Foundations of Physics, Vol. 20, No. 6, 1990

On the Inverse FPR Problem: Quantum is Classical

George Svetlichny i

Received January 17, 1990

The notion of quantum supports h~troduced by Foulis, Piron, and Randall can be used to construct combinatorial versions of contextualist hidden-variable models for finite quantum logics. The original logic can be uniquely recovered from appropriate such models as a solution of a combinatorial inverse problem. One can thus set up a classical ontology for a finite quantum logics that completely specifies it. Computer studies are used to explore the ideas.

In 1983, Foulis, Piron, and Randall (FPR) (1) introduced a new com- binatorial construct in quantum logics which generalized the notion of a probability measure. This construct, called a suppor t , was introduced to provide for a "realist" interpretation of quantum logics. It has many appealing characteristics, the major one being that it is non-numeric, thus freeing foundation studies of the reliance on real numbers. Many feel that the use of numerical quantities in science is a device, at times conventional and arbitrary, and that true foundations are qualitative in nature with the numerical aspects arising through appropriate constructs. The FPR sup- ports promise thus a combinatorial substitute for numerical probabilities. In many cases, if not all, these supports can take the place of older considerations in foundation questions. Thus Cohen and Svetlichny (2) showed that an axiom concerning minimal supports can replace Piron's covering axiom (3) in identifying those propositional systems that have generalized Hilbert-space models. Svetlichny (4~ showed that FPR supports are intimately relayed to Kripke models of sets of formulas in modal logics. Here we explore some further properties of these models showing that they furnish a combinatorial version of what in other contexts of the founda-

1Departamento de Matematica, Pontificia Universidade Catolica, Rua Marques de Silo Vincente 225, Rio de Janeiro, Brasil.

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0015-9018/90/0600-0635506.00/0 © 1990 Plenum Publishing Corporation

636 Svet|ichny

tions of quantum mechanics are called contextual hidden-variable theories, providing thus an explicit "realist" ontology for quantum logics. We shall refer to the cited article of Svetlichny as QSML (Quantum supports and modal logic). We shall not define Kripke models in this paper, and the reader who is not familiar with them should consult the mentioned article or other appropriate literature/5'6) On the whole we shall only use com- binatorial analogs of modal notions, thus making this article accessible even to those unfamiliar with modal logic.

A quantum logic is generally taken to be an orthomodular lattice, or more simply an orthomodular poset. We shall only consider finite logics. A block of such a logic is a maximal set of mutually orthogonal nonzero elements. The set of blocks forms a hypergraph which is nothing but a set of sets. The logic can be reconstructed from the hypergraph, though not every hypergraph leads to an orthomodular poset. In FPR such hyper- graphs are called quasi-manuals and are taken as the starting point for all constructs as they have an immediate operational interpretation. The elements of a quasi-manual, called operations, correspond to empirical procedures, and their elements in turn, called outcomes, correspond to the possible results upon terminating the procedure. All the quasi-manuals that we use as examples are to be found in the Appendix, along with various related constructs.

In what follows we shall generally denote by d a quasi-manual, by E, F, G,... its elements (the operations), and by 1all its union (the set of all outcomes). A subset S c Is¢l is called a (FPR) support if it is, first of all, a transversal of d , that is,

V E n d , Sc~Eva~

and, second, if it satisfies the exchange condition

VE, F e d , S ~ E ~ F = ~ S n F ~ E

In FPR, supports correspond to ontological states where the elements of the support are the possible observational outcomes in the state. The exchange condition corresponds to the requirement that the certainty of a set of outcomes be independent of the operation of which this set happens to be a subset ("realist" properties are independent of how one verifies them). Examples of supports are stochastic supports, that is, supports of weights, which are functions co: [dL --, [O, 1] such that V E n d , ~ x ~ e o ) ( x ) = l . The number ~o(x) is interpreted as the probability of getting outcome x. Needless to say, one generally has many FPR supports that are not stochastic. The simplest example of this is for the square, for which a nonstochastic support is {a, w, b, c, d}.

