A Comparison of Implications in Orthomodular Quantum Logic—Morphological Analysis of Quantum Logic

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Volume 2012, Article ID 259541,8pages doi:10.1155/2012/259541

Research Article

A Comparison of Implications in

Orthomodular Quantum Logic—Morphological

Analysis of Quantum Logic

Mitsuhiko Fujio

Department of Systems Design and Informatics, Faculty of Computer Science and Systems Engineering, Kyushu Institute of Technology, Iizuka-shi 820-8502, Japan

Correspondence should be addressed to Mitsuhiko Fujio,fujio@ces.kyutech.ac.jp

Received 26 June 2012; Accepted 3 August 2012

Academic Editor: Shigeru Kanemitsu

Copyrightq2012 Mitsuhiko Fujio. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Morphological operators are generalized to lattices as adjunction pairsSerra, 1984; Ronse, 1990; Heijmans and Ronse, 1990; Heijmans, 1994. In particular, morphology for set lattices is applied to analyze logics through Kripke semanticsBloch, 2002; Fujio and Bloch, 2004; Fujio, 2006. For example, a pair of morphological operators as an adjunction gives rise to a temporalization of normal modal logicFujio and Bloch, 2004; Fujio, 2006. Also, constructions of models for intuitionistic logic or linear logics can be described in terms of morphological interior and/or closure operatorsFujio and Bloch, 2004. This shows that morphological analysis can be applied to various non-classical logics. On the other hand, quantum logics are algebraically formalized as orhomodular or modular ortho-complemented latticesBirkhoﬀand von Neumann, 1936; Maeda, 1980; Chiara and Giuntini, 2002, and shown to allow Kripke semanticsChiara and Giuntini, 2002. This suggests the possibility of morphological analysis for quantum logics. In this article, to show an eﬃciency of morphological analysis for quantum logic, we consider the implication problem in quantum logics Chiara and Giuntini, 2002. We will give a comparison of the 5 polynomial implication connectives available in quantum logics.

1. Mathematical Morphology

Mathematical morphology is a method of non-linear signal processing using simple set-theoretic operations, which has the feasibility of extracting the characteristic properties of shapes 1, 2. In this paper we will adopt the formulation thereof generalized on lattices 3–7.

We identify a binary relationR⊆X×Aand the correspondence fromXtoA. Namely,

Rx {a∈A |x, a∈R}forx ∈X. We call the relationRwithXandAexchanged, the

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1.1. Dilation and Erosion

LetX,Abe partially ordered sets. If for any family{xλ} ⊆ X ofX which has a supremum

λxλinX, the image{fxλ} ⊆Ahas the supremumλfxλinAandfλxλ λfxλ

holds, then we call the mapping f : X → A a dilation from X to A. Similarly, by changing supremum by infimum, we may introduce an erosion. We call dilation and erosion

morphological operations. For two elements x ≤ yof X, we havex∨y y,x∧y x, the

morphological operations are monotone.

Example 1.1 morphology of set lattices7. Given sets X and A, consider the lattices of

their power setsX 2X,A 2A. LetRbe a binary relation inX×A. Then the mappings

DR :A → XandER:A → Xdefined by

DRB {x∈X|Rx∩B /∅},

ERB {x∈X|Rx⊆B}

1.1

are a dilation and an erosion, respectively.

From the transposetRwe may similarly define the dilation and erosionDt_R:X → A,

Et_R:X → A.

The importance of this example lies in the fact that all morphological operations between set lattices are expressed in this form, whence it follows that giving a framework of morphological operations and a binary relationRare equivalent. In particular, in the Kripke semantics, accessibility relationships being binary operations between possible worlds, we contend that giving the Kripke framework amounts to giving morphological operations.

1.2. Adjunctions

Suppose two mappingsf:X → Aandg:A → Xbetween partially ordered sets satisfy the condition

fx≤a⇐⇒x≤ga 1.2

for anyx∈X,a∈A. Then the mapping pairf, gis called an adjunction and is written as

fgandfis called the lower adjoint ofg, withgis the upper adjoint off. Notice that every adjoint is uniquely determined if exists.

