/ department of mathematics and computer science TSP: operational semantics / department of mathematics and computer science 3/15 / department of mathematics and computer science 4/15 Add action and termination rules for sequential composition. Exercise Process Algebra (2IMF10) Examples TSP: operational semantics TSP: sequential composition / department of mathematics and computer science 2/15 Lecture 5 http://www.win.tue.nl/~luttik s.p.luttik@tue.nl MF 6.072 Bas Luttik Theory of Sequential Processes We extend BSP( A ) with a binary operator · for sequential composition: The process p · q executes p and, upon successful termination of p , then executes q . ▶ a .1 · b .1 fjrst executes the action a and then executes the action b ; ▶ a .0 · b .1 executes the action a and then deadlocks; ▶ ( a .1 + b .1) · ( a .1 + b .1) fjrst executes either an a or a b , and subsequently executes, again, either an a or a b ; ▶ ( a .1 + 1) · b .1 either executes an a followed by a b , or immediately executes a b . a a a a y − → y ′ y − → y ′ x → x ′ x → x ′ − − a a a a a a → x ′ → y ′ → x ′ → y ′ a . x − → x x + y − x + y − a . x − → x x + y − x + y − x ↓ y ↓ x ↓ y ↓ 1 ↓ ( x + y ) ↓ ( x + y ) ↓ 1 ↓ ( x + y ) ↓ ( x + y ) ↓ a a a a → y ′ → y ′ x ↓ y − x ↓ y ↓ x ↓ y − x ↓ y ↓ → x ′ → x ′ x − x − a → x ′ · y a ( x · y ) ↓ a → x ′ · y a ( x · y ) ↓ x · y x · y → y ′ x · y x · y → y ′ − − − − These rules associate with TSP( A ) a transition-system space ( S , L , → , ↓ ) : ▶ S is the set of closed TSP( A ) -terms; ▶ L is the set A ; ▶ → ⊆ S × L × S is the least relation satisfying the rules above; ▶ ↓ ⊆ S is the least set satisfying the rules above.
5/15 If the operational rules of a process calculus are all in the path format, 6/15 / department of mathematics and computer science SOS meta-theory: the path format Defjnition holds that 1. the target of every transition in the premises is just a single variable; 2. the source of the conclusion is either a single variable, or it is of the 3. the variables in the targets of the transitions in the premises and the / department of mathematics and computer science Theorem then bisimilarity is a congruence for that process calculus. Exercise 7/15 / department of mathematics and computer science TSP: operational semantics Theorem Proof. Note that rules are all in the path -format, from which it immediately follows 8/15 / department of mathematics and computer science TSP: relevant questions about extension important questions to be addressed: Compute the transition systems associated with the following terms: variables in the source of the conclusion are all distinct. TSP: operational semantics a a → y ′ y − → x ′ x − a a a a . x − → x x + y − → x ′ x + y − → y ′ A collection of operational rules is in path format if for every rule in R it x ↓ y ↓ 1 ↓ ( x + y ) ↓ ( x + y ) ↓ a a x ↓ y − → y ′ x ↓ y ↓ x − → x ′ → x ′ · y a a ( x · y ) ↓ form f ( x 1 , . . . , x n ) with f an n -ary function symbol and the x i → y ′ x · y − x · y − ( 1 ≤ i ≤ n ) variables; ▶ ( a .1 + b .1) · ( a .1 + b .1) ; ▶ a .1 · b .1 ; ▶ a .0 · b .1 ; ▶ ( a .1 + 1) · b .1 ; and ▶ ( a .1 + 1) · 0 ; ▶ ( a .1 + 1) · ( a .1 + 1) . a a y − → y ′ x → x ′ − a a a → x ′ → y ′ a . x − → x x + y − x + y − Considering that TSP( A ) is an extension of BSP( A ) , there are two x ↓ y ↓ 1 ↓ ( x + y ) ↓ ( x + y ) ↓ a a 1. Is TSP( A ) more expressive than BSP( A ) ? → y ′ x ↓ y − x ↓ y ↓ → x ′ x − Can we express, in TSP( A ) , more behaviour than in BSP( A ) ? a → x ′ · y a ( x · y ) ↓ x · y x · y → y ′ − − 2. Does the notion of behaviour associated with BSP( A ) -terms by the operational semantics for TSP( A ) coincide with the behaviour Bisimilarity is a congruence on P (TSP( A )) = ( C (TSP( A )), +, · , a . ( a ∈ A ), 0, 1) . associated with them by the operational semantics for BSP( A ) ? that bisimilarity is a congruence on P (TSP( A )) .