The Inverse FPR Problem 637

One immediate difference that one notices between weights and sup- ports is that for co to be a weight one imposes a set of conditions, each one coming from one operation, while the exchange condition comes from two operations. This "relational" character of supports is a major source of dif- ficulty in dealing with them. However, to some extent this characteristic is only apparent. In QSML we showed that if co(x) is now no longer a number but a proposition in a Kripke model of the modal system T, and if for every E = {el,..., e ~ } ~ d , the following formulas are valid in the model:

(N~): 5(co(el) v co(e2) v --- v co(e~))

i . (BE). O~o(ei)-~ ~D(co(el) v co(e2) v --. v co(ei+l) v --.co(e~))

where [] is the necessity operator, O the possibility operator, and i runs from 1 to n, then the support of such an co, that is, the set of x for which Oco(x) is valid in the model, is an FPR support of d and any FPR sup- port can be so gotten. Thus, if instead of assigning "probabilites that sum to 1," we assign "possibilities that join to necessity," not only do we characterize all FPR supports but we also eliminate the "relational" character of the exchange condition seeing that the requirements now come from each operation singly. One does have an additional complication though due to the second displayed formula above, whose analog has no effect for stochastic weights since it would only state the obvious fact that if from a sum to 1 one removes a nonzero summand, then the rest no longer sum to 1.

A typical quasi-manual has many FPR supports. Hand calculation of them is prohibitive even in the simplest cases, and one must resort to machine generation. We have created computer routines 2 for calculating all the combinatorial objects that we mention in this paper, which is in part a report on these computer studies. The algorithmic structure of supports is of interest in its own right and we present it here. First we rewrite the exchange condition in the equivalent form (under transversality):

VE, F ~ d , such that E c ~ F ¢ ~ , S ~ ( E \ F ) ~ S ~ ( F \ E ) v ~ ; 2 ~

This can be paraphrased as follows: either S intersects only the common part of E and F, or it intersects both differences. Sets that satisfy just this condition, without imposing transversality, are called generalized supports. For connected quasimanuals, the only generalized support that is not a support is the empty set, so for all practical purposes it is enough to find

2 We use muLISP-87, a version of LISP, for IBM-PC compatible machines, commercialilzed by Soft Warehouse, Honolulu, Hawaii, USA.

638 Svetlichny

the generalized supports. Suppose now, E\F={al ..... ap}, F \E= { b l , . . . , bq}, and E ~ F = {q ..... cr}. Assume E c ~ F # ~ and associate to such a pair of operations the following formula in ordinary propositional calculus:

(V(a, Abj))V((yCk)A ~(ya,)A -(yb/)) Here we have associated to each outcome x a distinct elementary proposi- tion, still denoted by x, whose meaning is x "belongs to the generalized support S." We see that the above formula translates exactly the paraphrase of the exchange condition. Let now P be the conjunction of all the above formulas over all pairs E, F of operations with nonempty inter- section. This formula thus affirms the exchange condition for all pairs. Any formula can be put into a disjunctive normal form, that is, a disjunct of conjunctions of elementary propositions or their negations. Let thus (h~ A ... A h, A ~kl A ... A ~k~) be a conjunct in the disjunctive nor- mal form of P; then any set S which contains the hi and does not contain the kj in fact satisfies this conjunct, and thus also P, and is hence a generalized support. All generalized supports can thus be found by placing P into disjunctive normal form. Although there are many algorithms for calculating the disjunctive normal form, and so any system of logic programming can be used to find generalized FPR supports, we in fact have chosen a more direct approach. If H = {h~ .... h~}, K = {kj ..... k~,}, then the set of generalized supports that the above disjunct defines is an order interval [H, bsJl\K] in the power set ~'(1~t) of Is~t. The exchange condi- tion above can also be paraphrased to mean that the generalized support S must lie in the following union of order intervals:

( U [ { ai, bj}, ls~/l ] ) u [;2~, E c~ F]

The set of all generalized support is then precisely the intersection over all pairs E, F of operations with nonempty intersection, of the above union of order intervals. Now the intersection of two order intervals is another order interval: [A, B] ~ [C, D] -- [A ~ C, Bc~D]. The intersection of two unions of order intervals can thus be expressed as the union of the inter- sections of pairs of order intervals taken from each union separately. This is the essence of the algorithm that we use to calculate generalized support. We successively form the intersection of the unions of order intervals coming from each pair of operations with nonempty intersection. At each step we eliminate those intervals that are subsets of others (the elimination of those that are subsets of a union of others is too costly, and we do not

