Papers by Giulio Manzonetto
Proceedings of the 2009 International Symposium on Logical Foundations of Computer Science, Jan 3, 2009
We recently introduced an extensional model of the pure λ-calculus living in a canonical cartesia... more We recently introduced an extensional model of the pure λ-calculus living in a canonical cartesian closed category of sets and relations [6]. In the present paper, we study the non-deterministic features of this model. Unlike most traditional approaches, our way of interpreting non-determinism does not require any additional powerdomain construction: we show that our model provides a straightforward semantics of non-determinism (may convergence) by means of unions of interpretations as well as of parallelism (must convergence) by means of a binary, non-idempotent, operation available on the model, which is related to the mix rule of Linear Logic. More precisely, we introduce a λ-calculus extended with non-deterministic choice and parallel composition, and we define its operational semantics (based on the may and must intuitions underlying our two additional operations). We describe the interpretation of this calculus in our model and show that this interpretation is sensible with respect to our operational semantics: a term converges if, and only if, it has a non-empty interpretation.
... lattices? Let ChΣ be the Σ-Church algebra. The following term op-... Page 16. Is the variet... more ... lattices? Let ChΣ be the Σ-Church algebra. The following term op-... Page 16. Is the variety of Church algebras satisfying the above identi-ties useful for et-lattice? (Nurakunov 08) A lattice L is an et-lattice if, and only if, L ∼ = ConA for some et-monoid A (a monoid with two more ...
We recently introduced an extensional model of the pure λ-calculus living in a canonical cartesia... more We recently introduced an extensional model of the pure λ-calculus living in a canonical cartesian closed category of sets and relations [6]. In the present paper, we study the non-deterministic features of this model. Unlike most traditional approaches, our way of interpreting non-determinism does not require any additional powerdomain construction: we show that our model provides a straightforward semantics of non-determinism (may convergence) by means of unions of interpretations as well as of parallelism (must convergence) by means of a binary, non-idempotent, operation available on the model, which is related to the mix rule of Linear Logic. More precisely, we introduce a λ-calculus extended with non-deterministic choice and parallel composition, and we define its operational semantics (based on the may and must intuitions underlying our two additional operations). We describe the interpretation of this calculus in our model and show that this interpretation is sensible with respect to our operational semantics: a term converges if, and only if, it has a non-empty interpretation.
We define the class of relational graph models and study the induced order-and equational-theorie... more We define the class of relational graph models and study the induced order-and equational-theories. Using the Taylor expansion, we show that all λ-terms with the same Böhm tree are equated in any relational graph model. If the model is moreover extensional and satisfies a technical condition, then its order-theory coincides with Morris's observational pre-order. Finally, we introduce an extensional version of the Taylor expansion, then prove that two λ-terms have the same extensional Taylor expansion exactly when they are equivalent in Morris's sense.
Lecture Notes in Computer Science, 2008
21st Annual IEEE Symposium on Logic in Computer Science (LICS'06), 2006
ABSTRACT
Lecture Notes in Computer Science, 2011
Electronic Notes in Theoretical Computer Science, 2014
We define the class of relational graph models and study the induced order-and equational-theorie... more We define the class of relational graph models and study the induced order-and equational-theories. Using the Taylor expansion, we show that all λ-terms with the same Böhm tree are equated in any relational graph model. If the model is moreover extensional and satisfies a technical condition, then its order-theory coincides with Morris's observational pre-order. Finally, we introduce an extensional version of the Taylor expansion, then prove that two λ-terms have the same extensional Taylor expansion exactly when they are equivalent in Morris's sense.
Lecture Notes in Computer Science, 2010
We provide a strong normalization result for ML F , a type system generalizing ML with first-clas... more We provide a strong normalization result for ML F , a type system generalizing ML with first-class polymorphism as in system F. The proof is achieved by translating ML F into a calculus of coercions,
Lecture Notes in Computer Science, 2007
A longstanding open problem is whether there exists a nonsyntactical model of the untyped λ-calcu... more A longstanding open problem is whether there exists a nonsyntactical model of the untyped λ-calculus whose theory is exactly the least λ-theory λβ. In this paper we investigate the more general question of whether the equational/order theory of a model of the untyped λ-calculus can be recursively enumerable (r.e. for brevity). We introduce a notion of effective model of λ-calculus, which covers in particular all the models individually introduced in the literature. We prove that the order theory of an effective model is never r.e.; from this it follows that its equational theory cannot be λβ, λβη. We then show that no effective model living in the stable or strongly stable semantics has an r.e. equational theory. Concerning Scott's semantics, we investigate the class of graph models and prove that no order theory of a graph model can be r.e., and that there exists an effective graph model whose equational/order theory is the minimum one. Finally, we show that the class of graph models enjoys a kind of downwards Löwenheim-Skolem theorem.
Lecture Notes in Computer Science, 2009
We recently introduced an extensional model of the pure λ-calculus living in a canonical cartesia... more We recently introduced an extensional model of the pure λ-calculus living in a canonical cartesian closed category of sets and relations [6]. In the present paper, we study the non-deterministic features of this model. Unlike most traditional approaches, our way of interpreting non-determinism does not require any additional powerdomain construction: we show that our model provides a straightforward semantics of non-determinism (may convergence) by means of unions of interpretations as well as of parallelism (must convergence) by means of a binary, non-idempotent, operation available on the model, which is related to the mix rule of Linear Logic. More precisely, we introduce a λ-calculus extended with non-deterministic choice and parallel composition, and we define its operational semantics (based on the may and must intuitions underlying our two additional operations). We describe the interpretation of this calculus in our model and show that this interpretation is sensible with respect to our operational semantics: a term converges if, and only if, it has a non-empty interpretation.
Lecture Notes in Computer Science, 2007
Models of the untyped λ-calculus may be defined either as applicative structures satisfying a bun... more Models of the untyped λ-calculus may be defined either as applicative structures satisfying a bunch of first order axioms, known as "λ-models", or as (structures arising from) any reflexive object in a cartesian closed category (ccc, for brevity). These notions are tightly linked in the sense that: given a λ-model A, one may define a ccc in which A (the carrier set) is a reflexive object; conversely, if U is a reflexive object in a ccc C, having enough points, then C(½, U ) may be turned into a λ-model. It is well known that, if C does not have enough points, then the applicative structure C(½, U ) is not a λ-model in general. This paper: (i) shows that this mismatch can be avoided by choosing appropriately the carrier set of the λ-model associated with U ; (ii) provides an example of an extensional reflexive object D in a ccc without enough points: the Kleisli-category of the comonad "finite multisets" on Rel; (iii) presents some algebraic properties of the λ-model associated with D by (i) which make it suitable for dealing with non-deterministic extensions of the untyped λ-calculus.
2013 28th Annual ACM/IEEE Symposium on Logic in Computer Science, 2013
ABSTRACT
A longstanding open problem is whether there exists a non-syntactical model of the untyped -calcu... more A longstanding open problem is whether there exists a non-syntactical model of the untyped -calculus whose theory is exactly the least -theory . In this paper we investigate the more general question of whether the equational/order theory of a model of the untyped -calculus can be recursively enumerable (r.e. for brevity). We introduce a notion of eective model of -calculus,
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Papers by Giulio Manzonetto