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↓ c ≀ 1 ? ≀ tr β€ž Ξ±q ↓ c tr β€ž Ξ±q ↓ b

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Unformatted text preview: ) ↓ c ≀ 1 ∧ ≀ ( tr β€ž Ξ±Q ) ↓ c- ( tr β€ž Ξ±Q ) ↓ b ≀ 1) β‡’ ≀ tr ↓ a- tr ↓ b ≀ 2 34 ’ & $ % L2 If P and Q never stop and if ( Ξ±P ∩ Ξ±Q ) contains at most one event, then ( P || Q ) never stops. Example: The process ( P || Q ) defined in X1 will never stop, because Ξ±P ∩ Ξ±Q = c The proof rule for change of symbol is: L3 If P sat S ( tr ), then f ( P ) sat S ( f- 1 * ( tr )) 35 ’ & $ % Mathematical theory of deterministic processes In our description of processes, we have stated a large number of laws, and we have occasionally used them in proofs. The laws have been justified (if at all) by informal explanations of why we should expect and want them to be true. Question: β€’ Are these laws in fact true? β€’ Are you consistent? β€’ Are you complete? β€’ Could one manage with fewer and simpler laws? These are questions for which an answer must be sought in a deeper mathematical investigation. 36 ’ & $ % The basic definitions 1. From pair ( A,S ) to a process Consider now an arbitrary pair of sets ( A,S ) which satisfy these three laws (in 1.8.1, L6, L7, L8). This pair uniquely identifies a process P whose traces are S constructed according to the following definitions. Let P = { x |h x i ∈ S } and, for all x in P , let P ( x ) be the process whose traces are { t |h x i _ t ∈ S } Then Ξ±P = A and P = ( x : P β†’ P ( x )) 37 ’ & $ % 2. From a process to pair ( A,S ) Furthermore, the pair ( A,S ) can be recovered by the equations A = Ξ±P S = traces ( x : P β†’ P ( x )) 3. The one-one correspondence between a process and the corresponding pair Thus there is a one-one correspondence between each process P and the pairs of sets ( Ξ±P,traces ( P )). 38 ’ & $ % 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 βˆ€ s,t β€’ s _ t ∈ S β‡’ s ∈ S 39 ’ & $ % D1 STOP A = ( A, {hi} ) D2 RUN A = ( A,A * ) D3 ( x : B β‡’ ( A,S ( x ))) = ( A, {hi} βˆͺ{h x i s | x ∈ B ∧ s ∈ S ( x ) } ) provided B βŠ† A D4 ( A,S ) /s = ( A, { t | ( s _ t ) ∈ S } ) provided s ∈ S D5 ΞΌX : A β€’ F ( X ) = ( A, S n β‰₯ traces ( F n ( STOP A ))) provided F is a guarded expression D6 ( A,S ) || ( B,T ) = ( A βˆͺ B, { s | s ∈ ( A βˆͺ B ) * ∧ ( s β€ž A ) ∈ S ∧ ( s β€ž B ) ∈ T } ) D7 f ( A,S ) = ( f ( A ) , { f * ( s ) | s ∈ S } ) provided f is one-one 40 ’ & $ % 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., ΞΌX β€’ F ( X ) = F ( ΞΌX β€’ F ( X )) The treatment follows the fixed-point theory of Scott....
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↓ c ≀ 1 βˆ β‰€ tr β€ž Ξ±Q ↓ c tr β€ž Ξ±Q ↓ b ≀ 1...

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