v, i.e. I is a function I :

W0 x K ^ M[01] satisfying for any p € V, A,B € W0, a € K the following conditions:

(i) I(p,a) = v(p,a);

(ii) I (—A, a = 1 — x iff I (A, a*) = x;

(iii) I (A B,a) = inf(1,1 — x + y) iff for any b,c € K, Rabc and I (a, b) = x ^ I (B, c) = y.

(iv) I(QA, a) = 1 iff for any b € K(ROab ^ 3c € K(RObc ^

I (A,c) = 0)).

The following theorem was proved [8, p. 67]:

Theorem 2. The .system LQ is complete in respect to the ternary semantic that is for any A € W0, if I (A) = 1 for any intepretation I then h A.

We have the following finite model property. The junction of both semantics of LQ can be achieved via putting for any A e W0, dom(ID(A)) C K and rng(ID(A)) C {I(A, a) : a e K}, that is, treating the set ^ as K.

Proposition 1. A formula F is valid in LQ if and only if F is valid in all those L-frames {O, K, R,* ) where K is finite.

Proof. Let n = {O,K,R* ,I) and let VF = {p1,...,pn} be the propositional variables occurring in F. Moreover, let BF be the set of all bi-valued assignments IF : VF — {0,1}. We write IF if Vp € V : If(p) = I(p, a) and define a new model nf = {O, Kf, R'*', I') as follows:

• Kf = {If (£BF : la e K : IF = IaF}

• I'(p, If) = I(p, a),where IF = Ip

• R' c Kf x Kf x Kf where we take R'I (a) I (b)I (c) as corresponding to Rabc.

We can uniquely extend this to all subsets of Kf. It is straightforward to check that I(F,O) = I'(I%,F). Thus we have shown that in evaluating F it suffices to consider nf with at most 2p(F where p(F) is the number of different propositional variables occurring in F. □

The analysis shows that we can replace the rule (iv) with the rule (iv) without the loss of the generality :

(iv)' I(QA, a) = inf{I(A, c) : for any b € K(ROab ^ 3c € K(RObc ^ I(A,c) = 0}.

Turning back to the game theoretic semantic of L^0 it is worth to denote that recently some its extensions were obtained (cf. [2], [3]). It seems natural to adopt such an approach for producing this kind of semantics for LQ which is also, in fact, the specific extension of LHo.

2 Dialogue Game for LQ

In 2006 C. Fermuller and R. Kosik [2] proposed some game theoretic interpretation for the SL system that extends both Lukasiewicz logic Lx0 and modal logic S5. It was builded on ideas already introduced by Robin Giles in the 1970th to obtain a characterization of L in terms of a Lorenzen style dialogue game combined with bets on the results of binary experiments that may show dispersion. In [2] the experiments were replaced by random evaluations with respect to a given probability distribution over permissible precisifications. We will try to implement main ideas of interpretation proposed (respectively modifying it) for obtaining game theoretical semantic for the LQ.

Assume that two players agree to pay 1€ to the opponent player for each assertion of an atomic statement, which is false in any a € K according to a randomly chosen set of observation points. More formally, given a set of all observation points K the risk value (x)K associated with a propositional variable x is defined as (x)K = ID(x). In fact, (x)K corresponds to the probabilities of having to pay 1€, when asserting x.

Let x1,x2, ...,y1,y2... denote atomic statements, i.e. propositional variables. By [x]^, ...,xm\\y1, ...,yn] we denote an elementary state in the game where the 1st — the first player — asserts each of the yi in the multiset {y1,..., yn} of atomic statements and the 2nd — the second player — asserts each atomic statement xi € {x-\_, ...,xm}. The risk associated with a multiset X = {x-\_, ...,xm} of atomic formu-

m

las is defined as (x-\_, ...,xm)K = Yl (xi)K. The risk ()K associated

i=1

with the empty multiset is 0. (V)K respectively denotes the average amount of payoffs that the 1st player expects to have to pay to the 2nd player according to the above arrangements if he/she asserted the atomic formulas in V. The risk associated with an elementary state [x1, ...,xm\\y1, ...,yn ] is calculated from the point of view of the 1st player and therefore the condition (x1, ...,xm)K > (y1, ...,yn)K (success condition) expresses that the 1st player does not expect any loss (but possibly some gain) when betting on the truth of atomic statements.

