By Helmut Seidl
This ebook constitutes the refereed court cases of the tenth overseas convention on Foundations of software program technology and Computation constructions, FOSSACS 2007, held in Braga, Portugal in March/April 2007 as a part of ETAPS 2007, the eu Joint meetings on conception and perform of Software.
The 25 revised complete papers awarded including the summary of one invited speak have been rigorously reviewed and chosen from 103 submissions. The papers hide a vast spectrum on theories and strategies to aid research, synthesis, transformation and verification of courses and software program structures. specifically the subsequent subject matters are handled: algebraic types, automata and language thought, behavioral equivalences, express versions, computation methods over discrete and non-stop info, countless nation platforms computation buildings, logics of courses, modal, spatial, and temporal logics, versions of concurrent, reactive, allotted, and cellular platforms, strategy algebras and calculi, semantics of programming languages, software program specification and refinement, variety platforms and sort idea, basics of safety, semi-structured info, application correctness and verification.
Read or Download Foundations of Software Science and Computational Structures: 10th International Conference, FOSSACS 2007, Held as Part of the Joint European Conferences on Theory and Practice of Software, ETAPS 2007, Braga, Portugal, March 24-April 1, 2007. Proceedings PDF
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Additional info for Foundations of Software Science and Computational Structures: 10th International Conference, FOSSACS 2007, Held as Part of the Joint European Conferences on Theory and Practice of Software, ETAPS 2007, Braga, Portugal, March 24-April 1, 2007. Proceedings
Sample text
By induction, Γ LI t : Nat and Θ LI u : A0 . Therefore by VarIter and Exchange, Γ, Θ, x : ∧It(A0 , . . , An ) LI iter t u x : An . Again by induction, ∀Bi ∈ It(A0 , . . Δ LI v : Bi . Thus, by Iter Γ, Θ, Δ LI iter t u v : An . This last result shows that any term typable in System L is also typable in LI . e, system LI allows more typings than System L. In particular, when typing an open term of the form iter (Sm 0) u x, it allows x to have an iterative type It(A0 , . . , An ). For example, we can have the following derivation in System LI (consider for example, Γ = {x : ∧It(A −◦ Nat −◦ Nat ⊗ Nat, .
By induction on the type derivation. As a consequence of these lemmas, Γ | Γ L t : A implies Γ Γ | ∅ L t : A (since the elements in Γ are also in Γ ). The relationship between the multiplicative and the hybrid versions of System L is as follows: Theorem 1. – If Γ L t : A then Γ | ∅ L t : A for any permutation Γ . – If Γ | ∅ L t : A for some permutation Γ then Γ L t : A. Proof. ⇒) By induction on the type derivation, using the previous lemmas. t : A −◦ B Γ |Γ (−◦Intro) Γ |Γ L L t:A Γ |Δ Γ , x : A, y : B | Δ Γ |Γ L Γ |Γ (Zero) L t : Nat 0 : Nat Γ |Θ L Γ |Γ L u : A0 Γ |Δ L Θ|Δ L L L u:A tu : B (−◦Elim) (⊗Intro) L u:C let x, y = t in u : C L Numbers Γ |Γ u:B L Γ |Δ L t, u : A ⊗ B L t : A⊗B Γ |Δ t : A −◦ B Γ |Δ Γ |Δ Γ |Γ L n : Nat S n : Nat (⊗Elim) (Succ) v : It(A0 , .
An ) LI : iter t u x : An Δ LI v : It(A0 , . . , An ) Also, Γ, x : ∧It(A0 , . . , An ) LI : iter t u x : An if Γ LI t : Nat and Γ LI u : A0 where if t ≡ Sm 0 then n = m otherwise n = 0, and Γ = Γ , Γ . By induction hypothesis: Γ L t : Nat Γ L u : A0 Δ L v : It(A0 , . . , An ). Thus, by Iter Γ, Δ L iter t u v : An . In particular for closed terms, the two systems are equivalent. Corollary 1. 2 L e : τ iff LI e : τ. Rank 2 Intersection Types: System L2I Being able to type terms of the form iter (Sm 0) u x using intersections like we do in System LI , does not really give us more interesting terms, because we can not abstract on x, therefore it will never be replaced by the function to iterate.