9/15 11/15 A4 TSP: elimination of sequential composition / department of mathematics and computer science 12/15 Hence, by the soundness of equational logic (Prop. 2.3.9 in the book), it follows previous slide). applicable only to terms in which sequential compositions occur, and sequential Proof. Theorem TSP: soundness / department of mathematics and computer science 2. Verify that these axioms are valid in the algebra of behaviour A8 Exercises A10 A9 A8 Note that A7 / department of mathematics and computer science A5 10/15 / department of mathematics and computer science TSP: axioms A7 A5 A9 Exercise 6.2.2 TSP: operationally conservative extension A4 A10 To get the equational theory TSP( A ) we extend the equational theory BSP( A ) with the following axioms: a a y − → y ′ x − → x ′ a a a a . x − → x x + y − → x ′ x + y − → y ′ ( x + y ) · z = x · z + y · z ( · right-distributes over + ) x ↓ y ↓ ( x · y ) · z = x · ( y · z ) ( · is associative) 1 ↓ ( x + y ) ↓ ( x + y ) ↓ 0 · x = 0 ( 0 is a left-zero for · ) a a x ↓ y − → y ′ x ↓ y ↓ x · 1 = x ( 1 is a right-identity for · ) x − → x ′ a → x ′ · y a ( x · y ) ↓ x · y − x · y − → y ′ 1 · x = x ( 1 is a left-identity for · ) a . x · y = a .( x · y ) ( · right-distributes over a . ) ▶ the sources of the conclusions of all operational rules not stemming from BSP( A ) are not BSP( A ) -terms. ▶ the operational rules stemming from BSP( A ) are source-dependent . 1. Do Exercise 6.2.1 from the book. Therefore, according to Theorem 3.2.19 in the book, TSP( A ) is an operationally P (TSP( A )) / ↔ . conservative extension of BSP( A ) (i.e., p ↔ q in TSP( A ) ifg p ↔ q in BSP( A ) ). The equational theory TSP( A ) is sound for the algebra of behaviour ( x + y ) · z = x · z + y · z ( · right-distributes over + ) P (TSP( A )) / ↔ associated with TSP( A ) . ( x · y ) · z = x · ( y · z ) ( · is associative) 0 · x = 0 ( 0 is a left-zero for · ) Note that, since the operational semantics of TSP( A ) only adds rules that are x · 1 = x ( 1 is a right-identity for · ) 1 · x = x ( 1 is a left-identity for · ) compositions do not occur in axioms A1–A3 and A6 of BSP( A ) , their validity proofs can simply be repeated in the setting of TSP( A ) . a . x · y = a .( x · y ) ( · right-distributes over a . ) The validity of the new axioms of TSP( A ) can be established similarly (see Prove that for all closed BSP( A ) -terms p and q , there exists a closed that TSP( A ) is sound for P (TSP( A )) / ↔ . BSP( A ) -term r such that TSP( A ) ⊢ p · q = r .
13/15 Theorem Proof. Theorem TSP: elimination of sequential composition / department of mathematics and computer science 14/15 / department of mathematics and computer science 15/15 / department of mathematics and computer science TSP: ground-completeness Proof. Proof. Lemma TSP: elimination of sequential composition For every closed TSP( A ) -term p there exists a closed BSP( A ) -term q For every closed BSP( A ) -term p it holds that for every closed such that p = q . BSP( A ) -term q there exists a closed BSP( A ) -term r such that p · q = r . We prove the lemma with induction on the structure of p : The proof is by induction on the structure of p : ▶ If p ≡ 0 , then, by A7, p · q = 0 , which is a BSP( A ) -term. ▶ If p ≡ 0 or p ≡ 1 , then p is itself a BSP( A ) -term. ▶ If p ≡ 1 , then, by A9, p · q = q , which is a BSP( A ) -term. ▶ Suppose that p ≡ a . p ′ and suppose that there exists a BSP( A ) -term q ′ such that p ′ = q ′ (IH). Then p = a . q ′ , which is a BSP( A ) -term. ▶ Suppose that p ≡ a . p ′ , and suppose that for every BSP( A ) -term q there exists a BSP( A ) -term r ′ such that p ′ · q = r ′ (IH). Then, by A10 and IH, ▶ Suppose that p ≡ p 1 + p 2 and suppose that there exist BSP( A ) -terms q 1 p · q = a .( p ′ · q ) = a . r ′ , which is a BSP( A ) -term. and q 2 such that p 1 = q 1 and p 2 = q 2 . Then p = q 1 + q 2 , which is a BSP( A ) -term. ▶ Suppose that p ≡ p 1 + p 2 , and suppose that for every BSP( A ) -term q there exist BSP( A ) -terms r 1 and r 2 such that p 1 · q = r 1 and p 2 · q = r 2 . Then, ▶ Suppose that p ≡ p 1 · p 2 and suppose that there exists a BSP( A ) -terms q 1 by A4 and IH, p · q = p 1 · q + p 2 · q = r 1 + r 2 , which is a BSP( A ) -term. and q 2 such that p 1 = q 1 and p 2 = q 2 . Then p = q 1 · q 2 , so, by the lemma on the previous slide, there exists a BSP( A ) -term r such that p = r . The equational theory TSP( A ) is ground-complete for the algebra of behaviour P (TSP( A )) / ↔ associated with TSP( A ) . Recall that it suffjces to prove, for all closed TSP( A ) -terms p and q , that p ↔ q implies p = q . So, let p and q be closed TSP( A ) -terms such that p ↔ q . Then, by the elimination theorem, there exist BSP( A ) -terms p ′ and q ′ such that p = p ′ and q = q ′ . Since TSP( A ) is sound for P (TSP( A )) / ↔ , it follows that p ′ ↔ p ↔ q ↔ q ′ , so p ′ ↔ q ′ . Hence, since TSP( A ) is an operationally conservative extension of BSP( A ) , p ′ and q ′ are BSP( A ) -terms, and BSP( A ) is ground-complete for P (BSP( A )) / ↔ , it follows that p ′ = q ′ in BSP( A ) , and hence p ′ = q ′ in TSP( A ) . Thus we get p = p ′ = q ′ = q .
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