Now we accept the following dialogue rule for implication (cf. [2]):

(RIf the 1st player asserts A — B in point a then, whenever the 2nd player chooses to attack this statement by asserting A in point b, the 1st has to assert also B in point c (the points are choosing according to the condition (iii) above). (And vice versa, i.e., for the roles of 1st and the 2nd player switched.)

A player may also choose not to attack the opponent's assertions of A — B. The rule reflects the idea that the meaning of implication entails the principle that an assertion of 'If A then B' obliges one to assert also B if the opponent in a dialogue grants (i.e. asserts) A.

The dialogue rule for the negation involves a relativization to specific observation points:

(R—) If the 1st player asserts —A in point a then the 2nd player chooses to attack this statement by asserting A in point a* (the points are choosing according to the condition (iii) above). (And vice versa, i.e., for the roles of 1st and the 2nd player switched.)

The dialogue rule for the Q-modality also involves a relativization to specific observation points:

(RQ) If the 1st player asserts QA then the 1st also have to assert that A holds (its interpretation differs from 0) at any point that the 2nd may choose using the condition (iv) above (And vice versa, i.e., for the roles of the 1st and the 2nd switched.)

Henceforth we will use Aa as shorthand for 'A holds at the observation point a' and speak of A as a formula indexed by a, accordingly. Thus using rule (RQ) entails that we have to deal with indexed formulas also in rule (R^). However, we don't have to change the rule itself, which will turn out to be adequate independently of the kind of evaluation that we aim at in a particular context. We only need to stipulate that in applying (R^) the observation point index of A — B (if there is any) is used for defininig the respective indexes for the subformulas A and B. If, on the other hand, we apply rule (RQ) to an already indexed formula (QA)a then the index a is overwritten by whatever index b is chosen by the opponent

player; i.e., we have to continue with the assertion Ab and, of course, we also have to account for indices of formulas in elementary states. We augment the definition of risk by (xa)K = 1 — I(x, a). In other words, the probability of having to pay 1€ for claiming that x holds at the observation point a is 0 if x is true at a and 1 if x is false at a.

We use A1 ,..,Aam B1 ,..,Bbnn ] to denote an arbitrary (not necessarily elementary) state of the game, where {Aa1,..., Am? } is the multiset of formulas that are currently asserted by the 2nd player, and {B1,..., Bbn } is the multiset of formulas that are currently asserted by the 1st player. (We don't care about the order in which formulas are asserted.)

A move initiated by thelst player (1st-move) in state [r\\A] consists in his/her picking of some non-atomic formula Aa from the multiset r and proceeding as follows:

• If Aa = (A1 — A2)a then the 1st may either attack by asserting A1 in order to force the 2nd to assert A2 in accordance with (R^), or admit Aa. In the first case the successor state is [r', A2\\A, A1], in the second case it is [r'\\A], where r' = r — {Aa}.

• If Aa = (—A-])a then the 1st chooses the point a* thus forcing the 2nd to assert Aa . The successor state is [r,Aa \ \A'], where A' = A — {Aa}.

• If Aa = QBa then the 1st chooses an arbitrary b € K using the condition (iv) above thus forcing the 2nd to assert Bc. The successor state is [r',Bc\\A], where r' = r — {Aa}.

A move initiated by the 2nd player (2-move) is symmetric, i.e. with the roles of the 1st and the 2nd players interchanged. A run of the game consists in a sequence of states, each resulting from a move in the immediately preceding state, and ending in an elementary state [xa1, ..,xm\\ybl, ...,ytbT]. The 1st player succeeds in this run if this final state fulfills the success condition, i.e., if

n m

j=i i=i

The term at the left hand side of inequality is an expected loss of the 1st player at this state. In other words, the 1st succeeds if its expected loss is 0 or even negative, i.e., in fact a gain. The other connectives can be reduced to implication and negation.

