2,012,405 research outputs found
Logical Step-Indexed Logical Relations
Appel and McAllester's "step-indexed" logical relations have proven to be a
simple and effective technique for reasoning about programs in languages with
semantically interesting types, such as general recursive types and general
reference types. However, proofs using step-indexed models typically involve
tedious, error-prone, and proof-obscuring step-index arithmetic, so it is
important to develop clean, high-level, equational proof principles that avoid
mention of step indices. In this paper, we show how to reason about binary
step-indexed logical relations in an abstract and elegant way. Specifically, we
define a logic LSLR, which is inspired by Plotkin and Abadi's logic for
parametricity, but also supports recursively defined relations by means of the
modal "later" operator from Appel, Melli\`es, Richards, and Vouillon's "very
modal model" paper. We encode in LSLR a logical relation for reasoning
relationally about programs in call-by-value System F extended with general
recursive types. Using this logical relation, we derive a set of useful rules
with which we can prove contextual equivalence and approximation results
without counting steps
Logical Dreams
We discuss the past and future of set theory, axiom systems and independence
results. We deal in particular with cardinal arithmetic
Logical-function generator
Apparatus and technique for generating logical functions and circuits have been developed. They provide aid in designing and constructing hardware to generate logic circuits, by defining circuit connections required to generate these functions. With this method, it is possible quickly and automatically to design logic, while eliminating involved and time-consuming mathematical manipulations
Limiting logical pluralism
In this paper I argue that pluralism at the level of logical systems requires a certain monism at the meta-logical level, and so, in a sense, there cannot be pluralism all the way down. The adequate alternative logical systems bottom out in a shared basic meta-logic, and as such, logical pluralism is limited. I argue that the content of this basic meta-logic must include the analogue of logical rules Modus Ponens and Universal Instantiation. I show this through a detailed analysis of the âadoption problemâ, which manifests something special about MP and UI. It appears that MP and UI underwrite the very nature of a logical rule of inference, due to all rules of inference being conditional and universal in their structure. As such, all logical rules presuppose MP and UI, making MP and UI self-governing, basic, unadoptable, and required in the meta-logic for the adequacy of any logical system
Logical Bell Inequalities
Bell inequalities play a central role in the study of quantum non-locality
and entanglement, with many applications in quantum information. Despite the
huge literature on Bell inequalities, it is not easy to find a clear conceptual
answer to what a Bell inequality is, or a clear guiding principle as to how
they may be derived. In this paper, we introduce a notion of logical Bell
inequality which can be used to systematically derive testable inequalities for
a very wide variety of situations. There is a single clear conceptual
principle, based on purely logical consistency conditions, which underlies our
notion of logical Bell inequalities. We show that in a precise sense, all Bell
inequalities can be taken to be of this form. Our approach is very general. It
applies directly to any family of sets of commuting observables. Thus it covers
not only the n-partite scenarios to which Bell inequalities are standardly
applied, but also Kochen-Specker configurations, and many other examples. There
is much current work on experimental tests for contextuality. Our approach
directly yields, in a systematic fashion, testable inequalities for a very
general notion of contextuality.
There has been much work on obtaining proofs of Bell's theorem `without
inequalities' or `without probabilities'. These proofs are seen as being in a
sense more definitive and logically robust than the inequality-based proofs. On
the hand, they lack the fault-tolerant aspect of inequalities. Our approach
reconciles these aspects, and in fact shows how the logical robustness can be
converted into systematic, general derivations of inequalities with provable
violations. Moreover, the kind of strong non-locality or contextuality
exhibited by the GHZ argument or by Kochen-Specker configurations can be shown
to lead to maximal violations of the corresponding logical Bell inequalities.Comment: 12 page
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