# Simona Ronchi Della RoccaUniversità degli Studi di Torino | UNITO · Dipartimento di Informatica

Simona Ronchi Della Rocca

full professor

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126

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Introduction

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January 1987 - present

## Publications

Publications (126)

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)determinist...

Randomized higher-order computation can be seen as being captured by a $\lambda$-calculus endowed with a single algebraic operation, namely a construct for binary probabilistic choice. What matters about such computations is the \emph{probability} of obtaining any given result, rather than the \emph{possibility} or the \emph{necessity} of obtaining...

We define an observability property for a calculus with pattern matching which is inspired by the notion of solvability for the lambda-calculus. We prove that observability can be characterized by means of typability and inhabitation in an intersection type system P based on non-idempotent types. We show first that the system P characterizes the se...

In this paper an implicit characterization of the complexity classes k-EXP and k-FEXP, for k≥0, is given, by a type assignment system for a stratified λ-calculus, where types for programs are witnesses of the corresponding complexity class. Types are formulae of Elementary Linear Logic (ELL), and the hierarchy of complexity classes k-EXP is charact...

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 s...

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 refineme...

We present a type assignment system for the call-by-value \(\lambda \)-calculus, such that typable terms reduce to normal form in a number of steps which is elementary in the size of the term itself and in the rank of the type derivation. Types are built through non-idempotent and non-associative intersection, and the system is loosely inspired by...

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 define
essential
models (a new class of logical models) through a parametric type assignment system using non-idempotent intersection types. Under an interpretatio...

One of the aims of Implicit Computational Complexity is the design of
programming languages with bounded computational complexity; indeed,
guaranteeing and certifying a limited resources usage is of central importance
for various aspects of computer science. One of the more promising approaches
to this aim is based on the use of lambda-calculus as...

In this paper an implicit characterization of the complexity classes k-EXP and k-FEXP, for k≥0, is given, by a type assignment system for a stratified λ-calculus, where types for programs are witnesses of the corresponding complexity class. Types are formulae of elementary linear logic (ELL), and the hierarchy of complexity classes k-EXP is charact...

Non-idempotent intersection types are used in order to give a bound of the
length of the normalization beta-reduction sequence of a lambda term: namely,
the bound is expressed as a function of the size of the term.

The resource calculus is an extension of the lambda-calculus allowing to
model resource consumption. It is intrinsically non-deterministic and has two
general notions of reduction - one parallel, preserving all the possible
results as a formal sum, and one non-deterministic, performing an exclusive
choice at every step. We prove that the non-determ...

This volume contains the proceedings of the Sixth Workshop on Logical and
Semantic Frameworks with Applications (LSFA 2011). The workshop will be hold in
Belo Horizonte, on August 27th 2011.
Logical and semantic frameworks are formal languages used to represent
logics, languages and systems. These frameworks provide foundations for formal
specifica...

A new complete characterization of β-strong normalization is given, both in the classical and in the lazy λ-calculus, through the notion of potential valuability inside two suitable parametric calculi.

We present a type system for an extension of lambda calculus with a conditional construction, named STAB, 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 fo...

The resource calculus is an extension of the -calculus allow- ing to model resource consumption. Namely, the argument of a function comes as a nite multiset of resources, which in turn can be either linear or reusable, giving rise to non-deterministic choices, expressed by a for- mal sum. Using the -calculus terminology, we call solvable a term tha...

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 pessimistic (demoniac, must). We give a syntactical, operational and logical characterization for th...

The intersection type assignment system has been designed directly as deductive system for assigning formulae of the implicative and conjunctive fragment of the intuitionistic logic to terms of lambda-calculus. But its relation with the logic is not standard. Between all the logics that have been proposed as its foundation, we consider ISL, which g...

Using Soft Linear Logic (SLL) as case study, we analyze a method for transforming a light logic into a type assignment system
for the λ-calculus, inheriting the complexity properties of the logics. Namely the typing assures the strong normalization
in a number of steps polynomial in the size of the term, and moreover all polynomial functions can be...

The aim of this paper is to understand the interplay between intersection, universally quantified, and reference types. Putting toget her the standard typing rules for intersection, universally quantified, and reference types leads to loss of subject reduc- tion. The problem comes from the invariance of the reference type constructor and the rules...

Type assignment systems for λ-calculus based on intersection types are a general framework for building models of λ-calculus (known as filter-models) which are useful tools for reasoning in a finitary way about the denotational intepretation of terms. Indeed the denotation of a term is the set of types derivable for it and a type is a "finite piece...

Bellantoni and Cook have given a function-algebra characterization of the polynomial-time computable functions via an unbounded recursion scheme which is called safe recursion. Inspired by their work, we characterize the exponential-time computable functions ...

