arXiv subjects

Federico Aschieri

Publications and source records attributed to Federico Aschieri.

15 recordsLinked to original sources

arXiv2020

A typed parallel λ-calculus via 1-depth intermediate proofs

We introduce a Curry-Howard correspondence for a large class of intermediate logics characterized by intuitionistic proofs with non-nested applications of rules for classical disjunctive tautologies (1-depth intermediate proofs). The resulting calculus, we call it $λ_{\parallel}$, is a strongly normalizing parallel extension of the simply typed $λ$-calculus. Although simple, the $λ_{\parallel}$ reduction rules can model arbitrary process network topologies, and encode interesting parallel programs ranging from numeric computation to algorithms on graphs.

cs.LO↗

arXiv2019

$\unicode{8523}$ means Parallel: Multiplicative Linear Logic Proofs as Concurrent Functional Programs

Along the lines of the Abramsky ``Proofs-as-Processes'' program, we present an interpretation of multiplicative linear logic as typing system for concurrent functional programming. In particular, we study a linear multiple-conclusion natural deduction system and show it is isomorphic to a simple and natural extension of $λ$-calculus with parallelism and communication primitives, called $λ_{\unicode{8523}}$. We shall prove that $λ_{\unicode{8523}}$ satisfies all the desirable properties for a typed programming language: subject reduction, progress, strong normalization and confluence.

cs.LO↗

arXiv2019

Natural Deduction and Normalization Proofs for the Intersection Type Discipline

Refining and extending previous work by Retoré, we develop a systematic approach to intersection types via natural deduction. We show how a step of beta reduction can be seen as performing, at the level of typing derivations, Prawitz reductions in parallel. Then we derive as immediate consequences of Subject Reduction the main theorems about normalization for intersection types: for system D, strong normalization, for system Omega, the leftmost reduction termination for terms typable without Omega.

cs.LO↗

arXiv2018

On Natural Deduction for Herbrand Constructive Logics III: The Strange Case of the Intuitionistic Logic of Constant Domains

The logic of constant domains is intuitionistic logic extended with the so-called forall-shift axiom, a classically valid statement which implies the excluded middle over decidable formulas. Surprisingly, this logic is constructive and so far this has been proved by cut-elimination for ad-hoc sequent calculi. Here we use the methods of natural deduction and Curry-Howard correspondence to provide a simple computational interpretation of the logic.

math.LO↗

arXiv2018

Classical Proofs as Parallel Programs

We introduce a first proofs-as-parallel-programs correspondence for classical logic. We define a parallel and more powerful extension of the simply typed lambda calculus corresponding to an analytic natural deduction based on the excluded middle law. The resulting functional language features a natural higher-order communication mechanism between processes, which also supports broadcasting. The normalization procedure makes use of reductions that implement novel techniques for handling and transmitting process closures.

cs.LO↗

arXiv2018

Expansion Trees with Cut

Herbrand's theorem is one of the most fundamental insights in logic. From the syntactic point of view, it suggests a compact representation of proofs in classical first- and higher-order logic by recording the information of which instances have been chosen for which quantifiers. This compact representation is known in the literature as Miller's expansion tree proof. It is inherently analytic and hence corresponds to a cut-free sequent calculus proof. Recently several extensions of such proof representations to proofs with cuts have been proposed. These extensions are based on graphical formalisms similar to proof nets and are limited to prenex formulas. In this paper we present a new syntactic approach that directly extends Miller's expansion trees by cuts and covers also non-prenex formulas. We describe a cut-elimination procedure for our expansion trees with cut that is based on the natural reduction steps and show that it is weakly normalizing.

math.LO↗

arXiv2017

On Natural Deduction for Herbrand Constructive Logics II: Curry-Howard Correspondence for Markov's Principle in First-Order Logic and Arithmetic

Intuitionistic first-order logic extended with a restricted form of Markov's principle is constructive and admits a Curry-Howard correspondence, as shown by Herbelin. We provide a simpler proof of that result and then we study intuitionistic first-order logic extended with unrestricted Markov's principle. Starting from classical natural deduction, we restrict the excluded middle and we obtain a natural deduction system and a parallel Curry-Howard isomorphism for the logic. We show that proof terms for existentially quantified formulas reduce to a list of individual terms representing all possible witnesses. As corollary, we derive that the logic is Herbrand constructive: whenever it proves any existential formula, it proves also an Herbrand disjunction for the formula. Finally, using the techniques just introduced, we also provide a new computational interpretation of Arithmetic with Markov's principle.

cs.LO↗

arXiv2017

Gödel Logic: from Natural Deduction to Parallel Computation

Propositional Gödel logic extends intuitionistic logic with the non-constructive principle of linearity $A\rightarrow B\ \lor\ B\rightarrow A$. We introduce a Curry-Howard correspondence for this logic and show that a particularly simple natural deduction calculus can be used as a typing system. The resulting functional language enriches the simply typed lambda calculus with a synchronous communication mechanism between parallel processes. Our normalization proof employs original termination arguments and sophisticated proof transformations with a meaningful computational reading. Our results provide a computational interpretation of Gödel logic as a logic of communicating parallel processes, thus proving Avron's 1991 conjecture.

cs.LO↗

arXiv2016

On Natural Deduction for Herbrand Constructive Logics I: Curry-Howard Correspondence for Dummett's Logic LC