3 Adequacy of the game

To show that the considered game indeed characterizes logic LQ, we have to analyse all possible runs of the game starting with some arbitrarily complex assertion by the 1st player. A strategy for the 1st player will be a tree-like structure, where a branch represents a possible run resulting from particular choices made by the 1st player, taking into account all possible choices of the 2nd player in (2- or 1-moves) that are compatible with the rules. We will only have to look at strategies for the 2nd player and thus call a strategy winning if the 1st player succeeds in all corresponding runs (according to condition (2)).

Taking into account that by Theorem (2) we can assume that the set K of observation points (states) is finite. The construction of strategies can be viewed as systematic proof search in an analytic tableau calculus with the following rules:

[T\\A, (Ai — A2)a] ) [r,Abi\\A,A2] \ r\AV 2nd>

[r, (Ai — A2)*\\A] i ) r (Ai — A2r\\A] 2 ) [r,Ac2\\A,Ab1] (—ist) [r \\ A] {—ist)

{THAii—A0!( ) r (-A)a\\a] )

[r,Aa* \\A] y—2nd) [T\\A,Aa*] (—ist) [r\\A, (QA)a] Q [T,Qr\\A](Q )

f (Q2nd) —m — (Qist)

[r\\A,Aa ]\...\[r\\A,Acn [r,Ac\\A]

In all rules a can denote any index. In the rule (Q2nd) as well as in the rule (Q1st) we assume that indexes c1, ... , cn and c are defined by means of the condition (iv) above. Note that, in accordance with the definition of a strategy for the 2nd player, his/her choices in the moves induce branching, whereas for the 1st player choices a single successor state that is compatible with the dialogue rules is chosen.

Theorem 3. A formula F is valid in LQ if and only if for every set K of observation points (states) the 1st player have a winning strategy for the game starting in game state [||F].

Proof. Every run of the game is finite. For every final elementary state [x^1 ,...,xamillvl1 ,...,yhn] the success condition says that

n b m

we have to compute the risk ^ (y ? )K — Y^ (xT)k, where (ra)K =

3=1 i=1

I(r,a) if a </ dom(ID(r)) and (ra)K = 1 — ID(r)(a) otherwise, and check whether the resulting value (in the following denoted by (xT1, ...,xmmllvl, ■■■,ytbn)) is < 0 to determine whether the 1st player 'win' the game. To obtain the minimal final risk of the 1st player (i.e., his/her minimal expected loss) that the 1st can enforce in any given state S by playing according to an optimal strategy, we have to take into account the supremum over all risks associated with the successor states to S that you can enforce by a choice that you may have in a (2nd- or 1st-)move S. On the other hand, for any of the 1st player choices the 1st can enforce the infimum of risks of corresponding successor states. In other words, we prove that we can extend the definition of the 1st expected loss from elementary states to arbitrary states such that the following conditions are satisfied:

(3.1) (r, (A ^ B)allA)K = inf{(rilA)K, (r,Bc||Ab, A)k}

(3.2) (r, (-A)allA)K = sup{(rllA, Aa*)k}

for assertions by the 2nd player and, for assertions by the 1st player:

(3.3) (rll(A ^ B)a, A)k = sup{(r, AbllBc, A)k, (rllA)K}

(3.4) (TllA, (-A)t)k = inf{(r,Aa* ||A)k )} Furthermore we have

(3.5) (rllA, (QA)t)k = sup {(r||A,Ac)K}

c&K

ROab^RObc

(3.6) r (QA)a\\A)K = inf {(r, Ac\\A)k}

c&K

ROab^RObc

We have to check that (.||.)k is well-defined; i.e., that conditions above together with the definition of my expected loss (risk) for elementary states indeed can be simultaneously fulfilled and guarantee uniqueness. To this aim consider the following generalisation of the truth function for LQ to multisets G of indexed formulas:

