4,735 research outputs found
Smart matching
One of the most annoying aspects in the formalization of mathematics is the
need of transforming notions to match a given, existing result. This kind of
transformations, often based on a conspicuous background knowledge in the given
scientific domain (mostly expressed in the form of equalities or isomorphisms),
are usually implicit in the mathematical discourse, and it would be highly
desirable to obtain a similar behavior in interactive provers. The paper
describes the superposition-based implementation of this feature inside the
Matita interactive theorem prover, focusing in particular on the so called
smart application tactic, supporting smart matching between a goal and a given
result.Comment: To appear in The 9th International Conference on Mathematical
Knowledge Management: MKM 201
On the saturation of YAGO
YAGO is an automatically generated ontology out of Wikipedia
and WordNet. It is eventually represented in a proprietary
flat text file format and a core comprises 10 million facts
and formulas. We present a translation of YAGO into the
Bernays-Sch¨onfinkel Horn class with equality. A new
variant of the superposition calculus is sound, complete
and terminating for this class. Together with extended term
indexing data structures the new calculus is implemented in
Spass-YAGO. YAGO can be finitely saturated by Spass-YAGO in
about 1 hour.We have found 49 inconsistencies in the original
generated ontology which we have fixed. Spass-YAGO can then
prove non-trivial conjectures with respect to the resulting
saturated and consistent clause set of about 1.4 GB in less
than one second
Uniform Continuity and Br\'ezis-Lieb Type Splitting for Superposition Operators in Sobolev Space
Using concentration-compactness arguments we prove a variant of the
Brezis-Lieb-Lemma under weaker assumptions on the nonlinearity than known
before. An intermediate result on the uniform continuity of superposition
operators in Sobolev space is of independent interest
Contextual viewpoint to quantum stochastics
We study the role of context, complex of physical conditions, in quantum as
well as classical experiments. It is shown that by taking into account
contextual dependence of experimental probabilities we can derive the quantum
rule for the addition of probabilities of alternatives. Thus we obtain quantum
interference without applying to wave or Hilbert space approach. The Hilbert
space representation of contextual probabilities is obtained as a consequence
of the elementary geometric fact: -theorem. By using another fact from
elementary algebra we obtain complex-amplitude representation of probabilities.
Finally, we found contextual origin of noncommutativity of incompatible
observables
Rewriting Modulo \beta in the \lambda\Pi-Calculus Modulo
The lambda-Pi-calculus Modulo is a variant of the lambda-calculus with
dependent types where beta-conversion is extended with user-defined rewrite
rules. It is an expressive logical framework and has been used to encode logics
and type systems in a shallow way. Basic properties such as subject reduction
or uniqueness of types do not hold in general in the lambda-Pi-calculus Modulo.
However, they hold if the rewrite system generated by the rewrite rules
together with beta-reduction is confluent. But this is too restrictive. To
handle the case where non confluence comes from the interference between the
beta-reduction and rewrite rules with lambda-abstraction on their left-hand
side, we introduce a notion of rewriting modulo beta for the lambda-Pi-calculus
Modulo. We prove that confluence of rewriting modulo beta is enough to ensure
subject reduction and uniqueness of types. We achieve our goal by encoding the
lambda-Pi-calculus Modulo into Higher-Order Rewrite System (HRS). As a
consequence, we also make the confluence results for HRSs available for the
lambda-Pi-calculus Modulo.Comment: In Proceedings LFMTP 2015, arXiv:1507.0759
De-linearizing Linearity: Projective Quantum Axiomatics from Strong Compact Closure
Elaborating on our joint work with Abramsky in quant-ph/0402130 we further
unravel the linear structure of Hilbert spaces into several constituents. Some
prove to be very crucial for particular features of quantum theory while others
obstruct the passage to a formalism which is not saturated with physically
insignificant global phases.
First we show that the bulk of the required linear structure is purely
multiplicative, and arises from the strongly compact closed tensor which,
besides providing a variety of notions such as scalars, trace, unitarity,
self-adjointness and bipartite projectors, also provides Hilbert-Schmidt norm,
Hilbert-Schmidt inner-product, and in particular, the preparation-state
agreement axiom which enables the passage from a formalism of the vector space
kind to a rather projective one, as it was intended in the (in)famous Birkhoff
& von Neumann paper.
Next we consider additive types which distribute over the tensor, from which
measurements can be build, and the correctness proofs of the protocols
discussed in quant-ph/0402130 carry over to the resulting weaker setting. A
full probabilistic calculus is obtained when the trace is moreover linear and
satisfies the \em diagonal axiom, which brings us to a second main result,
characterization of the necessary and sufficient additive structure of a both
qualitatively and quantitatively effective categorical quantum formalism
without redundant global phases. Along the way we show that if in a category a
(additive) monoidal tensor distributes over a strongly compact closed tensor,
then this category is always enriched in commutative monoids.Comment: Essential simplification of the definitions of orthostructure and
ortho-Bornian structure: the key new insights is captured by the definitions
in terms of commutative diagrams on pages 13 and 14, which state that if in a
category a (additive) monoidal tensor distributes over a strongly compact
closed tensor, then this category is always enriched in commutative monoid
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