Dummett's logic LC is intuitionistic logic extended with Dummett's axiom: for every two statements the first implies the second or the second implies the first. We present a natural deduction and a Curry-Howard correspondence for first-order and second-order Dummett's logic. We add to the lambda calculus an operator which represents, from the viewpoint of programming, a mechanism for representing parallel computations and communication between them, and from the viewpoint of logic, Dummett's axiom. We prove that our typed calculus is normalizing and show that proof terms for existentially quantified formulas reduce to a list of individual terms forming an Herbrand disjunction.

cs.LO↗

arXiv2016

Game Semantics and the Geometry of Backtracking: a New Complexity Analysis of Interaction

We present abstract complexity results about Coquand and Hyland-Ong game semantics, that will lead to new bounds on the length of first-order cut-elimination, normalization, interaction between expansion trees and any other dialogical process game semantics can model and apply to. In particular, we provide a novel method to bound the length of interactions between visible strategies and to measure precisely the tower of exponentials defining the worst-case complexity. Our study improves the old estimates on average by several exponentials.

math.LO↗

arXiv2013

Strong Normalization for HA + EM1 by Non-Deterministic Choice

We study the strong normalization of a new Curry-Howard correspondence for HA + EM1, constructive Heyting Arithmetic with the excluded middle on Sigma01-formulas. The proof-term language of HA + EM1 consists in the lambda calculus plus an operator ||_a which represents, from the viewpoint of programming, an exception operator with a delimited scope, and from the viewpoint of logic, a restricted version of the excluded middle. We give a strong normalization proof for the system based on a technique of "non-deterministic immersion".

cs.LO↗

arXiv2012

Interactive Realizability and the elimination of Skolem functions in Peano Arithmetic

We present a new syntactical proof that first-order Peano Arithmetic with Skolem axioms is conservative over Peano Arithmetic alone for arithmetical formulas. This result - which shows that the Excluded Middle principle can be used to eliminate Skolem functions - has been previously proved by other techniques, among them the epsilon substitution method and forcing. In our proof, we employ Interactive Realizability, a computational semantics for Peano Arithmetic which extends Kreisel's modified realizability to the classical case.

cs.LO↗

arXiv2011

Learning, Realizability and Games in Classical Arithmetic

In this dissertation we provide mathematical evidence that the concept of learning can be used to give a new and intuitive computational semantics of classical proofs in various fragments of Predicative Arithmetic. First, we extend Kreisel modified realizability to a classical fragment of first order Arithmetic, Heyting Arithmetic plus EM1 (Excluded middle axiom restricted to Sigma^0_1 formulas). We introduce a new realizability semantics we call "Interactive Learning-Based Realizability". Our realizers are self-correcting programs, which learn from their errors and evolve through time. Secondly, we extend the class of learning based realizers to a classical version PCFclass of PCF and, then, compare the resulting notion of realizability with Coquand game semantics and prove a full soundness and completeness result. In particular, we show there is a one-to-one correspondence between realizers and recursive winning strategies in the 1-Backtracking version of Tarski games. Third, we provide a complete and fully detailed constructive analysis of learning as it arises in learning based realizability for HA+EM1, Avigad's update procedures and epsilon substitution method for Peano Arithmetic PA. We present new constructive techniques to bound the length of learning processes and we apply them to reprove - by means of our theory - the classic result of Godel that provably total functions of PA can be represented in Godel's system T. Last, we give an axiomatization of the kind of learning that is needed to computationally interpret Predicative classical second order Arithmetic. Our work is an extension of Avigad's and generalizes the concept of update procedure to the transfinite case. Transfinite update procedures have to learn values of transfinite sequences of non computable functions in order to extract witnesses from classical proofs.

math.LO↗

arXiv2011

Interactive Learning Based Realizability and 1-Backtracking Games

We prove that interactive learning based classical realizability (introduced by Aschieri and Berardi for first order arithmetic) is sound with respect to Coquand game semantics. In particular, any realizer of an implication-and-negation-free arithmetical formula embodies a winning recursive strategy for the 1-Backtracking version of Tarski games. We also give examples of realizer and winning strategy extraction for some classical proofs. We also sketch some ongoing work about how to extend our notion of realizability in order to obtain completeness with respect to Coquand semantics, when it is restricted to 1-Backtracking games.

cs.LO↗

arXiv2010

Interactive Learning-Based Realizability for Heyting Arithmetic with EM1

We apply to the semantics of Arithmetic the idea of ``finite approximation'' used to provide computational interpretations of Herbrand's Theorem, and we interpret classical proofs as constructive proofs (with constructive rules for $\vee, \exists$) over a suitable structure $\StructureN$ for the language of natural numbers and maps of Gödel's system $\SystemT$. We introduce a new Realizability semantics we call ``Interactive learning-based Realizability'', for Heyting Arithmetic plus $\EM_1$ (Excluded middle axiom restricted to $Σ^0_1$ formulas). Individuals of $\StructureN$ evolve with time, and realizers may ``interact'' with them, by influencing their evolution. We build our semantics over Avigad's fixed point result, but the same semantics may be defined over different constructive interpretations of classical arithmetic (Berardi and de' Liguoro use continuations). Our notion of realizability extends intuitionistic realizability and differs from it only in the atomic case: we interpret atomic realizers as ``learning agents''.

cs.LO↗