In this work we present a proof-theoretical justification for the intersection type assign-ment system (IT) by means of the logical system Intersection Synchronous Logic (ISL). ISL builds classes of equivalent deductions of the implicative and conjunctive fragment of the intuitionistic logic (NJ). ISL results from decomposing intuitionistic conjunc...

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 connection...

We describe some results inspired to Lafont's Soft Linear Logic (SLL) which is a subsystem of second-order linear logic with restricted rules for exponentials, correct and complete for polynomial time computations. SLL is the basis for the design of type assignment systems for lambda-calculus, characterizing the complexity classes PTIME, PSPACE and...

We study the type inference problem for the Soft Type As- signment system (STA) for -calculus introduced in (1), which is correct and complete for polynomial time computations. In particular we design an algorithm which, given in input a -term, provides all the constraints that need to be satised in order to type it. For the propositional frag- men...

We propose a characterization of PSPACE by means of a type 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. We extend STA by means of a ground type and terms for Booleans. The key point is tha...

In this paper, we present , a fully typed λ-calculus based on the intersection-type system discipline, which is a counterpart à la Church of the type assignment system as invented by Coppo and Dezani. The relationship between and the intersection type assignment system is the standard isomorphism between typed and type assignment system, and so the...

Aim of this paper is to understand the interplay between intersection and reference types. Putting together the standard typing rules for intersection types and reference types leads to loss of subject reduction. The problem comes from the invariance of the reference type constructor and the rule of intersection elimination, which is essentially a...

The denotational semantics of the call-by-value -calculus in a categorical setting is given. Furthermore, a particular model based on coherence domains is studied.

This paper introduces the Φ-calculus, a new call-by-value version of the λ-calculus, following the spirit of Plotkin’s λβ
v
-calculus. The Φ-calculus satisfies some interesting properties, in particular that its set of solvable terms coincides with the set of β-strongly normalizing terms in the classical λ-calculus.

We study the cube of type assignment systems, as introduced in [10]. This cube is obtained from Barendregt's typed -cube [1] via a natural type erasing function E, that erases type information from terms. We prove that the systems in the former cube enjoy good computational properties, like subject reduction and strong normalization. We study the r...

Among all the reduction strategies for the untyped ‚-calculus, the so called lazy fl-evaluation is of particular interest due to its large applicability to functional programming languages (e.g. Haskell). This strategy reduces only redexes not inside a lambda abstraction. The lazy strongly fl- normalizing terms are the ‚-terms that don't have infln...

In this paper, we presents a comfortable fully typed lambda calculus based on the well-known intersection type system discipline where proof are not only feasible but easy; the present system is the counterpart à la Church of the type assignment system as invented by Coppo and Dezani.

The so-called light logics [1,2,3] 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, as discussed in [4].
In this paper we show that shifting from usual call-by-name to call-by-value lambda calculus allow...

Elementary Affine Logic (EAL) is a variant of Linear Logic characterizing the computa- tional power of the elementary bounded Turing machines. The EAL Type Inference problem is the problem of automatically assigning to terms of -calculus EAL formulas as types. This problem, re- stricted to the propositional fragment of EAL, is proved to be decidabl...

In this paper, we presents a comfortable fully typed lambda calculus based on the well-known intersection type system discipline where proof are not only feasible but easy; the present system is the counterpart à la Church of the type assignment system as invented by Coppo and Dezani.

The goal of this work is to present a proof-theoretical justification for IT. In particular, we discuss the relationship between the intersection connective and the intuitionistic conjunction. For this purpose, we define a new logical system called Intersection Synchronous Logic (ISL), that proves properties of sets of deductions of the implication...

The lazy evaluation of the -calculus, both in call-by-name and in call-by-value setting, is studied. Starting from a logical descriptions of two topological models of such calculi, a pre-order relation on terms, stratied by types, is dened, which grasps exactly the two operational semantics we want to model. Such a relation can be used for building...

A λ-calculus is defined, which is parametric with respect to a set V of input values and subsumes all the different λ-calculi given in the literature, in particular the classical one and the call-by-value λ-calculus of Plotkin. It is proved that it enjoy the confluence property, and a necessary and sufficient condition is given, under which it enjo...

There is an analogy between λΔ-filter models and λΔ-models that are ω-algebraic lattices, which was first noticed in [28] and further developed in [1] and [3]. This analogy lies in the fact that type symbols in a λΔ-filter model play the role of names for compact elements in the corresponding ω-algebraic lattice. It is out of the aim of this book t...

In the Introduction we claimed that both the λΛ-calculus and the λΓ-calculus can be seen as paradigms for programming languages in the call-by-name and call-by-value settings respectively. In this chapter this claim will be justified. In fact, we will show that both the call-by-name and the call-by-value λ-calculi have the computational power of Tu...

