10 research outputs found
Toward the interpretation of non-constructive reasoning as non-monotonic learning
AbstractWe study an abstract representation of the learning process, which we call learning sequence, aiming at a constructive interpretation of classical logical proofs, that we see as learning strategies, coming from Coquand’s game theoretic interpretation of classical logic. Inspired by Gold’s notion of limiting recursion and by the Limit-Computable Mathematics by Hayashi, we investigate the idea of learning in the limit in the general case, where both guess retraction and resumption are allowed. The main contribution is the characterization of the limits of non-monotonic learning sequences in terms of the extension relation between guesses
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
) over a suitable structure \StructureN for the language of
natural numbers and maps of G\"odel'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
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''
Knowledge Spaces and the Completeness of Learning Strategies
We propose a theory of learning aimed to formalize some ideas underlying
Coquand's game semantics and Krivine's realizability of classical logic. We
introduce a notion of knowledge state together with a new topology, capturing
finite positive and negative information that guides a learning strategy. We
use a leading example to illustrate how non-constructive proofs lead to
continuous and effective learning strategies over knowledge spaces, and prove
that our learning semantics is sound and complete w.r.t. classical truth, as it
is the case for Coquand's and Krivine's approaches
Proofs as stateful programs: A first-order logic with abstract Hoare triples, and an interpretation into an imperative language
We introduce an extension of first-order logic that comes equipped with
additional predicates for reasoning about an abstract state. Sequents in the
logic comprise a main formula together with pre- and postconditions in the
style of Hoare logic, and the axioms and rules of the logic ensure that the
assertions about the state compose in the correct way. The main result of the
paper is a realizability interpretation of our logic that extracts programs
into a mixed functional/imperative language. All programs expressible in this
language act on the state in a sequential manner, and we make this intuition
precise by interpreting them in a semantic metatheory using the state monad.
Our basic framework is very general, and our intention is that it can be
instantiated and extended in a variety of different ways. We outline in detail
one such extension: A monadic version of Heyting arithmetic with a wellfounded
while rule, and conclude by outlining several other directions for future work.Comment: 29 page
Learning, realizability and games in classical arithmetic
PhDAbstract. 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 modi ed realizability to a classical fragment of rst order
Arithmetic, Heyting Arithmetic plus EM1 (Excluded middle axiom restricted to 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, thanks to their ability of perpetually questioning, testing and
extending their knowledge. Remarkably, that capability is entirely due to classical principles
when they are applied on top of intuitionistic logic.
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 G odel that provably total functions of PA can be represented
in G odel'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 trans nite case. Trans-
nite update procedures have to learn values of trans nite sequences of non computable
functions in order to extract witnesses from classical proofs