The Inverse FPR Problem 639

necessarily get a minimal representation of the generalized supports as a union of order intervals). The results of this calculation is a representation of all generalized supports as a union of order intervals. Further simplifica- tion is possible. The union of a set of generalized supports is itself a generalized support. A generalized support is called irredundant if it is not the union of properly contained generalized supports. Each generalized support is a union of irredundant generalized supports which thus form a set of generators. Once one has the set of generalized supports represented as a union of order intervals, the irredundant ones can be found easily. Let [A, B] be an order interval, and for each x e B \ A consider the set A • {x}. Any element of the interval other than A is obviously a union of such sets. Thus, the irredundant elements of a union of order intervals are to be found among the bottoms A of the intervals, and the sets A u {x} as above. Making a list of these ordered by decreasing cardinality, and eliminating those that are equal to the union of those further on the list that are a subset of the given one, defines an algorithm that provides the set of irredundant elements. Most of the other algorithms we use are fairly straightforward and we will gladly supply any interested reader with appropriate details.

The union of all supports is the maximum support. This may or may not be Id l . If it is not, then the quasi-manual contains outcomes that are impossible to realize in any ontological state. Such dummy outcomes thus play no part in the interpretation and only complicate the combinatorics. They can be simply excised and the investigation reduced to the resulting quasi-manual with no dummy outcomes. In what follows we shall always assume that the maximum support is I~'1. What this also rules out, by the exchange condition, is one operation properly containing another. With this assumption, the set of irredundant supports cover tall.

For FPR a physical system consists of a quasi-manual and a set of supports whose union is INf. One can ask to what extent the quasi-manual is determined by the sety of supports and formulate the following inverse problem: given 5 ~ c ~ ( X ) , find all quasi-manuals d for which each Se 6; is a support. Obviously, the solultions only depend on the irredundant elements of 5Q so assume that no redundant members are present. Com- puter studies show that in general one has a large set of solutions. This is basically due to the "relational" characteristic of the exchange condition. Suppose ~ is one such quasi-manual; then, first of all, each operation is a transversal of 5 P, and second of all, pairs of operations satisfy the exchange condition for each putative support. Thus, all quasi-manuals can be found by, first, finding all transversals of c~9° and, second, extracting, from this set, maximal susets whose members pairwise satisfy the exchange condition for each S e Y. This provides us with so many generally incompatible ways of

640 Sveflichny

viewing 5 e as a set of supports. (There is, of course, no guarantee that for each such quasi-manual, 5 P will be the set of irredundant supports.) A simple example of this phenomenon is for the set of supports of the' triangle which is invariant under the exchanges a ~ x, b ~-* y, c ~ z, and so the image of the triangle under this permutation, also a triangle, has the same set of support as the original one. These two quasi-manuals are onto- logically incompatible since in the original one there is an ontological state in which x is a necessary outcome but not such state for the outcome a. This phenomenon is rather common. The following computer study was performed for many quasi-manuals: calculate the irredundant supports, then calculate all the maximal quasi-manuals consistent with these sup- ports. Typically there are many solutions, the majority of which are quasi- manuals with fewer operations and a larger set of supports. In all cases, however, there were a few (and often just one) additional quasi-manuals with exactly the same set of irredundant supports. These "siblings" are, in general, just as for the triangle, incompatible with the original quasi- manual in not containing any of the original operations. The existence of these siblings is one fo the most fascinating discoveries of these computer studies but whose meaning is still unclear. For all cases tested, all the siblings also have the same stochastic supports in common (though with different weights), that is, a support is stochastic for all or nonstochastic for all. If this is a general phenomenon, then one could possibly identify the set of stochastic supports as a subset of the set of all supports by some structural property without any rfeference to the quasi-manual.

For the most part, FPR deals with special quasi-manuals known as manuals. A manual is a quasi-manual that satisfies the following condition: if A w B, B w C, and C w D are three operations with each union disjoint, then A and D are disjoint and A u D is an operation. A quasi-manual d is a pre-manual if it is contained in a manual having the same outcome set. There is a smallest such containing manual which is called the manual closure of d . This manual closure can be obtained by successively adding pairs of the form A u D as described above (utilizing the newly created operations as new sources of new disjoint pairs). A pre-manual and its manual closure have the same stochastic state space (set of weights). This is obvious for from co(A)+co(B)= 1, ~o(B)+co(C)= 1, and co(C)+~o(D)= 1 one readily concludes that ~o(A)+co(D)=l [here co(A)=Zx~Ae)(X), etc.]. Thus, for the stochastic theory, there seems to be no real objection in going to the manual closure and dealing exclusively with the manual. The combinatorial situation is quite different, for the modal equivalent of the above argument, which would be the deduction of [] (a v d) from the conjunction of [ ] (a v b), [2](b v c), and [](c v d), is not valid, and even if in some Kripke model we force the conclusion by removing all