Proposition 1.2. For two monotone mappings f : X → A andg : A → X between partially

ordered sets to satisfyfg, it is necessary and suﬃcient that for anyx∈X,a∈A, the relation

fga≤a, x≤gfx 1.3

holds.

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Proposition 1.3. LetX,Abe partially ordered sets andf:X → A,g:A → X.

1If f has the upper adjoint, then it is a dilation. Conversely, if A is a complete upper

semilattice, then a dilation has the upper adjoint.

2If g has the lower adjoint, then it is an erosion. Conversely, if X is a complete lower

semilattice, then a dilation has the lower adjoint.

Example 1.4adjunctions of set lattices7. InExample 1.1, we haveDREt_RandDt_RE_R.

Note that in each adjunction pair, dilation and erosion,RandtRare to be interchanged.

1.3. Interior and Closure Operators

An idempotent monotone mappingf :X → Xon a partially ordered setX is called a filter mapping. A filter mapping with extensibilityx≤fxis called a closure operator and one with antiextensibilityfx≤xis called an interior operator.

Proposition 1.5. LetX,Abe partially ordered sets andf :X → A,g :A → Xandf g. Then

g◦f :X → Xis a closure operator ofXandf◦g:A → Ais an interior operator ofA.

Example 1.6closing and opening. The closure operatorER ◦ Dt_R and the interior operator

DR ◦ Et_R on X which are induced by the adjunctions in Example 1.4 are called closing

and opening byR, respectively. Similarly, we may define the closing and opening by tR as operators onA.

In any complete lattice, closure operators are characterized by the notion of Moore family8, where a Moore family is a subsetMof a partially ordered setXwhich satisfies the following condition. For any subsetS ⊆ M, ifS has the infimumSinX, thenS ∈ M holds true.

Proposition 1.7see7,8. LetXbe a partially ordered set.

1For any closure operatorϕ:X → X, the totality of allϕ-closed setsFϕ {x∈X|ϕx

x}forms a Moore family.

2IfXis a complete lattice, then for any Moore familyM⊆X, there exists a unique closure

operator onXsuch thatMFϕholds.

We may establish similar properties of interior operators by appealing to the duality of a Moore family7.

2. Quantum Logic

We refer to9,10for quantum logic and lattice theory associated to it and we assemble here the minimum requisites for the subsequent discussions.

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2.1. OL and OQL

An ortho-complemented lattice L is a lattice which has an involutive and complementary operation·:L → Lreversing the order:

1a∧a0,a∨a1. 2 aa.

3a≤b⇒b≤a.

If, moreover, foraand its complementa, the modular relation

b≤a_⇒_a_∨_b_∧_a_b _2.1

holds, thenLis called an orthomodular lattice.

An ortho-complemented lattice satisfying the modular relation

c≤b⇒c∨a∧bc∨a∧b 2.2

for any a, b is an orthomodular, but not conversely. A Boolean lattice is an ortho-complemented lattice satisfying the modular relation. The inclusion order among these classes of lattices is

Boolean⊂modular ortho-complemented⊂orthomodular⊂ortho-complemented. 2.3

In general, we call collectively quantum logic QLboth orthologic OLmodelled on an ortho-complemented lattice and orthomodular logicOQLmodelled on an orthomodular lattice. An orthomodular lattice being an ortho-complemented lattice, we mostly work with OL, with additional mentioning of some special features intrinsic of OQL.

The language of QL consists of a countable number of propositional variables

p, q, r, . . ., and logical connectives ¬ negation, ∧ conjunction. Denote by α, β, . . . the formulae withΦtheir totality. The disjunction∨is defined as an abbreviation of¬¬α∧ ¬β.

2.2. Kripke Semantics

The pairF Ω, Rof the set of all possible worldsΩ/∅and the reflexive and accessibility relationsR⊆Ω×Ωis called a Kripke frame or orthogonal frame of OL. Intuitionally, the binary relation ∈Rωmeans thatωandare “not orthogonal”. Indeed, definingω⊥by /∈

Rω, then we see that the reflexiveness corresponds toω⊥ω, while symmetry to⊥ω ⇒

ω⊥.