I (T)k =def £ I (A, a) + £ ID (A)(a) Aer Aer

a/dom(ID (A)) aedom(ID (A))

Note that

I ({A})k = I (A)k = £ I (A, a) + £ ID (A)(a) = 1 iff (U)k < 0,

aedom(ID (A)) aedom(ID (A))

that is, A is valid in LQ iff my risk in the game starting with my assertion of A is non-positive. Moreover, for elementary states we have

« ,..,xam llyl1, ■~,ybn )k = n - m + I (xa1 ,..,xam )k - I (yb1,-, Vbn" )k ■

We generalize the risk function to arbitrary observation states

by

(rilA)K =def |A|-|r| + I(r)K - I(A)k

and check that it satisfies conditions (3.1)-(3.6). We only spell out two cases. In order to avoid case distinctions let I (Aa)K = I (A, a). For condition (3.1) we have

(r, (A ^ B)allA)K = |A| -iri- 1 + I (r)K + I (A ^ B, a)K -I (A)k = (r||A)K - 1 + I (A ^ B,a) = (r||A)K - 1 + inf {1,1 -I (A,b)+I (B,c)} = (rilA)K-1+inf {1,1 + (BcllAb)K } = (rilA)K + inf{0, (BcllAb)K} = inf{(r||A)K, (r, BcllAb, A)K}■ For condition (3.5) we have

(rilA, (QA)a)K = |A| -iri- 1 + I (r)K - I (A)k - I ((QA)a)K = (r||A)K + 1 - I (QA, a) = (r||A)K + 1 - inf {I (A,c) : for any b G K(ROab ^ 3c £ K(RObc ^ I(A,c) = o} = (rllA)*K +

sup{I (A, c) for any b e K (ROab ^3c e K (RObc ^ I (A, c) =0} = sup {(r|| A, Ac)K} □

c&K

ROab^RObc

Let us define a regulation as assignment of labels 'the 2nd player moves next' and 'the 1st player moves next' to game states that obviously constrain the possible runs of the game. A regulation is consistent if the label '2nd(1st) move next' is only assigned to states where such a move is possible, i.e., where 1st player (2nd player) have asserted a non-atomic formula. As a corollary to our proof of Theorem (3), we obtain:

Corollary 1. The total expected loss (r||A)K that the 1st player can enforce in a game over K starting in state [r || A] only depends on r, A and K. In particular, it is the same for every consistent regulation that may be imposed on the game.

References

[1] Dishkant, H., An Extension of the Lukasiewicz Logic to the Modal Logic of Quantum Mechanics, Studia Logica 37(2):149—155, 1976.

[2] Fermuller, C. G., and R. Kosik, Combining supervaluation and degree based reasoning under vagueness, Proceedings of LPAR 2006, volume 4246 of LNAI. Springer, 2006, pp. 212-226.

[3] Fermuller, C. G., Revisiting Giles — connecting bets, dialogue games, and fuzzy logics, in O. Majer, A. V. Pietarinen and T. Tulenheimo (eds.), Games: Unifying Logic, Language, and Philosophy, Springer, 2009, pp. 209-227.

[4] Giles, R., A non-classical logic for physics, Studia Logica 33(4):399-417, 1974.

[5] Giles, R., A non-classical logic for physics, in R. Wojcicki, G. Mali-nowski (eds.), Selected Papers on Lukasiewicz Sentential Calculi, Polish Academy of Sciences, 1977, pp. 13-51.

[6] Mackey, G. W., The Mathematical Foundations of Quantum Mechanics, New York, 1963.

[7] Vasyukov, V. L., The Completeness of the Factor Semantics for Lukasiewicz's Infinite-valued Logics, Studia Logica 52(1):143-167, 1993.

[8] Vasyukov, V. L., Quantum Logics, Moscow: Per Se, 2005 (in Russian).