The Inverse FPR Problem 641

worlds in which it is not true, the other modal requirements pi could E by this be destroyed. In fact, typically, the manual closure has fewer supports. A simple example of this is the pre-manual we have called squaloo ("square less one outcome") whose manual closure contains one more operation {a, c, k, l} and whose set of irredundant supports contains three less elements. The nonstochastic supports have been eliminated in this process, but this is coincidental. It seem to be quite rare for a quasi- manual to properly contain another and have the same set of supports, as computer studies show up very few containing siblings. A simple exception is the hook, whose unique sibling is its manual closure containing one addi- tional element. One must thus conclude that in the combinatorial case the "manual axiom" has less motivation and general quasi-manuals are of interest. It should be mentioned, thnough, that the hypergraph of blocks of an orthomodular lattice is a manual, and thus form the "quantum logic" viewpoint the axiom is still interesting.

The stochastic case contrasts with the combinatorial one in not only allowing naturally for the manual axiom but also in having a much clearer inverse problem. A physical system for the stochastic case is a quasi- manual s¢ along with a convex subset A of the set of weights usually satisfying some additional conditional to the effect of being sufficiently large to distinguish enough elements of the corresponding quantum logic. In general, one cannot extract d from A, but there is a unique canonical quasi-manual associated to A to which sg is naturally related. Each out- come x of ~4 defines a frequency functional on A, that is, an affine map fx to [0, 1 ], via f x ( 2 ) = 2(x) for 2 ~ A. An operation of ~ thus defines a set of frequency functionals that sum to the constant functional 1. As was argued in Svetlichny, (v) there is a quasi-manual (finite if A is a polytope) formed from the extreme sets of frequency functionals that sum to one, and the operations of ~ can then be considered as stochastic mixtures of the operations of the constructed quasi-manual.

What this suggests is that in the combinatorial case, the idea of a "state space" prior to any quasi-manual is not adequately represented by a set of putative supports. Kripke models suggest themselves as candidates since then one can avoid multiple incompatible solutions to the quasi- manual construct eliminating the "relational" aspect of the exchange condi- tion. In the proof of Theorem 2 of QSML we construct a Kripke model from a support and the transversals of ~¢ contained in the support.

Definition 1. A (combinatorial) phase space 22 in a finite set X is composed of a cover of X, which we also denote by X, and for each ele- ment S~22 of a cover J o f S.

Actually the requirement of being covers is not essential to the idea

642 Svetlichny

behind the construction, but as we shall only consider the special cases where we do have covers, we adopt this condition as it simplifies further considerations.

This definition is in essence a combinatorial paraphrase of certain types of Kripke models. For completeness sake, the next definition defines these models explicitly.

Definition 2. Given a phase spaced Z" the Kripke model Y ( X ) associated to X is the $5 model defined as follows: (1) The elementary propositions are identified with the elements of X; (2) The worlds are pairs (S, T) where S e 5 ° and T e Y-s; (3) The accessibility relation between two worlds (S,, T,) and ($2, T2) is the equivalence relation which holds just when $ 1= $2; (4) An elementary proposition x is held to be true in the world (S, T) just in the case c E T.

Definition 3. Given a quasi-manual ~ , the maximal phase space Z ' ~ ( ~ ) associated to d is the one for which X consists of the supports of sd and Ys of the transversals of d that are contained in S.

Lemmal . Let S b e a support o f d and E e d . Suppose x e E c ~ S ; then there is a transversal T of ~4 contained in S such that Ec~ T = {x}.

Proof This is explicitly proved in QSML in the proof of Theorem 2. If the Kripke model of the proof of that theorem is used, the present lemma is a paraphrase of the validity of one of the modal formulas P~ in that model.

The property stated by this lemma happens to be the essential one that allows for the unique recovery of a quasi-manual from an appropriate phase space.

