arXiv subjects

Paolo Pistone

Publications and source records attributed to Paolo Pistone.

27 records · Page 2Linked to original sources

arXiv2021

On Measure Quantifiers in First-Order Arithmetic (Long Version)

We study the logic obtained by endowing the language of first-order arithmetic with second-order measure quantifiers. This new kind of quantification allows us to express that the argument formula is true in a certain portion of all possible interpretations of the quantified variable. We show that first-order arithmetic with measure quantifiers is capable of formalizing simple results from probability theory and, most importantly, of representing every recursive random function. Moreover, we introduce a realizability interpretation of this logic in which programs have access to an oracle from the Cantor space.

cs.LO↗

arXiv2021

The naturality of natural deduction (II). Some remarks on atomic polymorphism

In a previous paper (of which this is a prosecution) we investigated the extraction of proof-theoretic properties of natural deduction derivations from their impredicative translation into System F. Our key idea was to introduce an extended equational theory for System F codifying at a syntactic level some properties found in parametric models. In a recent series of papers a different approach to extract proof-theoretic properties of natural deduction derivations was proposed by defining predicative variants of the usual translation, embedding intuitionistic propositional logic into the atomic fragment of System F. In this paper we show that this approach finds a general explanation within our equational study of second-order natural deduction, and a clear semantic justification provided by parametricity.

math.LO↗

arXiv2020

The Yoneda Reduction of Polymorphic Types (Extended Version)

In this paper we explore a family of type isomorphisms in System F whose validity corresponds, semantically, to some form of the Yoneda isomorphism from category theory. These isomorphisms hold under theories of equivalence stronger than beta-eta-equivalence, like those induced by parametricity and dinaturality. We show that the Yoneda type isomorphisms yield a rewriting over types, that we call Yoneda reduction, which can be used to eliminate quantifiers from a polymorphic type, replacing them with a combination of monomorphic type constructors. We establish some sufficient conditions under which quantifiers can be fully eliminated from a polymorphic type, and we show some application of these conditions to count the inhabitants of a type and to compute program equivalence in some fragments of System F.

cs.LO↗

arXiv2019

On completeness and parametricity in the realizability semantics of System F

We investigate completeness and parametricity for a general class of realizability semantics for System F defined in terms of closure operators over sets of $λ$-terms. This class includes most semantics used for normalization theorems, as those arising from Tait's saturated sets and Girard's reducibility candidates. We establish a completeness result for positive types which subsumes those existing in the literature, and we show that closed realizers satisfy parametricity conditions expressed either as invariance with respect to logical relations or as dinaturality. Our results imply that, for positive types, typability, realizability and parametricity are equivalent properties of closed normal $λ$-terms.

cs.LO↗

arXiv2019

The naturality of natural deduction

Developing a suggestion by Russell, Prawitz showed how the usual natural deduction inference rules for disjunction, conjunction and absurdity can be derived using those for implication and the second order quantifier in propositional intuitionistic second order logic $NI^2$. It is however well known that the translation does not preserve the relations of identity among derivations induced by the permutative conversions and immediate expansions for the definable connectives, at least when the equational theory of $NI^2$ is assumed to consist only of $β$ and $η$ equations. On the basis of the categorial interpretation of $NI^2$, we introduce a new class of equations expressing what in categorial terms is a naturality condition satisfied by the transformations interpreting $NI^2$-derivations. We show that the Russell-Prawitz translation does preserve identity of proof with respect to the enriched system by highlighting the fact that naturality corresponds to a generalized permutation principle. We show that these result generalize some facts which have gone so far unnoticed, namely that the Russell-Prawitz translation maps particular classes of instances of the equations governing disjunction (and the other definable connectives) onto equations which are already included in the $βη$ equational theory of $NI^2$. Finally, we compare our approach with the one proposed by Ferreira and Ferreira and show that the naturality condition suggests a generalization of their methods to a wider class of formulas.

math.LO↗

arXiv2019

Proof Nets, Coends and the Yoneda Isomorphism

Proof nets provide permutation-independent representations of proofs and are used to investigate coherence problems for monoidal categories. We investigate a coherence problem concerning Second Order Multiplicative Linear Logic (MLL2), that is, the one of characterizing the equivalence over proofs generated by the interpretation of quantifiers by means of ends and coends. We provide a compact representation of proof nets for a fragment of MLL2 related to the Yoneda isomorphism. By adapting the "rewiring approach" used in coherence results for star-autonomous categories, we define an equivalence relation over proof nets called "re-witnessing". We prove that this relation characterizes, in this fragment, the equivalence generated by coends.

cs.LO↗

arXiv2018

Proof nets and the instantiation overflow property

Instantiation overflow is the property of those second order types for which all instances of full comprehension can be deduced from instances of atomic comprehension. In other words, a type has instantiation overflow when one can type, by atomic polymorphism, "expansion terms" which realize instances of the full extraction rule applied to that type. This property was investigated in the case of the types arising from the well-known Russell-Prawitz translation of logical connectives into System F, but is not restricted to such types. Moreover, it can be related to functorial polymorphism, a well-known categorial approach to parametricity in System F. In this paper we investigate the instantiation overflow property by exploiting the representation of derivations by means of linear logic proof nets. We develop a geometric approach to instantiation overflow yielding a deeper understanding of the structure of expansion terms and Russell-Prawitz types. Our main result is a characterization of the class of types of the form $\forall XA$, where $A$ is a simple type, which enjoy the instantiation overflow property, by means of a generalization of Russell-Prawitz types.

cs.LO↗

arXiv2017

Polymorphism and the obstinate circularity of second order logic: a victims' tale

The investigations on higher-order type theories and on the related notion of parametric polymorphism constitute the technical counterpart of the old foundational problem of the circularity (or impredicativity) of second and higher order logic. However, the epistemological significance of such investigations, and of their often non trivial results, has not received much attention in the contemporary foundational debate. The results recalled in this paper suggest that the question of the circularity of second order logic cannot be reduced to the simple assessment of a vicious circle. Through a comparison between the faulty consistency arguments given by Frege and Martin-Löf, respectively for the logical system of the Grundgesetze (shown inconsistent by Russell's paradox) and for the intuitionistic type theory with a type of all types (shown inconsistent by Girard's paradox), and the normalization argument for second order type theory (or System F), we indicate a bunch of subtle mathematical problems and logical concepts hidden behind the hazardous idea of impredicative quantification, constituting a vast (and largely unexplored) domain for foundational research.

math.LO↗

arXiv2014

Logic Programming and Logarithmic Space

We present an algebraic view on logic programming, related to proof theory and more specifically linear logic and geometry of interaction. Within this construction, a characterization of logspace (deterministic and non-deterministic) computation is given via a synctactic restriction, using an encoding of words that derives from proof theory. We show that the acceptance of a word by an observation (the counterpart of a program in the encoding) can be decided within logarithmic space, by reducing this problem to the acyclicity of a graph. We show moreover that observations are as expressive as two-ways multi-heads finite automata, a kind of pointer machines that is a standard model of logarithmic space computation.

cs.LO↗