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1 Ver I 3 1 / ( ) : II : :

2 Ver 1.21 Page 2 0 ( ) [35] : 2

3 Ver 1.21 Page 3 I 1 / [18] * 4 5 * 6 6 [37]

4 Ver 1.21 Page * 8 9 * 10 * 11 dynamical system * 13 * X S F : X S X s S F s : X X s 14 *

5 Ver 1.21 Page * 16 *. 17 * [27, ] * [10, p188] ( )

6 Ver 1.21 Page [37] * [22] : folk psychology

7 Ver 1.21 Page [1] [3, ] 27 : *

8 Ver 1.21 Page 8 [41] [38] [42]

9 Ver 1.21 Page [43] 35 35

10 Ver 1.21 Page ( ) : [31].. (AP1) 36

11 Ver 1.21 Page 11 (AP2) [21] π- [28] [29], [24] 2.4

12 Ver 1.21 Page 12 Chemical abstract machine [4] T T ( T - ) T T T T T T T T T T 2.6 T 38 ([9, p154]) 39

13 Ver 1.21 Page 13 T [21] 2.6 ([14]) 40 Moore 41 [12] (1) a 1,a 2,,a n b a 1,,a n,b a 1,,a n b a 1,,a n b a 1,,a n b *

14 Ver 1.21 Page 14 a 1,,a n b a 1,,a n b X (1) (1) a 1,,a n b (1) (1) Γ a 1,,a n Γ b Γ 3 a, b, c b, c a a b b, c a a b c a, b (H1) a (H2) a 1,,a n b a a, a 1,,a n b (H3) : Γ i b i i =1,,n b 1,,b n c Γ 1, Γ 2,, Γ n c X C pow (X) : X A CA (Cl1) A CA (A 42 ) (Cl2) A B CA CB ( ) (Cl3) CCA = CA A A. C(A B) =CA CB X A B 43 Moore A A A Moore Moore A meet a b b c a c a b b a a = b 42 43

15 Ver 1.21 Page 15 a, b a b a b a b a, b a, b c a b a, b a b 2 a, b c, d, e c d a, b e, c e = a b X X 5 a, b, c 186 [12]. [12] 6 7 [14] a b V L l l A l V Moore L Moore V L A l l l b d a { c, d, b } a a { a, c }

16 Ver 1.21 Page 16 II [9] [9, p142] 45 [9, p ] [48] 3 46 : ( [9, p310]). ( [9, p359])

17 Ver 1.21 Page 17 P P A P B P [33, p152] 3.2 (cf ) 47 (cf ) :: 47 [13] active system

18 Ver 1.21 Page (impredicative) [8, p69] ([38]). [6] 49 [8] [8, p68] 49

19 Ver 1.21 Page incompletability( ) inexhaustibility() [7] incompletablility inexhaustibility 50 ( ) p86 52 [ ] [ ] [ ]

20 Ver 1.21 Page 20 [48, 201] [23, p40] x, y 57 x y = x + y x y = [45, ] (a) (b) [23, p108] [13, Ch1] 55 [45, ] [11, ]

21 Ver 1.21 Page ( 3.2.4) 57 [ ]. [ ].

22 Ver 1.21 Page 22 [17,. ] p107

23 Ver 1.21 Page 23 [32, p ] a b (a = b) a = b b = c a = c [23]

24 Ver 1.21 Page 24 [23, p117] [23, p151] [23, p188-9] ( 3.3.3)

25 Ver 1.21 Page [44] ( 4.5) : m m 1 1 m m

26 Ver 1.21 Page

27 Ver 1.21 Page [20] [8] 4.5 ([19])

28 Ver 1.21 Page 28 4 : 67 [10, p108] 4.1 [25, p45]. 67 [39] 68 [16] 68

29 Ver 1.21 Page 29 [34, p58] [26, p81] [10, p160] B C B C [15, p160] 4.3 [20, p8] 69 [46]

30 Ver 1.21 Page 30 [40, p188] 4.4 DNA : [9, p309]

31 Ver 1.21 Page 31 (cf ) 74 [9] [9, 1995 p313] 8 A B B A C [36]

32 Ver 1.21 Page X S 79 R X x S s R- x, s R x = s cf. 9 X S x = s x s x s x S x Y 78 M. Barr Chu autonomous category *-autonomous category *-autohomous category Pratt Wille (Formal Concept analysis) [5] Barwise [2] 79 X S = { (x, s) x X, s S }

