8 research outputs found
On Extensions of AF2 with Monotone and Clausular (Co)inductive Definitions
This thesis discusses some extensions of second-order logic AF2 with primitive constructors representing least and greatest fixed points of monotone operators, which allow to define predicates by induction and
coinduction. Though the expressive power of second-order logic has been well-known for a long time and suffices to define (co)inductive predicates by means of its (co)induction principles, it is more user-friendly to have a direct way of defining predicates inductively. Moreover recent applications in computer science oblige to consider also coinductive definitions useful for handling infinite objects, the most prominent example being the data type of streams or infinite lists. Main features of our approach are the use clauses in the (co)inductive definition mechanism, concept which simplifies the syntactic shape of the predicates, as well as the inclusion of not only (co)iteration but also primitive (co)recursion principles and in the case of coinductive definitions an inversion principle.
For sake of generality we consider full monotone, and not only positive definitions, after all positivity is only used to ensure monotonicity.
Working towards practical use of our systems we give them realizability interpretations where the systems of realizers are strongly normalizing extensions of the second-order polymorphic lambda calculus, system F in
Curry-style, with (co)inductive types corresponding directly to the logical systems via the Curry-Howard correspondence. Such realizability interpretations are therefore not reductive: the definition of realizability for a (co)inductive definition is again a (co)inductive definition. As main application of realizability we extend the so-called programming-with-proofs paradigm of Krivine and Parigot to our logics, by means of which a correct program of the lambda calculus can be extracted from a proof in the logic
Mendler-style Iso-(Co)inductive predicates: a strongly normalizing approach
We present an extension of the second-order logic AF2 with iso-style
inductive and coinductive definitions specifically designed to extract programs
from proofs a la Krivine-Parigot by means of primitive (co)recursion
principles. Our logic includes primitive constructors of least and greatest
fixed points of predicate transformers, but contrary to the common approach, we
do not restrict ourselves to positive operators to ensure monotonicity, instead
we use the Mendler-style, motivated here by the concept of monotonization of an
arbitrary operator on a complete lattice. We prove an adequacy theorem with
respect to a realizability semantics based on saturated sets and
saturated-valued functions and as a consequence we obtain the strong
normalization property for the proof-term reduction, an important feature which
is absent in previous related work.Comment: In Proceedings LSFA 2011, arXiv:1203.542
Two extensions of system F
This paper presents two extensions of the second order polymorphic
lambda calculus, system F, with monotone (co)inductive types supporting
(co)iteration, primitive (co)recursion and inversion principles as
primitives. One extension is inspired by the usual categorical
approach to programming by means of initial algebras and final
coalgebras; whereas the other models dialgebras, and can be seen as an extension of Hagino's
categorical lambda calculus within the framework of parametric
polymorphism. The systems are presented in Curry-style, and are proven to be terminating and
type-preserving. Moreover
their expressiveness is shown by means of several programming
examples, going from usual data types to lazy codata types such as streams
or infinite trees
Conjuntos y modelos: curso avanzado
El libro Conjuntos y modelos: curso avanzado, está dedicado a los alumnos de posgrado o del último año de licenciatura que en sus estudios utilicen la teoría de conjuntos o la lógica matemática. En este libro se cubren los fundamentos de la teoría de conjuntos y de la lógica matemática para llegar a temas avanzados como la combinatoria infinita, la teoría de modelos y el universo construible de Gödel. Se pone especial énfasis en las aplicaciones principalmente en álgebra. Esta obra pretende remediar en buena medida la ausencia de material didáctico en español e incluso en otros idiomas que cubra los temas presentados sistemáticamente en este trabajo