The quest for a fully abstract model of the call-by-value λ-calculus remains crucial in programming language theory, and constitutes an ongoing line of research. While a model enjoying this property has not been found yet, this interesting problem acts as a powerful motivation for investigating classes of models, studying the associated theories and capturing operational properties semantically. We study a relational model presented as a relevant intersection type system, where intersection is in general non-idempotent, except for an idempotent element that is injected in the system. This model is adequate, equates many λ-terms that are indeed equivalent in the maximal observational theory, and satisfies an Approximation Theorem w.r.t. a system of approximants representing finite pieces of call-by-value Böhm trees. We show that these tools can be used for characterizing the most significant properties of the calculus - namely valuability, potential valuability and solvability - both semantically, through the notion of approximants, and logically, by means of the type assignment system. We mainly focus on the characterizations of solvability, as they constitute an original result. Finally, we prove the decidability of the inhabitation problem for our type system by exhibiting a non-deterministic algorithm, which is proven sound, correct and terminating.

Randomized higher-order computation can be seen as being captured by a λ-calculus endowed with a single algebraic operation, namely a construct for binary probabilistic choice. What matters about such computations is the probability of obtaining any given result, rather than the possibility or the necessity of obtaining it, like in (non)deterministic computation. Termination, arguably the simplest kind of reachability problem, can be spelled out in at least two ways, depending on whether it talks about the probability of convergence or about the expected evaluation time, the second one providing a stronger guarantee. In this paper, we show that intersection types are capable of precisely characterizing both notions of termination inside a single system of types: the probability of convergence of any λ-term can be underapproximated by its type , while the underlying derivation’s weight gives a lower bound to the term’s expected number of steps to normal form. Noticeably, both approximations are tight—not only soundness but also completeness holds. The crucial ingredient is non-idempotency, without which it would be impossible to reason on the expected number of reduction steps which are necessary to completely evaluate any term. Besides, the kind of approximation we obtain is proved to be optimal recursion theoretically: no recursively enumerable formal system can do better than that.

The notion of solvability, crucial in the λ-calculus, is conservatively extended to a probabilistic setting, and a complete characterization of it is given. The employed technical tool is a type assignment system, based on non-idempotent intersection types, whose typable terms turn out to be precisely the terms which are solvable with nonnull probability. We also supply an operational characterization of solvable terms, through the notion of head normal form, and a denotational model of Λ_⊕, itself induced by the type system, which equates all the unsolvable terms.

We introduce two extensions of the λ -calculus with a probabilistic choice operator, Λ⊕cbvand Λ⊕cbn, modeling respectively call-by-value and call-by-name probabilistic computation. We prove that both enjoys confluence and standardization, in an extended way: we revisit these two fundamental notions to take into account the asymptotic behaviour of terms. The common root of the two calculi is a further calculus based on Linear Logic, Λ⊕!, which allows us to develop a unified, modular approach.

Despite the fact that call-by-value λ-calculus was defined by Plotkin in 1977, we believe that its theory of program approximation is still at the beginning. A problem that is often encountered when studying its operational semantics is that, during the reduction of a λ-term, some redexes remain st uck (waiting for a value). Recently, Carraro and Guerrieri proposed to endow this calculus with permutation rules, naturally arising in the context of linear logic proof-nets, that succeed in unblocking a certain number of such redexes. In the present paper we introduce a new class of models of call-by-value λ-calculus, arising from non-idempotent intersection type systems. Beside satisfying the usual properties as soundness and adequacy, these models validate the permutation rules mentioned above as well as some reductions obtained by contracting suitable λI-redexes. Thanks to these (perhaps unexpected) features, we are able to demonstrate that every model living in this class satisfies an Approximation Theorem with respect to a refined notion of syntactic approximant. While this kind of results often require impredicative techniques like reducibility candidates, the quantitative information carried by type derivations in our system allows us to provide a combinatorial proof.

The inhabitation problem for intersection types in the lambda-calculus is known to be undecidable. We study the problem in the case of non-idempotent intersection, considering several type assignment systems, which characterize the solvable or the strongly normalizing lambda-terms. We prove the decidability of the inhabitation problem for all the systems considered, by providing sound and complete inhabitation algorithms for them.

