COMETA 03 LOGICAL SEMANTICS

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COMETA 03 LOGICAL SEMANTICS

• Fabio Alessi (UD)
• Stefano Berardi (TO)
• Felice Cardone (MI-Bicocca)
• Ugo de Liguoro (TO)
• Mariangiola Dezani (TO)
• Furio Honsell (UD)
• Marina Lenisa (UD)
• Ines Margaria (TO)
• Simona Ronchi Della Rocca (TO)
• Antonino Salibra (VE)

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INTERSECTION TYPES TO CHARACTERIZE EVALUATON
PROPERTIES OF TERMS

• The goal will be to understand better how to use
  intersection types to characterize,
  compositionally, evaluation properties of terms.

Can we characterize in some significant way
the class of evaluation properties
which we can model using intersection
types?
Is there a method for going from a logical
specification of a property to the appropriate
intersection type theory? More examples are
needed.
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lambda -calculusFILTER MODELS CHARACTERIZING

• easyness of terms (Alessi, Lusin, ENTCS 2002)
  (Alessi, Dezani,Lusin, TCS 2003) (Dezani, Lusin,
  WIT 2002)

A term M is easy if ?N. MN is consistent.
Given an easy term E and a term M, a filter model
is built, which equates the interpretation of M
and E. The interpretation of an easy term is any
filter which can be described by a continuous
predicate.
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lambda -calculusFILTER MODELS CHARACTERIZING

• ß-(strong)-(head)-(weak head)-normalization and
  their persistent versions (Dezani, Ghilezan,
  Likavec, TCS 2003 )(Dezani, Ghilezan, Schedae
  Inf. 2003) (Dezani, Ghilezan, LNCS 2003)

.
A term enjoys persistently the property P, if
it enjoys P and its application of terms
enjoying P enjoys P too. Two filter models are
built, characterizing all the before listed
properties.
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lambda -calculusFILTER MODELS CHARACTERIZING

• lazy evaluation (both for cbn and cbv ?-calculus)
  - a uniform treatment of the full-abstraction
  problem (Paolini, Ronchi Della Rocca, COMETA 03)

Lazy evaluation can be characterized by
intersection types in a very uniform way in both
cbn and cbv case. A preorder relation on terms is
defined, through intersection types, that allows
to build in a uniform way a fully abstract model
for both calculi.
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lambda -calculusINTERSECTION TYPES
CHARACTERIZING

• Strongly normalizing terms in the lambda calculus
  with explicit substitution (van Bakel, Dezani,
  LNCS, 2002)

A new cut-rule which allows to forget the context
of the minor premise when the context of the
main premise does not have an assumption for the
cut variable.
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lambda-calculusCHARACTERIZATION OF THE
INTERSECTION TYPE
THEORIES

• characterizing meaning invariance of terms
  (Alessi, Barbanera, Dezani,ENTCS 2003)

Typing invariance of terms w.r.t. various notion
of reduction/expansion (?, ? and restrictions)
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lambda-calculusCHARACTERIZATION OF THE
INTERSECTION TYPE
THEORIES

• inducing complete intersection type assignment
  systems w.r.t. the (inference)(simple)(F)-semantic
  s of types (Alessi, Dezani, Honsell, ACM TOCL,
  2003)

The interpretation of the ?? constructor is à la
Scott, w.r.t. either any possible functionality
set, or the larger one, or the least one.
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lambda-calculusLAMBDA MODELS AND LAMBDA THEORIES

• SFPm, a category of compositional domain-models
  for separable Stone spaces (Alessi, Baldan,
  Honsell, TCS, 2003)

SFPm is subcategory of SFPep, closed under direct
limits and some constructors (lifting, sum,
product, and Plotkin powerdomain but not
function space). This category provides very
satisfactory domain-models Of 2-Stone spaces.
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lambda-calculusLAMBDA MODELS AND LAMBDA THEORIES

• Investigation on lambda-models using algebraic
  and topological methods (Salibra, ACM TOCL, 2003)
  (Salibra, AMiLp, 2003)(Lusin, Salibra,
  200?)(Bucciarelli, Salibra,, 2003)

The minimal graph model, whose equational theory
is exactly the set of equations true in each
graph model. The lattice structure of the lambda
theories. Topological incompleteness of lambda
calculus.
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lambda-calculusLAMBDA MODELS AND LAMBDA THEORIES

• Wave-style GoI lambda-models (Honsell, Lenisa,
  int.report, 2002)

An axiomatisation of the GoI construction, which
captures a class of graph-like models.
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ANALYZING VARIOUS TYPE DISCIPLINES

• We will start by analyzing various (intersection)
  type disciplines from the viewpoint of principal
  typings. The key benefit of having principal
  typings is that they seems to be essential for
  having compositional semantics.
• A goal will be to study the expressive power of
  type systems with union types, recursive types,
  and other type constructors.

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ELEMENTARY AFFINE LOGIC

• Type inference and principal typing for
  Elementary Affine Logic (Coppola, Martini, TOCL
  200?)(Coppola, Ronchi Della Rocca, LNCS,2003)

Elementary Affine Logic (EAL) is a restriction of
Linear Logic characterizing elementary
computations. Formulas of EAL can be assigned to
?-calculus. The typabily is decidable. A term
has a finite numbers of principal typings.
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INTERSECTION AND UNION TYPES

• Subtyping as minimal relevant logic (Dezani,
  Frisch, Giovannetti, Motohama, ENTCS, 2002)

Semantics based subtyping relation of Frisch,
Castagna and Benzanken coincides with the
minimal relevant logic of Meyer and Routley.
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INTERSECTION TYPED LAMBDA CALCULUS

• Intersection Typed Lambda Calculus (Ronchi della
  Rocca, ENTCS, 2002)

Intersection Logic (IL) has been proposed as
logical foundation of intersection types (Ronchi
Della Rocca, Roversi). A proof in IL describes a
set of syncronous proofs in LJ. An intersection
typed lambda calculus is built by decorating IL.
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LOGICAL SEMANTICS FOR MODELLING PARTICULAR
PROGRAMMING FEATURES

• Moreover we want to model, by means of logical
  semantics, programming features presenting
  particular problems.
• The main problem to address in giving the
  denotational semantics of languages with names
  (like the pi-calculus and the ambient calculus)
  is to extend the mathematical framework so that
  static scoping of name restrictions can be
  correctly modelled.
• Moreover we want to address various form of
  mobility.

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FILTER MODELS OF MOBILE AMBIENTS

• Filter models for higher order mobile ambients
  (Coppo, Dezani, LNCS, 2002) (Margaria, Zacchi,
  COMETA 03).

Two filter models are built for a calculus with
mobility and higher-order value passing. The
second one models also the feature of safety,
i.e., it models a calculus where to every
mobility action a corresponding co-action is
added.
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FILTER MODELS OF MOBILE OBJECTS

• A filter model for a calculus combining mobility
  and object oriented features (Barbanera, De
  Liguoro, COMETA 03)

The aim is to reason about languages with a
sequential core, and primitives for mobility and
concurrency and communication. The idea for
typing objects is the late typing of the self
(the check on the pre-condition on the self
variable is checked at invocation time.
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SOMETHING THAT YET MUST BE DONE

• We shall extend existing completeness results for
  intersection types to the case of Kripke
  semantics on Kripke applicative structures.
• We shall address also the problem of giving
  general approximation theorems for filter models.
• We will try to describe in an equational way,
  through suitable algebraic operators, the
  properties of the approximants in a given model.