Travaux Dirigés n 2 Modèles des langages de programmation Master Parisien de Recherche en Informatique Paul-André Melliès

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1 Travaux Dirigés n 2 Modèles des langages de programmation Master Parisien de Recherche en Inormatique Paul-André Melliès Problem. We saw in the course that every coherence space A induces a partial order (D A, A ) whose elements are the cliques o the coherence space A, ordered by inclusion. a. Show that the partial order (D A, A ) deines a domain, that is: (1) D A contains a least element denoted A, (2) every increasing sequence o elements (x n ) n N in the partial order D A has a least upper bound x = n N x n. b. Every clique : A&S B&S induces a amily o unary (sorted) operations b s t b a a s s one or each pair o elements a A, b B, and s, t S such that a[]b a[]s s[]t s[]b respectively. Let trees () denote the set o linear trees o the orm: b s n s 1 a 1

2 containing at least one node. Deine the relation trace () : A B as the set o elements a b such that there exists a linear tree in trees () with input sort a and output sort b. Show that the relation trace () deines a clique o the coherence space A B. Remark: this construction deines a «trace operator» on the category o coherence spaces equipped with the cartesian product. c. Describe explicitly the diagonal clique : S S & S characterized as the unique morphism making the diagram id S π 1 S S S & S id S π 2 S commute in the category o coherence spaces. d. Every clique induces a clique : A & S S ixpoint () = trace ( ) : A S deined as the trace operator applied to the composite A & S S S & S. Show that ixpoint () coincides with the set o all elements a s such that there exists a linear tree in trees () with input sort a and output sort s. 2

3 e. Show that the clique ixpoint () makes the diagram A & A A & ixpoint () A & S A A ixpoint () commute in the category o coherence spaces. S. At this stage, we are ready to lit to stable unctions what we have just achieved or linear unctions. We have seen during the course that every stable unction : D A D S D B D S may be seen as a clique :! ( A & S ) B & S This clique induces in turn an n-ary (sorted) operation b s x 1 x i x n x 1 x i x n whenever {x 1,, x n } [] b {x 1,, x n } [] s where each x i is an element o A & S, that is, either an element o A or an element o S. Note that the descendents o a node are unordered. An element o trees () is deined as a tree constructed with these n-ary operations, and containing at least a node. The relation is then deined as the set o elements trace () :!A B {a 1,..., a n } b such that there exists a tree in trees () with output sort b and with {a 1,, a n } as set o input sorts. Show that trace () deines a clique o the coherence space!a B. [Hint: proceed as in question b.] 3

4 /g. Every clique induces a clique :! ( A & S ) S ixpoint () = trace ( ) :! A S deined as the trace operator applied to the composite! ( A & S ) S S & S. Show that ixpoint () coincides with the set o all elements {a 1,..., a n } s such that there exists a tree in trees () with input sorts {a 1,..., a n } and output sort s, where each a i is an element o the web A. g/h. Show that the clique ixpoint () is a parametric ixpoint o in the sense that the diagram! A!! A! A δ! A! A iso! A! ixpoint ()! A! S iso! ( A & S )! ( A & A )! A! A ixpoint () S commutes in the category o coherence spaces. h/i. Explain how to compute the actorial unction actorial : N N 4

5 as a ixpoint o the clique deined as the set o elements act :! (N N) N N! ( N N ) N N { p q } p + 1 (p + 1) q 1 1 where a b is an alternative notation or a b. Explain in particular why the tree 5 ( ) establishes that is an element o the clique act 4 ( ) act 3 ( ) act 2 ( 2 1 ) act 1 1 act 5 ( ) actorial : N N. 5

6 i/j. We have seen in the course how to see a pair o morphisms (=cliques) in the category o coherence spaces :! A B g :! B C as a pair o morphisms : A k B g : B k C in the Kleisli category associated to the comonad!. Recall that the composite g o the two morphisms in the Kleisli category is deined as ollows:! A δ A!! A!!B g C Express the set trees (g ) directly rom the sets trees (g) and trees (). j/k. Explain (ormally or inormally) why the stable unction D A D S associated to the clique ixpoint () :! A S deines the least ixpoint o the stable unction D A D S D S associated to the clique :! ( A & S ) S. 6