Definition 4. Let d be a quasi-manual. A phase space for N is any phase space in l d l for which elements of Z are supports and for which each Ys is a set of transversal of sJ that satisfies the condition of Lemma 1, that is, E e d , SeX, and x e E ~ S implies there is a TeY- s such that

Definition 5. Let X be any phase space in X; the hypergraph ~ ( Z ) of Z" consists of those subsets E of X that satisfy the following conditions. (1) E is a transversal of all the Y-s for all the S e X , and (2). If S e X and x6Ec~S, then there is a T ~ Y s such that Ec~ T = {x}.

Theorem 1. Let 2 be a phase space i n X. If ~ ( X ) cover X, then Z" is a set of FPR supports for it.

The Inverse FPR Problem 643

Proof Interpretinig the construct within the Kripke model JU(Z), what this means is that for each element E~ ~ ( X ) the modal formulas Ne and P~ hold. By Theorem 1 of QSML, each S e X is a support.

This construct of a quasi-manual from a phase space thus avoids the comparison of two putative operations to check the exchange condition. We need only check each one singly and the condition is automatically taken care of.

Corollary 1. Lety .4 be a quasi-manual and Z" any phase space of it; then H(Z) contains d and shares with it at least those supports that are elements of ~.

Theorem 2. For any quasi-manual d , let Z'tAI(~4 ) be the phase space { ( ] d ] , J ) } , where J- consists of all the transversals of .J. One has YC'(Z'IAI(S¢')) = S~', that is, the quasi-manual is uniquely recoverable from at least one of its phase spaces.

ProoJl This is essentially contained in the proof of Theorem 5 of QSML.

Now the superset of any transversal is also a transversal. Thus, all transversals can be generated from the minimal ones. This often results in considerable simplification.

Theorem 3. Let Z" be any phase space in X, and let 22'. be obtained from _r replacing each J s by the 3-1s consisting of the minimal elements (under set inclusion) of ~s. One has ~ (Z : ' )= H(X).

Proof Note that as covers of X, Z:, and _r' are the same. The inclu- sion ~¢¢'(±~')~ ~ ( X ) is immediate. Let Ee ~g(~(Z'); then obviously E satisfies the transversality condition for ~'. So suppose xeEc~S for SeZ"; then there is an element T in ~s with Ec~ T= {x}. Such a T contains a minimal element T ' e ,~s. Now by construction of E, it must intersect T' and by T' c T the only possibility is again E ~ T' = {x}. This shows Ee ~(Z").

Combining this result with Theorem 2, we come to the remarkable conclusion that any quasi-manual for which X-41 is the maximum suport is uniquely recoverable from the set gmi~ of its minimal transversals. All one needs to do is to construct the hypergraph of the phase space {(l~I, -Y-mi=)}- Among these transversals are the supports of dispersion-free weights, whose set in certain cases is taken to be a dual object to the quasi- manual. Note that for the triangle the set of minimal transversals no longer has the permutation symmetry of the set of irredundant supports, thus eliminating the sibling.

644 Svetlichny

Definition 6. The irredundant phase space, 27ir~ed(~ ¢) is the phase space in which 27 consists of the irredundant suports and for each S e Z, Ys consists of all the transversals of d contained in S. The minimal irredundant phase space consists of the same cover Z, but the ~ consists of the minimal transversals contained in S.

The minimal irredundant phase space along with the set of all minimal transversals are very convenient objects to have at hand when studying the phase space structures of a quasi-manual. See the Appendix for examples.

For almost all the quasi-manuals in our computer study, one has ~ = ~(£:irred(d)), but this is not a theorem. The simplest counterexampte is thje hook for which the hypergraph of the irredundant phase space is its manual closure.

A phase space 27 of a quasi-manual can be interpreted as representing a realist ontology compatible with the operational interpretations of the quasi-amanual. In contrast to PFR, a support S E 27 is no longer to be con- sidered as the ultimate ontological state but as some kind of "macrostate." The "microstates" which are the ultimate ontological entities are repre- sented by the elements of the ,Y-s. One can adopt two possible views as to the relation between the macrostates and the microstates. One would be that due to unsurmountable intrinsic restrictions on the procedures that prepare states, one cannot repeatedly prepare the same microstate. Repeti- tions of the procedure prepare different microstates, but such variation can be confined to the microstates contained within a given macrostate. Thus macrostates would correspond to "preparable states" pretty much in accordance with the usual ideas of hidden variables in quantum mechanics. The other view is that the dynamics of ontological states is such that they uncontrollably change with time, being confined, however, to fit about within one macrostate, eventually passing through all the microstates contained therein. Thus, macrostates would behave as some sort of thermodynamic entities.