We study an extension of Plotkin's call-by-value lambda-calculus via two commutation rules (sigma-reductions). These commutation rules are sufficient to remove harmful call-by-value normal forms from the calculus, so that it enjoys elegant characterizations of many semantic properties. We prove that this extended calculus is a conservative refinement of Plotkin's one. In particular, the notions of solvability and potential valuability for this calculus coincide with those for Plotkin's call-by-value lambda-calculus. The proof rests on a standardization theorem proved by generalizing Takahashi's approach of parallel reductions to our set of reduction rules. The standardization is weak (i.e. redexes are not fully sequentialized) because of overlapping interferences between reductions. Comment: 27 pages

Intersection type assignment systems can be used as a general framework for building logical models of λ-calculus that allow to reason about the denotation of terms in a finitary way. We defineessentialmodels (a new class of logical models) through a parametric type assignment system using non-idempotent intersection types. Under an interpretation of terms based on typings instead than the usual one based on types, every suitable instance of the parameters induces a λ-model, whose theory is sensible. We prove that this type assignment system provides a logical description of a family of λ-models arising from a category of sets and relations.

The eleventh Ackermann Award is presented at CSL'15 in Berlin, Germany. This year, again, the EACSL Ackermann Award is generously sponsored by the Kurt Gödel Society. Besides providing financial support for the Ackermann Award, the Kurt Gödel Society has also committed to inviting the recipients of the Award for a special lecture to be given to the Society in Vienna.

In this work we present a proof-theoretical justification for the intersection type assignment system (IT) by means of the logical system Intersection Synchronous Logic (ISL). ISL builds classes of equivalent deductions of the implicative and conjunc

We present a type system for an extension of lambda calculus with a conditional construction, named STA B , that characterizes the PSPACE class. This system is obtained by extending STA, a type assignment for lambda-calculus inspired by Lafont’s Soft Linear Logic and characterizing the PTIME class. We extend STA by means of a ground type and terms for Booleans and conditional. The key issue in the design of the type system is to manage the contexts in the rule for conditional in an additive way. Thanks to this rule, we are able to program polynomial time Alternating Turing Machines. From the well-known result APTIME = PSPACE, it follows that STA B is complete for PSPACE. Conversely, inspired by the simulation of Alternating Turing machines by means of Deterministic Turing machine, we introduce a call-by-name evaluation machine with two memory devices in order to evaluate programs in polynomial space. As far as we know, this is the first characterization of PSPACE that is based on lambda calculus and light logics.

We study the notion of solvability in the resource calculus, an extension of the λ-calculus modelling resource consumption. Since this calculus is non-deterministic, two different notions of solvability arise, one optimistic (angelical, may) and one

No abstract available.

We propose a characterization of PSPACE by means of atype assignment for an extension of lambda calculus with a conditional construction. The type assignment STAB is an extension of STA, a type assignment for lambda-calculus inspired by Lafont's Soft Linear Logic.

The so-called light logics have been introduced as logical systems enjoying quite remarkable normalization properties. Designing a type assignment system for pure lambda calculus from these logics, however, is problematic. In this paper we show that shifting from usual call-by-name to call-by-value lambda calculus allows regaining strong connections with the underlying logic. This will be done in the context of Elementary Affine Logic (EAL), designing a type system in natural deduction style assigning EAL formulae to lambda terms.

No abstract available.

There are two results in this paper. We first prove that alpha-conversion on types can be eliminated from the second-order λ -calculus F of Girard and Reynolds without affecting the typing power of the system. On the other hand we show that it is impossible to eliminate alpha-conversion on universally quantified variables in the higher-order λ -calculus F ω of Girard, by exhibiting a term which is typable in F ω with alpha-conversion but not typable in F ω without alpha-conversion.

In this paper we give a big-step Structured Operational Semantics (SOS), in the style of Plotkin, Kahn and Milner, of a significant fragment of the functional programming language Scheme , including quote, eval, quasiquote and unquote. The SOS formalism allows us to discuss incrementally the various features of the language and to keep a low mathematical overhead, thus producing a rigorous account of the semantics of a ‘real’ programming language, which nonetheless has a pedagogical value. More specifically, we formalize four strictly increasing fragments of Scheme, using a number of formal systems which express the evaluation of expressions, the display of output results, and the handling of errors.

The polymorphic type assignment system F/sub 2/ is the type assignment counterpart of Girard's and Reynolds' (1972) system F. Though introduced in the early seventies, both the type inference and the type checking problems for F/sub 2/ remained open for a long time. Recently, an undecidability result was announced. Consequently, it is considerably interesting to find decidable restrictions of the system. We show a bounded type inference and a bounded type checking algorithm, both based on the study of the relationship between the typability of a term and the typability of terms that "properly" /spl eta/-reduce to it.>

Polymorphic type discipline for lambda -calculus is an extension of H.B. Curry's (1969) classical functionality theory, in which types can be universally quantified. An algorithm that, given a term M, builds a set of constraints, is satisfied. Moreover, all the typings for M (if any) are built from the set of constraints by substitutions. Using the set of constraints, some properties of polymorphic type discipline are proved.>