For any set of possible worldsX⊆Ω, we define its ortho-complement set by

X_{_ω_∈_Ω_|_ω_⊥_ξ_∀_ξ_∈_X_}_. _2.4

Then in view of this, the power set lattice 2Ω of Ω becomes ortho-complemented. The orthogonality of a setXand the possible worldsωis defined by

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Expressing the orthogonality in terms of morphological operations

ω⊥X⇐⇒ω∈DRXER

X_. _2.6

In an orthogonal frame F Ω, R, we consider a special class of subsets called

propositions inF, that is,X is a proposition inFmeans thatX X holds. As we shall see

below, it immediately follows from the definition that formulae in OL may be interpreted by assigning propositions in an orthogonal frame.

Proposition 2.1. In an orthogonal frameF Ω, R, forX ⊆Ωto be a proposition, it is necessary

and suﬃcient that it is anR-closed set (ERDRX X) in the sense of morphology.

(Note thatRbeing symmetric, we haveDRDt_R.)

Proof. By2.6, we haveXDRX, whence

X_D_R_D_R_X_E

RDRX. 2.7

Corollary 2.2. The totality PF of all propositions of F forms a lower semi-complete

ortho-complemented sublattice of2Ω.

Proof. Note thatDR∅ ∅, and that from reflexibility of R, we have DRX X, so that

∅ _∅_, _X _X_{. Hence,} _∅_, _X _∈ _P

F. Since PF is a Moore family, and a fortiori lower

semi-complete. Also, since DR ◦ ER ◦ DR DR, we have for X ∈ PF, X X

DRERDRX DRX X, whence we obtainX ∈ PF. Hence,PF is closed with

respect to the complementation ’.

LetΠbe a lower semi-complete ortho-complemented sub-lattice ofP_Fand letρ:Φ → Πbe a mapping such that

1ρ¬α ρα

2ρα∧β ρα∧ρβ.

We call the setM Ω, R,Π, ρa Kripke model of OL, consisting of both this and the Kriplke frameF Ω, R.

If ω ∈ ρα holds true, we write ωMαand say that the formula α is true in the

possible worldα. We call the formulaαsuch thatρα Ωis true in the modelMand write Mα. More generally, if for anyβbelonging to a setTof formulae, we haveρβ⊆ρα, then

we say thatαis a consequence ofT in the modelMand writeTMα. If further, these hold

true in any models, then we say that they are logically true in or logical consequences of, OL respectively.

The Kripke semantics of the orthomodular logic OQL may be defined by considering asΠonly those satisfying the orthomodular condition

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3. Morphological Analysis of Implication Connectives

3.1. Implication Problem in QL

In quantum logic QL, the implication problem is important10. Not only those in quantum logic, but in general, an implication connective → is required to satisfy, for any modelM, at least the conditions

1_Mα → α,

2IfMαandMα → β, thenMβ.

In QL, this condition may be stated as follows. For any Kripke modelM, we have

Mα−→β⇐⇒ρα⊆ρβ. 3.1

Thus, we take3.1as a requirement for an implication connective → in QL10. Then we note that the forumulaα → β:¬α∨βin classical logic is not an implication connective in the sense of QL.

On the other hand, there are several candidates for implication connectives. However, there are only 5 polynomial ones in the sense that they are expressed in finitely many¬,∨,∧ 10:

iα →

1 β:¬α∨α∧β,

iiα →

2 β:β∨¬α∧ ¬β,

iiiα →

3 β: ¬α∧β∨α∧β∨¬α∧ ¬β,

ivα →

4 β: ¬α∧β∨α∧β∨¬α∨β∧ ¬β,

vα →

5 β: ¬α∧β∨¬α∧ ¬β∨α∧¬α∨β.

These are the all candidates for polynomial implications in the free orthomodular lattice generated by two elements satisfying

a≤b⇐⇒a−→

∗ b1. 3.2

There is a distinction between OL and OQL in that they are really implications in the respective logic.

Theorem 3.1see10. The polynomial implications →

i 1 ≤i≤ 5are all implications in OQL

but none of them are so in OL.

Proof. Proof depends on the fact that for3.1to hold for eachi, it is necessary and suﬃcient

thatsatisfies the orthomodular condition2.8. For details we refer to10.