The FPR ontology admits the notion of ontological indeterminacy; thus if Ee,~¢ and S • E has more than one element, the which particular element of this intersection is the result in a particular execution of the operation E is considered to be ontologically indeterminate, much in the spirit of the usual quantum indeterminacy. Just as quantum indeterminacy can be eliminated by recourse to hidden variable theories, so this FPR indeterminacy can be eliminated by considering the microstates (transver- sals in ~ ) as representing the value of a classical hidden variable. Each such transversal can be considered as a definite predisposition of the ontological state to produce results in a given operation. An ontological state has to respond to each operation and produce an outcome. This defines a transversal. In the FPR ontology this transversal is not a realist

The Inverse FPR Problem 645

object; each response is undetermined until it happens, and one should not even talk about the counterfactual transversal of the results that would be obtained if each operation [were to be performed. This is all in accord with the usual quantum creed. Hidden-variable theories try to substitute such ontological indeterminacies by epistemic ones. In the phase space ontology that we introduced, at each instance, the system is characterized by a definite value of the hidden variable which is a transversal within the sup- port. This still does not eliminate totaliy all indeterminacies, and a final "contextual" interpretation is needed to do so. To illustrate this point of view better, consider again the triangle and, in particular, let us focus on the support {a, b, c} which is nonclassical in the sense of not being the support of a stochastic mixture of dispersion-free weight. If the value of the hidden variable is {a, c} and the operation performed is {a, z, b}, then of course the result is uniquely determined to be a. If the operation is {a, y, c}, however, the outcome is still undetermined, and it seems that one has to continue invoking ontological indeterminacy. This is nothing more than the manifestation of the general nonexistence of noncontextual hidden variable theories. In quantum mechanics the theorems of von Neuman, ~8) Gleason, (9) and Kochen and Specker (1°) established the nonexistence of hidden variable theories that are now known as noncontextual. This non- existence is a consequence of the nonexistence of dispersion-free weights on the quantum logic of all subspaces of a Hilbert space (the Kochen and Specker proof utilizes a finite sublogic.). Historically, these theorems contrasted paradoxically with an abundance of existing hidden-variable theories. Bell (H) clarified the situation by what has been labelled by Shimony (12) as a "judo-like maneuver." Noncontextual theories try to posit well-defined values for all quantum observables existing simultaneously at all times and independently of any observations that may be made on the system. This was shown to be impossible under certain conditions (relating the value of an observable and the value of a function of the observable). Bell introduced the idea, however, that the value of a degenerate observ- able could possibly depend on which other compatible observable was being measured along with it. Nondegenerate observables wouold thus have well-defined values at all times, but the value of degenerate ones would depend on the "context" in which they were being measured. Such theories are called contextual. Although this constitutes a sort of an ironic and inevitable return to Bohrian philosophy of the indivisibility of quan- tum events, which only make sense when the whole experimental setup is specified, it does point out a loophole in the nonexistence proofs through which the existent theories escape. In the stochastic case, one can generally construct a contextual hidden-variable theory for any weight. In the FPR situation, the phase spaces introduced above provide explicit constructs of

646 S~etlichny

combinatorial contextual hidden-variable theories. This is due to the truth of Lemma 1. If for each operation E e ~4 and for each S e ~ we choose a T e ~ s for which E n T is a singleton (and such exists by Lemma 1), then all observables have well-defined contextual values. Similarly, by Lemma 1 such a choice can be made to exhibit any element of S so the whole sup- port can be realized through variation on this choice. Thus, in our triangle example for the macrostate {a, b, c} we can maintain our original hidden- variable value {a, c} for the operations {a, z, b} and {b, x, c}, but for the operation {a, z, c} we must choose a different hidden-variable value, say, {a, b}. The results of every operation is now uniquely determined. The contextual nature is apparent, for whether c is a result or not depends on the operation performed. If {b, x, c} is performed, then c is the result, but if {a, z, c} is performed, then a is the result.

