38 d0 a deterministic process is a pair a s where a

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& $ % D0 A deterministic process is a pair ( A, S ) where A is any set of symbols and S is any subset of A * which satisfies the two conditions C0 hi ∈ S C1 s, t s _ t S s S 39
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& $ % D1 STOP A = ( A, {hi} ) D2 RUN A = ( A, A * ) D3 ( x : B ( A, S ( x ))) = ( A, {hi} ∪ {h x i s | x B s S ( x ) } ) provided B A D4 ( A, S ) /s = ( A, { t | ( s _ t ) S } ) provided s S D5 μX : A F ( X ) = ( A, S n 0 traces ( F n ( STOP A ))) provided F is a guarded expression D6 ( A, S ) || ( B, T ) = ( A B, { s | s ( A B ) * ( s A ) S ( s B ) T } ) D7 f ( A, S ) = ( f ( A ) , { f * ( s ) | s S } ) provided f is one-one 40
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& $ % Fixed point theory 1. Aim: The purpose of this section is to give an outline of a proof of the fundamental theorem of recursion, that a recursively defined process (2.8.1 D5) is indeed a solution of the corresponding recursive equation, i.e., μX F ( X ) = F ( μX F ( X )) The treatment follows the fixed-point theory of Scott. 41
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& $ % 2. v relation : An ordering relationship among processes D1 ( A, S ) v ( B, T ) = ( A = B S T ) 3. Properties of v (partial order) L1 P v P L2 P v Q Q v P P = Q L3 P v Q Q v R ) P v R 42
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& $ % 4. Chain A chain in a partial order is an infinite sequence of elements { P 0 , P 1 , P 2 , ... } such that P i v P i +1 for all i We define the limit (least upper bound) of such a chain t i 0 P i = ( αP 0 , S i 0 traces ( P i )) 43
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& $ % 5. Complete Partial Order A partial order is said to be complete if it has a least element, and all chains have a least upper bound. The set of all processes with a given alphabet A forms a complete partial order (c.p.o.), since it satisfies the laws L4 STOP A v P provided αP = A L5 P i v t i 0 P i L6 ( i 0 P i v Q ) ( t i 0 P i ) v Q Remark: Furthermore the definition of μ (2.8.1 D5) can be reformulated in terms of a limit L7 μX : A F ( X ) = t i 0 F i ( STOP A ) 44
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& $ % 6. Continuous Complete Partial Order A function F from one c.p.o. to another one (or the same one) is said to be continuous if it distributes over the limits of all chains, i.e., F ( t i 0 P i ) = t i 0 F ( P i ) if { P i | i 0 } is a chain Remark: (1) All continuous functions are monotonic in the sense that P v Q F ( P ) v F ( Q ) for all P and Q (2) A function G of several arguments is defined as continuous if it is continuous in each of its arguments separately (for examples, please see page 76). (3) The composition of continuous function is also continuous. 45
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& $ % All the operators (except / ) defined in D3 to D7 are continuous in the sense defined above L8 ( x : B ( t i 0 P i ( x ))) = t i 0 ( x : B P i ( x )) L9 μX : A F ( X, ( t i 0 P i )) = t i 0 μX : A F ( X, P i ) provided F is continuous L10 ( t i 0 Pi ) || Q = Q || ( t i 0 P i ) = t i 0 ( Q || P i ) L11 f ( t i 0 P i ) = t i 0 f ( P i ) 46
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& $ % 7. The Proof of basic fixed-point theorem Consequently if F ( X ) is any expression constructed solely in terms of these operators, it will be continuous in X . Now it is possible to prove the basic fixed-point theorem F ( μX : A F ( X )) { Def of μ } = F ( t i 0 F i ( STOP A )) { Continuity F } = t i 0 F ( F i ( STOP A )) { Def F i +1 } = t i 1 F i ( STOP A ) { STOP A v F ( STOP A ) } = t i 0 F i ( STOP A ) { Def μ } = μX : A F ( X ) 47
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& $ % Unique solution for fixed point 1. Aim: In this section we treat more formally the reasoning given in Section 1.1.2 to show that an equation defining a process by guarded recursion has only one solution.
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