19,596 research outputs found
A new coinductive confluence proof for infinitary lambda calculus
We present a new and formal coinductive proof of confluence and normalisation
of B\"ohm reduction in infinitary lambda calculus. The proof is simpler than
previous proofs of this result. The technique of the proof is new, i.e., it is
not merely a coinductive reformulation of any earlier proofs. We formalised the
proof in the Coq proof assistant.Comment: arXiv admin note: text overlap with arXiv:1501.0435
No-Signalling Is Equivalent To Free Choice of Measurements
No-Signalling is a fundamental constraint on the probabilistic predictions
made by physical theories. It is usually justified in terms of the constraints
imposed by special relativity. However, this justification is not as clear-cut
as is usually supposed. We shall give a different perspective on this condition
by showing an equivalence between No-Signalling and Lambda Independence, or
"free choice of measurements", a condition on hidden-variable theories which is
needed to make no-go theorems such as Bell's theorem non-trivial. More
precisely, we shall show that a probability table describing measurement
outcomes is No-Signalling if and only if it can be realized by a
Lambda-Independent hidden-variable theory of a particular canonical form, in
which the hidden variables correspond to non-contextual deterministic
predictions of measurement outcomes. The key proviso which avoids contradiction
with Bell's theorem is that we consider hidden-variable theories with signed
probability measures over the hidden variables - i.e. negative probabilities.
Negative probabilities have often been discussed in the literature on quantum
mechanics. We use a result proved previously in "The Sheaf-theoretic Structure
of Locality and Contextuality" by Abramsky and Brandenburger, which shows that
they give rise to, and indeed characterize, the entire class of No-Signalling
behaviours. In the present paper, we put this result in a broader context,
which reveals the surprising consequence that the No-Signalling condition is
equivalent to the apparently completely different notion of free choice of
measurements.Comment: In Proceedings QPL 2013, arXiv:1412.791
Relational Graph Models at Work
We study the relational graph models that constitute a natural subclass of
relational models of lambda-calculus. We prove that among the lambda-theories
induced by such models there exists a minimal one, and that the corresponding
relational graph model is very natural and easy to construct. We then study
relational graph models that are fully abstract, in the sense that they capture
some observational equivalence between lambda-terms. We focus on the two main
observational equivalences in the lambda-calculus, the theory H+ generated by
taking as observables the beta-normal forms, and H* generated by considering as
observables the head normal forms. On the one hand we introduce a notion of
lambda-K\"onig model and prove that a relational graph model is fully abstract
for H+ if and only if it is extensional and lambda-K\"onig. On the other hand
we show that the dual notion of hyperimmune model, together with
extensionality, captures the full abstraction for H*
- …