33 Ver 1.21 Page 33 S Y 8182 s X s T X T ( 9 ) F y F y F F F F F F F T s T s T T T T T T T T T T 2.6 F F F F F = F T T 81 S Y S y (y Y ) Y 82 F, G F G 83 X 2.6 X X 84 S F S F T X T S P Q def P Q 1. P Q X P X Q 2. P Q X P X Q 3. P Q X P X Q F G F, G F, G F G F G F G F G 84 i F i = { F F Fi i }. 85 P Q = P Q X P X Q = X P X Q P Q = P Q X P X Q = X P X Q

34 Ver 1.21 Page 34 X S X, S : a, b, c d a, b, c d a, b, c d a, b, c d a b a b a b X a X b X a X b = X F F a b a b = S a, b X a a a a X b X a b b a b a S b a a b 86 X o X (X, S, =) (X, S) (X o,s, =) (X o, S o X o X o S o x x S x S o x S o 86

35 Ver 1.21 Page 35 x S x S x z z S z z 87 S o X S S o (X, S o, =) (X o, S o ) s X s 87 F S F x, y x y = X o T x y T x, y S o x, y s x, y s x, y s x, y s x, y s s x, y s X s x, y

36 Ver 1.21 Page 36 X s S o E (X, S, =) S E C =(X, S) X o (X o,s, =) 88 C o C ( ) C o C E X o X s X o X o E ( ) X o E C o C (*) 88 ( ) X, S = = = (X 1,S 1, = 1 ) (X 2,S 2, = 2 )

37 Ver 1.21 Page PQ1 PQ2 PQ3 5.. [PQ1] [PQ2-3] [9, 10] X X ([10, p110]) 90 [9] [47]

38 Ver 1.21 Page 38 [48, 18] 1997 [11] [45] [36] ( )

39 Ver 1.21 Page 39 (cf.[49]) ( )

40 Ver 1.21 Page 40 [1] [2] J. Barwise and J. Seligman. Information flow in distributed sysmtems. to appear in the C.U.P. series Tracts in theoretical computer science,1996. [3] Beeson M.J. Foundations of constructive Mathematics, Springer [4] Berry, The chemincal abstract machine. Theoretical computer science 96 (1992), [5] B.A.Davey and H.A.Priestley. Introduction to lattices and order. Cambridge Univ. Press 1990, ISBN [6] J.W.Dauben. Georg Cantor, His mathematics and philosophy of the infinite, Princeton Univ Press [7] Godel,K. Some basic theorems on the foundations of mathematics and their philosophical implications, 1951( ). [8] p [9] ( ), ( ), ( ),1995.4( ),1995.5( ),1995.8( ), ( ),1996.6( ),1996.9( ), ( ). [10],p98 231,1997. [11] [12] A. Higuchi. Lattices of closure operators, to appear in Discrete Mathematics. [13] D.R. Hofstadter. Godel, Eshser, Bach: An eternal golden braid. Harvest Press, [14] A. Higuchi, K. Matsuo and T. Tsujishtia. Deductive hyperdigraphs, A method of describing diversity of coherence, preprint [15], p 7 49, [16] [17] II. [18] [19] [20] [21] [22].,1985. [23].,1983.

41 Ver 1.21 Page 41 [24] Yves Lafont. Interaction combinators. preprint July [25] p8 50, [26] p51 96, [27]. [28] Robin Milner. The polyadic pi-calculus: a tutorial. ( ). [29] Robin Milner. Calculi for Interaction, preprint April ( ) [30] Humberto Maturana and Francisco J. Varela. Autopoiesis:Autopoiesis and cognition. Reidel [31] ISBN [32]. [33],1995. ISBN [34], ,p [35]. DCC,(Y. Oono. Complex systems study as biology. to appear in International Journal of Modern Physics B, Vol 12 (1998). ) [36] [37] 1985, ISBN [38] [39] [40].. [41] Vol.31, 4 25, [42] [43] Vol 64(2), ,1995. [44] Computer Today, , p [45], [46] ( ) [47] [48] [49] ( )

42 Ver 1.21 Page 42 1:

43 Ver 1.21 Page 43 e c c d d e a b f g b a h 2: a, b a, e e 3: G 1 4: G 2. a b, d c, b, d

44 Ver 1.21 Page 44 c a b 5:.

45 Ver 1.21 Page 45 6: -1.(0) (6) (0 1) a, b, c d (1 2) a, b, d c (2 3) a, d c (5 6) a, d (10 11) a

46 Ver 1.21 Page 46 φ 7: -2. (15)

47 Ver 1.21 Page 47 8:.

48 Ver 1.21 Page 48 9:. R X S s S X s s S Xs { s } { s } X s S Xs X s S R. S S Xs. t s s t { t } X s R

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