1,155 research outputs found
A Direct Version of Veldman's Proof of Open Induction on Cantor Space via Delimited Control Operators
First, we reconstruct Wim Veldman's result that Open Induction on Cantor
space can be derived from Double-negation Shift and Markov's Principle. In
doing this, we notice that one has to use a countable choice axiom in the proof
and that Markov's Principle is replaceable by slightly strengthening the
Double-negation Shift schema. We show that this strengthened version of
Double-negation Shift can nonetheless be derived in a constructive intermediate
logic based on delimited control operators, extended with axioms for
higher-type Heyting Arithmetic. We formalize the argument and thus obtain a
proof term that directly derives Open Induction on Cantor space by the shift
and reset delimited control operators of Danvy and Filinski
Perspectives for proof unwinding by programming languages techniques
In this chapter, we propose some future directions of work, potentially
beneficial to Mathematics and its foundations, based on the recent import of
methodology from the theory of programming languages into proof theory. This
scientific essay, written for the audience of proof theorists as well as the
working mathematician, is not a survey of the field, but rather a personal view
of the author who hopes that it may inspire future and fellow researchers
An interpretation of the Sigma-2 fragment of classical Analysis in System T
We show that it is possible to define a realizability interpretation for the
-fragment of classical Analysis using G\"odel's System T only. This
supplements a previous result of Schwichtenberg regarding bar recursion at
types 0 and 1 by showing how to avoid using bar recursion altogether. Our
result is proved via a conservative extension of System T with an operator for
composable continuations from the theory of programming languages due to Danvy
and Filinski. The fragment of Analysis is therefore essentially constructive,
even in presence of the full Axiom of Choice schema: Weak Church's Rule holds
of it in spite of the fact that it is strong enough to refute the formal
arithmetical version of Church's Thesis
Delimited control operators prove Double-negation Shift
We propose an extension of minimal intuitionistic predicate logic, based on
delimited control operators, that can derive the predicate-logic version of the
Double-negation Shift schema, while preserving the disjunction and existence
properties
Answer-Type Modification without Tears: Prompt-Passing Style Translation for Typed Delimited-Control Operators
The salient feature of delimited-control operators is their ability to modify
answer types during computation. The feature, answer-type modification (ATM for
short), allows one to express various interesting programs such as typed printf
compactly and nicely, while it makes it difficult to embed these operators in
standard functional languages.
In this paper, we present a typed translation of delimited-control operators
shift and reset with ATM into a familiar language with multi-prompt shift and
reset without ATM, which lets us use ATM in standard languages without
modifying the type system. Our translation generalizes Kiselyov's direct-style
implementation of typed printf, which uses two prompts to emulate the
modification of answer types, and passes them during computation. We prove that
our translation preserves typing. As the naive prompt-passing style translation
generates and passes many prompts even for pure terms, we show an optimized
translation that generate prompts only when needed, which is also
type-preserving. Finally, we give an implementation in the tagless-final style
which respects typing by construction.Comment: In Proceedings WoC 2015, arXiv:1606.0583
A Type-Theoretic Foundation of Delimited Continuations
International audienceThere is a correspondence between classical logic and programming language calculi with first-class continuations. With the addition of control delimiters, the continuations become composable and the calculi become more expressive. We present a fine-grained analysis of control delimiters and formalise that their addition corresponds to the addition of a single dynamically-scoped variable modelling the special top-level continuation. From a type perspective, the dynamically-scoped variable requires effect annotations. In the presence of control, the dynamically-scoped variable can be interpreted in a purely functional way by applying a store-passing style. At the type level, the effect annotations are mapped within standard classical logic extended with the dual of implication, namely subtraction. A continuation-passing-style transformation of lambda-calculus with control and subtraction is defined. Combining the translations provides a decomposition of standard CPS transformations for delimited continuations. Incidentally, we also give a direct normalisation proof of the simply-typed lambda-calculus with control and subtraction
An Intuitionistic Formula Hierarchy Based on High-School Identities
We revisit the notion of intuitionistic equivalence and formal proof
representations by adopting the view of formulas as exponential polynomials.
After observing that most of the invertible proof rules of intuitionistic
(minimal) propositional sequent calculi are formula (i.e. sequent) isomorphisms
corresponding to the high-school identities, we show that one can obtain a more
compact variant of a proof system, consisting of non-invertible proof rules
only, and where the invertible proof rules have been replaced by a formula
normalisation procedure.
Moreover, for certain proof systems such as the G4ip sequent calculus of
Vorob'ev, Hudelmaier, and Dyckhoff, it is even possible to see all of the
non-invertible proof rules as strict inequalities between exponential
polynomials; a careful combinatorial treatment is given in order to establish
this fact.
Finally, we extend the exponential polynomial analogy to the first-order
quantifiers, showing that it gives rise to an intuitionistic hierarchy of
formulas, resembling the classical arithmetical hierarchy, and the first one
that classifies formulas while preserving isomorphism
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