Independence in D-posetsChovanec Ferdinand, Drobná Eva
Department of Natural Sciences, Armed Forces Academy, Liptovský Mikuláš, Slovakia
Nánásiová OľgaDepartment of Mathematics and Descriptive Geometry, Faculty of Civil Engineering
Slovak Technical University, Bratislava, Slovakia
Mathematical Structures for Nonstandard Logics Prague, Czech Republic, December 10-11, 2009
Classical approach
• Kolmogorov, A. N.
Grundbegriffe der Wahrscheikchkeitsrechnung. Springer, Berlin, 1933.
• De Finetti
• Rényi, A. On a new axiomatic theory of probability. Acta Math Acad Sci
Hung 6: 285–335, 1955.
• Bayes
Kolmogorov
( Ω , S , P )
(Ω ∩ E , SE , PE ) E , A S , P(E) > 0
)(
)()(
EP
EAPAPE
De Finetti, Rényi
S0 S
f: S × S 0 → [0, 1]
1. f ( E, E ) = 1 for every E S 0
2. f ( . , E ) σ – additive measure
3. f ( A ∩ B, C ) = f ( A, B ∩ C ) f ( B, C )
for every A, B S , C, B ∩ C S 0
Comparison
• These approaches give the same result
• Independence of random events
)(
)()(
EP
EAPAPE
)()(
)()( AP
EP
EAPAPE
Algebraic structures
Boolean Algebras
Multivalued Algebras
D-posets
Orthoalgebras
Orthomodular Posets
Orthomodular Lattices
D-lattices
• Beltrametti E, Bugajski S (2004) Separating classical and quantum correlations. Int J Theor Phys 43:1793–1801
• Beltrametti E, Cassinelli G (1981) The logic of quantum mechanics. Addison-Wesley, Reading
• Cassinelli G, Truini P (1984) Conditional probabilities on orthomodular lattices. Rep Math Phys 20:41–52
• Dvurečenskij A, Pulmannová S (2000) New trends in quantum structures. Kluwer/Ister Science, Dordrecht/Bratislava
• Gudder SP (1984) An extension of classical measure theory. Soc Ind Appl Math 26:71–89
• Khrennikov A Yu (2003) Representation of the Kolmogorov model having all distinguishing features of quantum probabilistic model. Phys Lett A 316:279–296
• Nánásiová O (2003) Map for simultaneous measurements for a quantum logic. Int J Theor Phys 42:1889–1903 572
• Nánásiová O (2004) Principle conditioning. Int J Theor Phys 43(7– 8):1757–1768
D-posetKôpka F, Chovanec F (1994) D-posets, Mathematica Slovaca, 44
(P , , 0P , 1P ) bounded poset
⊖ partial binary operation – difference on P b ⊖ a exists iff a b
(D1) a ⊖ 0P = a for any a P (D2) a b c implies c ⊖ b c ⊖ a and (c ⊖ a) ⊖ (c ⊖ b) = b ⊖ a
(P , , 0P , 1P , ⊖) D-poset
(P , , , , 0P , 1P , ⊖) D-lattice
dual partial binary operation to a difference – orthogonal sum
a b = ( a ⊖ b ) for a b
where x = 1P ⊖ x – orthosupplement
⊙ partial binary operation – product
a ⊙ b = a ⊖ b for b a
• Chovanec F, Kôpka F (2007) D-posets, handbook of quantum logic and quantum structures: quantum structures. Elsevier B.V.,Amsterdam, pp 367–428
• Chovanec F, Rybáriková E (1998) Ideals and filters in D-posets. Int J Theor Phys 37:17–22
Conditional state on a D-poset
Let P be a D-poset and P 0 P be its nonempty subset.
f: P × P 0 → [0, 1] is said to be a conditional state on P iff
(CS1) f(a, a) = 1 for every a P 0
(CS2) If b, bn P for n = 1, 2, ..., and bn b then f(bn , a) f(b, a)
(CS3) If b, c P , b c then
f(c ⊖b , a) = f(c, a) – f(b, a) for every a P 0
(CS4) If bP 0 , b a and a ⊖ b P 0 then for every x P
f(x, a) = f(x, b) f(b, a)
(CS5) If b, a ⊖ b P 0 then for every x P
f(x, a) = f(x, b) f(b, a) + f(x, a ⊖ b) f(a ⊖ b, a)
Example 1P 0 = { a, a , b, b , 1 }
0
a
b
b a
1s \ t a a b b 1
a 1CS1
a 0=1 1CS1
b 1CS1
b 1CS1
1 1CS3 1CS3 1CS3 1CS3 1CS1
Filter in a D-poset
A non-empty subset F of a D-poset P is said to be a filter in P iff (F1) a F , b P , a b b F
(F2) a F , b P , b a and (a ⊖ b) F b F
(F2*) a F , b F , b a a ⊙ b F
F is a proper filter in a D-poset P iff 0P F a F a F
Example 2F 1 = { b, a , 1 }
s \ t b a 1
a 0 0 0
b 0 0 0
b 1 1 1
a 1 1 1
1 1 1 10
a
b
a
1
b
Example 3F 2 = { b , 1 }
s \ t b 1
a 1/2 1/2
b 1 1
b 0 0
a 1/2 1/2
1 1 10
a
b
a
1
b
Example 4P0 = { b, b , a , 1 }
0
a
b
b
s \ t b a b 1
a 0 0 1/2 0
b 0 0 1 0
b 1 1 0 1
a 1 1 1/2 1
1 1 1 1 1
a
1
Maximal conditional system is a union of all
proper filters in a D-poset.
( Ω , S , P )
E S , P(E) > 0
SE = { A S ; E A} is a proper filter in S
S 0 = SE
E
}0)(;{ APA
Independence in D-posets
Let P be a D-poset bP , a P 0
and f be a conditional state on P .
b is said to be independent of an element a with respect to f iff
f(b, a) = f(b, 1P )
b ↪ a
)()()(iff)()/( BPAPBAPBPABP
B ↪ A A ↪ Biff
Orthomodular lattices, MV-algebras, D-posets
Boolean algebras
B ↪ A A ↪ B⇏
Chovanec F, Drobná E, Kôpka F, Nánásiová OConditional states and independence in D-posets. Soft Computing (2010) DOI 10.1007/s00500-009-0487-0
Example 5
0
a
b
b
s \ t b a b 1
a 0 0 1/2 0
b 0 0 1 0
b 1 1 0 1
a 1 1 1/2 1
1 1 1 1 1
a
1
f(a,b) = 1/2 f(a,1P ) = 1 a is not ↪ b
f(b,a) = 0 f(b,1P ) = 0 b ↪ a