In this part we will study the evaluation of terms and the induced operational semantics. Our notion of operational semantics is inspired by the structured operational semantics (SOS) developed by Plotkin [80] and by Kahn [55].

H ∈ ε (Λ, Λ-HNF) is the first evaluation relation that we will study; it is the universal evaluation relation u
Λ-HNFΛ
(see Example 5.1.5).

The more usual programming languages are such that parameters must be evaluated in order to be supplied to a function, and moreover the body of a function is evaluated only when parameters are supplied. The first policy is the so called call-by-value parameter passing, and the second policy is called lazy-evaluation. In order to mimic this kind of...

In order to represent a numerical function with respect to an evaluation relation O, it is necessary to exhibit a term mimicking the behaviour of the function itself. More precisely the reduction machine, taken as input this term applied to a sequence of terms representing natural numbers, gives as output the term representing the result, if it exi...

For modeling the λΓ-calculus, we must reflect in the model the fact that the set Γ of input values is a proper subset of the whole set Λ. In the setting of filter λΓ-models, this implies that every type system ∇ inducing a filter λΓ-model must be such that I(∇) ⊂ F(∇).

A calculus is a language equipped with some reduction rules. All the calculi we consider in this book share the same language, which is the language of λ-calculus, while they differ each other in their reduction rules. In order to treat them in an uniform way we define a parametric calculus, the λΔ-calculus, which gives rise to different calculi by...

In Sect. 1.3, the notion of full extensionality was introduced.

A parameter passing policy is said to be call-by-name if the parameters need not be evaluated in order to be supplied to the function. In our setting, this means that all terms can be considered as input values. So, in order to mimic this policy with the parametric λΔ-calculus, it is sufficient to define Δ = Λ. Then all terms are input values, and...

To study the operational behaviour of λ-terms, we will use the denotational (mathematical) approach. A denotational semantics for a language is based on the choice of a space of semantics values, or denotations, where terms are to be interpreted. Choosing a space with nice mathematical properties can help in proving the semantic properties of terms...

As proved in Property 5.0.3, the set of Γ-lazy blocked normal forms (Γ-lbnf’s), namely Γ-LBNF = {λx.M | M ∈ Λ} ∪ {xM
1...M
m
| M
i
∈ Λ, m ∈ ℕ} ∪ {(λx.P)QM
1...M
m
| P, M
i
∈ Λ, Q ∉ Γ, Q ∈ Γ-LBNF, m ∈ ℕ}, is a set of output values with respect to Γ. Notice that Γ-LBNF0 = Γ
0.

To model a λΛ-theory, it is necessary to reflect in the model the fact that the set of input values coincides with the whole set Λ. So it is natural to ask that in every model the set D and the set of semantic input values I must coincide. In a filter model, it must be F(C, ∇) = I(C, ∇), so T(C) = I(C).

The lambda-calculus was invented by Church in the 1930s with the purpose of supplying a logical foundation for logic and mathematics.
Its use by Kleene as a coding for computable functions makes it the first programming language, in an abstract sense, exactly as the Turing machine can be considered the first computer machine. The lambda-calculus ha...

Elementary Affine Logic (EAL) is a variant of the Linear Logic characterizing the computational power of the elementary bounded Turing machines. The EAL Type Inference problem is the problem of automatically assigning to terms of lambda-calculus EAL formulas as types. This problem is proved to be decidable, and an algorithm is showed, building, for...

The intersection-types discipline for λ-calculus was introduced in Coppo and Dezani-Ciancaglini (1980) as an extension of Curry's type assignment system. The motivation was essentially to increase the class of terms possessing types. Indeed, it turned out that this discipline can assign types to all and only the strongly normalising terms. This is...

The aim of this paper is to discuss the design of an explicitly typed #-calculus corresponding to the Intersection Type Assignment System (IT ), which assigns intersection types to the untyped #-calculus. Two di#erent proposals are given. The logical foundation of all of them is the Intersection Logic IL. 1

We study the cube of type assignment systems, as introduced in Giannini et al. (Fund. Inform. 19 (1993) 87–126), and confront it with Barendregt's typed gl-cube (Barendregt, in: Handbook of Logic in Computer Science, Vol. 2, Clarenden Press, Oxford, 1992). The first is obtained from the latter through applying a natural type erasing function E to d...

Intersection types are well-known to type theorists mainly for two reasons. Firstly, they type all and only the strongly normalizable lambda terms. Secondly, the intersection type operator is a meta-level operator, that is, there is no direct logical counterpart in the Curry- Howard isomorphism sense. In particular, its meta-level nature implies th...

The notion of solvability in the call-by-value λ-calculus
is defined and completely characterized, both from an operational and a logical
point of view. The operational characterization is given through a reduction
machine, performing the classical β-reduction, according to an innermost strategy.
In fact, it turns out that the call-by-value
reduct...