Our phase space ontology has a few unusual features. Consider the case of a redundant support S = S1 u $2 u ... w S, say, where the Si are properly contained supports. Such a situation can at face value be interpreted as describing a state of epistemic indeterminacy. One would maintain that at any time the system is actually in one of the macrostates Si but that either we do not know which one, or that the macrostate shifts around in time among these. This is the usual interpretation of mixed states in quantum mechanics (unions of stochastic supports are supports of stochastic mixtures of the corresponding weights). Our phase space model show that this is not always a correct interpretation. There may be minimal transversals contained in S that are not contained in any of the Si and so one can have an ontological situation in a mixed macrostate that is not present in the irreducible components of the mixture. This is intriguing, for if there is a microreality underlying quantum mechanics, then highly mixed states such as macroscopic objects could act in ways unforseseen by the usual quantum formalism, yet still somehow implicit in it. The simplest example of this sort of phenomenon is for the hook which has {b, c} as a minimal transversal that is not contained in any irredundant support. This situation is present in some quasi-manuals and not in others. For the triangle and the square, a minimal transversal is contained in some irredundant support, but the pentagon has five minimal transversals that are not. The occasional existence of this minimal transversal '°excess" is another interesting discovery of the computer study.

In our view, hidden-variable theories provide the most detailed insights into the structure of quantum phenomena. Any alternative inter- pretation to the conventional Copenhagen one can be viewed as one sort of hidden variable theory or another. As one does not a priori say anything about the nature of the hidden variable, such theories are catch-all situa- tions. They provide convenient starting points for generalizations. In this

The Inverse FPR Problem 647

spirit, the phase space of quasi-manuals can provide us with detailed notions of their structure and interpretations. In the end, little is left of what is truly "quantum." Modal logic and their implicit hidden-variable interpretations can provide a classical account of quantum mechanics. This classical world is certainly strange and contextual, but it is probably this extreme level of detail that is necessary in order to progress into new territory. Quantum mechanics is thus totally stripped of any original philosophical content. What survives as original is the realization that some age-old ideas apply to the physical world, from which the historical classical development has banished them. Struck down by Bell's judo-like maneuver, we are tempted to say that quantum is classical with a twist.

The problem of constructing a hypergraph of a combinatorial phase space is a particular case of the problem of finding within a Kripke model all sets of propositions E = {Pl ..... p ,} for which the modal formulas NE and P~ are valid, or of the analogous deductive problem of finding all sets of such formulas that can be deduced within some modal logic system from a given initial set. This problem can be viewed as a model for the systematizeation of knowledge. The initial set can be viewed as unorganized "raw" knowledge, and each set E as a "law," for what NE says is that, under any condition, some Pt is true, and what P~ says is that, if Pi is possible, then it is possible for only p~ (among the elements of E) to be true. Thus, such E express necesary divisions of the world, and so are definite regularities. The inverese FPR problem is thus a procedure for knowledge systematization.

APPENDIX

This appendix contains computer-generated data for the quasi- manuals mentioned in the main body of the paper. We do not present all the data computed for all the quasi-manuals investigated, but the mentioned data and what we have found interesting or instructive. Generally we present here: the minimal irredundant phase space (Defini- tion 6); the minimal-transversal excess, that is, those minimal transversals not contained in any irredundant support; and the siblings. For the case of the wadge, we have also included the set of all the maximal quasi-manuals consistent with the set of supports.

HOOK: {{a, b}, {b, c), {c, d}} Minimal irredundant phase space: { { {a, c} }, { {b, d} } } Minimal-transversal excess: { {b, c} } Sibling: {{a, b}, {b, c}, {a, d}, {c, d}}

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650 Svetlichny

{{a,b,c,d,e}, {a,b,c,f,g}, {a,d,e,f, g}}, {{a,b,c}, {a,d,e}, {c,e,g}, {a, f g}}, {{b,c,d,e}, {a,b,d,f}, {b,c,f,g}, {d,e,f,g}}}.

ACKNOWLEDGMENTS

Thanks go to the quantum-logic research group at the Mathematics Department of the University of Massachusetts at Amherst, whose work has greatly influenced mine. Special thanks go to the late professor Charles Randall whom I had known for too short a time. His wealth of ideas has been a constant source of inspiration and his insistence on elegence, both conceptual and mathematical, has kept me from straying too far from the true path.

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