6,558 research outputs found
Reasoning about modular datatypes with Mendler induction
In functional programming, datatypes a la carte provide a convenient modular
representation of recursive datatypes, based on their initial algebra
semantics. Unfortunately it is highly challenging to implement this technique
in proof assistants that are based on type theory, like Coq. The reason is that
it involves type definitions, such as those of type-level fixpoint operators,
that are not strictly positive. The known work-around of impredicative
encodings is problematic, insofar as it impedes conventional inductive
reasoning. Weak induction principles can be used instead, but they considerably
complicate proofs.
This paper proposes a novel and simpler technique to reason inductively about
impredicative encodings, based on Mendler-style induction. This technique
involves dispensing with dependent induction, ensuring that datatypes can be
lifted to predicates and relying on relational formulations. A case study on
proving subject reduction for structural operational semantics illustrates that
the approach enables modular proofs, and that these proofs are essentially
similar to conventional ones.Comment: In Proceedings FICS 2015, arXiv:1509.0282
Homotopy Batalin-Vilkovisky algebras
This paper provides an explicit cofibrant resolution of the operad encoding
Batalin-Vilkovisky algebras. Thus it defines the notion of homotopy
Batalin-Vilkovisky algebras with the required homotopy properties.
To define this resolution we extend the theory of Koszul duality to operads
and properads that are defind by quadratic and linear relations. The operad
encoding Batalin-Vilkovisky algebras is shown to be Koszul in this sense. This
allows us to prove a Poincare-Birkhoff-Witt Theorem for such an operad and to
give an explicit small quasi-free resolution for it.
This particular resolution enables us to describe the deformation theory and
homotopy theory of BV-algebras and of homotopy BV-algebras. We show that any
topological conformal field theory carries a homotopy BV-algebra structure
which lifts the BV-algebra structure on homology. The same result is proved for
the singular chain complex of the double loop space of a topological space
endowed with an action of the circle. We also prove the cyclic Deligne
conjecture with this cofibrant resolution of the operad BV. We develop the
general obstruction theory for algebras over the Koszul resolution of a
properad and apply it to extend a conjecture of Lian-Zuckerman, showing that
certain vertex algebras have an explicit homotopy BV-algebra structure.Comment: Last version before publication. To appear in Journal of
Noncommutative Geometry. 57 page
The minimal model for the Batalin-Vilkovisky operad
The purpose of this paper is to explain and to generalize, in a homotopical
way, the result of Barannikov-Kontsevich and Manin which states that the
underlying homology groups of some Batalin-Vilkovisky algebras carry a
Frobenius manifold structure. To this extent, we first make the minimal model
for the operad encoding BV-algebras explicit. Then we prove a homotopy transfer
theorem for the associated notion of homotopy BV-algebra. The final result
provides an extension of the action of the homology of the
Deligne-Mumford-Knudsen moduli space of genus 0 curves on the homology of some
BV-algebras to an action via higher homotopical operations organized by the
cohomology of the open moduli space of genus zero curves. Applications in
Poisson geometry and Lie algebra cohomology and to the Mirror Symmetry
conjecture are given.Comment: New section added containing applications to Poisson geometry, Lie
algebra cohomology and to the Mirror Symmetry conjecture. [36 pages, 4
figures
Monads, partial evaluations, and rewriting
Monads can be interpreted as encoding formal expressions, or formal
operations in the sense of universal algebra. We give a construction which
formalizes the idea of "evaluating an expression partially": for example, "2+3"
can be obtained as a partial evaluation of "2+2+1". This construction can be
given for any monad, and it is linked to the famous bar construction, of which
it gives an operational interpretation: the bar construction induces a
simplicial set, and its 1-cells are partial evaluations.
We study the properties of partial evaluations for general monads. We prove
that whenever the monad is weakly cartesian, partial evaluations can be
composed via the usual Kan filler property of simplicial sets, of which we give
an interpretation in terms of substitution of terms.
In terms of rewritings, partial evaluations give an abstract reduction system
which is reflexive, confluent, and transitive whenever the monad is weakly
cartesian.
For the case of probability monads, partial evaluations correspond to what
probabilists call conditional expectation of random variables.
This manuscript is part of a work in progress on a general rewriting
interpretation of the bar construction.Comment: Originally written for the ACT Adjoint School 2019. To appear in
Proceedings of MFPS 202
The capacity of hybrid quantum memory
The general stable quantum memory unit is a hybrid consisting of a classical
digit with a quantum digit (qudit) assigned to each classical state. The shape
of the memory is the vector of sizes of these qudits, which may differ. We
determine when N copies of a quantum memory A embed in N(1+o(1)) copies of
another quantum memory B. This relationship captures the notion that B is as at
least as useful as A for all purposes in the bulk limit. We show that the
embeddings exist if and only if for all p >= 1, the p-norm of the shape of A
does not exceed the p-norm of the shape of B. The log of the p-norm of the
shape of A can be interpreted as the maximum of S(\rho) + H(\rho)/p (quantum
entropy plus discounted classical entropy) taken over all mixed states \rho on
A. We also establish a noiseless coding theorem that justifies these entropies.
The noiseless coding theorem and the bulk embedding theorem together say that
either A blindly bulk-encodes into B with perfect fidelity, or A admits a state
that does not visibly bulk-encode into B with high fidelity.
In conclusion, the utility of a hybrid quantum memory is determined by its
simultaneous capacity for classical and quantum entropy, which is not a finite
list of numbers, but rather a convex region in the classical-quantum entropy
plane.Comment: 10 pages, 1 figures. Major revision; extra material could have been a
new paper. Has a much better treatment of noiseless coding and a new Holder
inequality for memory squeezing. To appear in IEEE Trans. Inf. Theor
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