Theorem 3.2. In OQL,α →

i βi 1,2,4,5are logical consequences ofα →3 β, that is, in any

Kripke modelM Ω, R,Π, ρ, we have

ρ

α−→ 3 β ⊆ρ

α−→

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Proof. We fix a Kripke modelM Ω, R,Π, ρ. Interpretation of each implication is as follows, where we denote the interpretations of the formulaeα,βbyAρα,Bρβ, respectively.

1ρα →

1 β A∩A∩B

,

2ρα →

2 β B

_∩_A_∩_B

,

3ρα →

3 β A

_∩_B_∩_A_∩_B_∩_A_∩_B_,

4ρα →

4 β A

_∩_B_∩_A_∩_B_∩_A_∩_B_∩_B_,

5ρα →

5 β A

_∩_B_∩_A_∩_B_∩_A_∩_A_∩_B

.

Proof of ρα →

3 β ⊆ ρα →1 β. It suﬃces to proveA ⊆ A

_∩_B_∩_A_∩_B_{, which}

reads in morphological operations,

A⊆DR

DRA∩B

∩DR

DRA∩DRB

. 3.4

Taking complements of bothsides, we obtain

A_⊇_D_R_D_R_A_∩_B_∪_D R

DRA∩DRB

DR

DRA∩B

∪DRA∩DRB

3.5

by the definition of dilation. Then by adjunction, this is equivalent to the following:

ER

A_⊇_D_R_A_∩_B_∪_D

RA∩DRB

ER

A_∩_B_∪_E_R_A_∩_D_R_B_. 3.6

The last equality follows from the duality between DR and ER. However, this inclusion

relation is always true.

Proof ofρα →

3 β⊆ρα →2 β. It suﬃces to proveB

_⊆_A_∩_B_∩_A_∩_B_{, which can}

be done as in the proof ofρα →

3 β⊆ρα →1 β.

Proof of ρα →

3 β ⊆ ρα →4 β.A

_∩_B _⊇ _A_∩_B_∩_B_{, or what amounts to the}

same thing, it is enough to showA∩B ⊆ A∩B∩B. Further for this to be true, it is necessary and suﬃcient thatA⊆A∩B, which is true, being equivalent toA⊇A∩B.

Proof of ρα →

3 β ⊆ ρα →5 β. Enough to proveA∩B

_⊇ _A_∩_A_∩_B

and the proof can be done in the same way as the proof ofρα →

3 β⊆ρα →4 β.

4. Conclusion

By applying morphological analysis to a Kripke model in quantum logics, we have shown that →

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sees the result, one may feel that one could do without morphological analysis. However, the point lies in whether by just looking at the defining equationi∼vor its interpretation, one could recognize the conclusion. Thus, the merit of morphological analysis seems to be its intuitive lucidness as “Calculus.”

We would like to return to the analysis of connectives other than → 3 .

References

1 G. Matheron, Random Sets and Integral Geometry, John Wiley & Sons, 1975.

2 J. Serra, Image Analysis and Mathematical Morphology, vol. 1, Academic Press, London, UK, 1984.

3 J. Serra, Image Analysis and Mathematical Morphology, vol. 2, Academic Press, London, UK, 1988.

4 C. Ronse, “Why mathematical morphology needs complete lattices,” Signal Processing, vol. 21, no. 2, pp. 129–154, 1990.

5 H. J. A. M. Heijmans and C. Ronse, “The algebraic basis of mathematical morphology. I. dilations and erosions,” Computer Vision Graphics Image Processing, vol. 50, pp. 245–295, 1990.

6 H. J. A. M. Heijmans, Morphological Image Operators, Academic Press, Boston, Mass, USA, 1994.

7 M. Fujio and I. Bloch, “Non-classical logic via mathematical morphology,” École Nationale Supérieure des Télécommunications, Technical Report ENST 2004D010, 2004.

8 G. Birkhoﬀ, Lattice theory, vol. 25 of American Mathematical Society Colloquium Publications, American Mathematical Society, Providence, RI, USA, 3rd edition, 1967.

9 S. Maeda, Lattice Theory and Quantum Logic, Maki-Shoten, 1980.

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