In this paper the notion of extensionality is studied, for theories of of -calculi which arise from operational semantics. A new definition of extensionality is introduced, parameterized with respect to the particular operational semantics we want to study, and it is proved that to be extensional is equivalent to be closed under a generalized j-red...

The intersection type assignment system IT uses the formulas of the negative fragment of the predicate calculus (LJ) as types
for the λ-terms. However, the deductions of IT only correspond to the proper sub-set of the derivations of LJ, obtained by
imposing a meta-theoretic condition about the use of the conjunction of LJ. This paper proposes a log...

There are two results in this paper. We first prove that alpha-conversion on types can be eliminated from the second-order lambda-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 la...

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 mathema...

The paper explores different approaches for modeling the lazy -calculus, which is a paradigmatic language for studying the operational behaviour of programming languages, like Haskell, using a call-by-name and lazy evaluation mechanism. Two models for lazy -calculus in the coherence spaces setting are built. They give a new insight in the behaviour...

The introduction of Linear Logic extends the Curry-Howard Isomorphism to intensional aspects of the typed functional programming. In particular, every formula of Linear Logic tells whether the term it is a type for, can be either erased/duplicated or not, during a computation. So, Linear Logic can be seen as a model of a computational environment w...

In this work we present a categorical approach for modeling the pure (i.e., without constants) call-by-value -calculus, defined by Plotkin as a restriction of the classical one. In particular, we study the properties a category must enjoy for give rise to a model of such a language. This definition is enough general for grasping models in different...

The paper is organized as follows. In Section 2 the polymorphic type assignment and a new type assignment based on a containment relation between types are introduced, and their equivalence is proved. In Section 3 the type inference problem is discussed. Namely the complete stratification is introduced; then the notion of scheme is defined, which m...

The polymorphic type assignment system F<sub>2</sub> 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</sub> remained open for a long time.
Recently, an undecidability result was announced. Consequently, it is
con...

In this paper we investigate the type inference problem for a large class of type assignment systems for the λ-calculus. This is the problem of determining if a term has a type in a given system. We discuss, in particular, a collection of type assignment systems which correspond to the typed systems of Barendregt’s “cube”. Type dependencies being s...

It is well known that a reflexive object in the Cartesian closed category of complete partial orders and Scott-continuous functions is a model of λ-calculus (briefly a topological model). A topological model, through the interpretation function, induces a λ-theory, i.e., a congruence relation on λ-terms closed under α- and β-reduction. It is natura...

The functional fragment of Landin’s ISWIM as implemented by the SECD machine is the paradigm of the procedural kernel of many programming languages. We investigate and compare operational, denotational and logical descriptions of the ISWIM-SECD system. Our goal is to illustrate how to derive from each of these descriptions logical tools for resonin...

A hierarchy of type assignment systems is defined, which is a complete stratification of the polymorphic type assignment system. For each of such systems a type inference algorithm is given.

The intersection type discipline for the λ-calculus (ITD) is an extension of the classical functionality theory of Curry. In the ITD a term satisfying a given property has a principal type scheme in an extended meaning, i.e., there is a type scheme deducible for it from which all and only the type schemes deducible for it are reachable, by means of...

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...

We consider the extension of Curry's basic functionality theory presented by Barendregt et al. (to appear), and we define, for any term X, a principal type scheme (p.t.s.). We prove that all and the only type schemes deducible for X can be obtained from the p.t.s. of X by suitable operations.

The filter λ-model is a model of the,λ-calculus, based on a system of type assignment which extends the basic functionality theory of Curry, invented in order to give a completeness proof for Scott's semantics of Curry's type assignment. In this paper the local structure of this λ-model, i.e., the syntactical characterization of the equality and in...

A finite set {F1,…,Fn} of λ-terms is said to be discriminable if, given n arbitrary λ-terms X1,…,Xn, there exists a λ-term Δ such that: ΔFi ⩾ Xifor 1 ⩽ i ⩽ n. In the present paper each finite set of normal combinators which are pairwise non α-η-convertible is proved to be discriminable. Moreover a discrimination algorithm is given.

A finite set {F1, ...,Fn} of terms of λ-calculus is said to be:
separable iff, given n arbitrary terms X1, ..., Xn, there exists a context C [ ] such that C[Fi]=Xi for 1≤i≤n
semi-separable iff, given n−1 arbitrary terms X1, ..., Xn−1 there exists a context C [ ] such that C [Fi]=Xi for 1≤i ≤n−1 and C [Fn] is unsolvable.
In the present paper the con...

First S-questionnaires are introduced as mechanical devices which permit the representation of recursively definable families of information structures by means of sets of sequences of non-negative integers. The subclass of L-questionnaires is then defined enabling: i) a bijection between such a family F and the set of non-negative integers ii) the...

## Projects

Project (1)