516 research outputs found
Linear lambda terms as invariants of rooted trivalent maps
The main aim of the article is to give a simple and conceptual account for
the correspondence (originally described by Bodini, Gardy, and Jacquot) between
-equivalence classes of closed linear lambda terms and isomorphism
classes of rooted trivalent maps on compact oriented surfaces without boundary,
as an instance of a more general correspondence between linear lambda terms
with a context of free variables and rooted trivalent maps with a boundary of
free edges. We begin by recalling a familiar diagrammatic representation for
linear lambda terms, while at the same time explaining how such diagrams may be
read formally as a notation for endomorphisms of a reflexive object in a
symmetric monoidal closed (bi)category. From there, the "easy" direction of the
correspondence is a simple forgetful operation which erases annotations on the
diagram of a linear lambda term to produce a rooted trivalent map. The other
direction views linear lambda terms as complete invariants of their underlying
rooted trivalent maps, reconstructing the missing information through a
Tutte-style topological recurrence on maps with free edges. As an application
in combinatorics, we use this analysis to enumerate bridgeless rooted trivalent
maps as linear lambda terms containing no closed proper subterms, and conclude
by giving a natural reformulation of the Four Color Theorem as a statement
about typing in lambda calculus.Comment: accepted author manuscript, posted six months after publicatio
A correspondence between rooted planar maps and normal planar lambda terms
A rooted planar map is a connected graph embedded in the 2-sphere, with one
edge marked and assigned an orientation. A term of the pure lambda calculus is
said to be linear if every variable is used exactly once, normal if it contains
no beta-redexes, and planar if it is linear and the use of variables moreover
follows a deterministic stack discipline. We begin by showing that the sequence
counting normal planar lambda terms by a natural notion of size coincides with
the sequence (originally computed by Tutte) counting rooted planar maps by
number of edges. Next, we explain how to apply the machinery of string diagrams
to derive a graphical language for normal planar lambda terms, extracted from
the semantics of linear lambda calculus in symmetric monoidal closed categories
equipped with a linear reflexive object or a linear reflexive pair. Finally,
our main result is a size-preserving bijection between rooted planar maps and
normal planar lambda terms, which we establish by explaining how Tutte
decomposition of rooted planar maps (into vertex maps, maps with an isthmic
root, and maps with a non-isthmic root) may be naturally replayed in linear
lambda calculus, as certain surgeries on the string diagrams of normal planar
lambda terms.Comment: Corrected title field in metadat
Refined floor diagrams from higher genera and lambda classes
We show that, after the change of variables , refined floor
diagrams for and Hirzebruch surfaces compute generating series
of higher genus relative Gromov-Witten invariants with insertion of a lambda
class. The proof uses an inductive application of the degeneration formula in
relative Gromov-Witten theory and an explicit result in relative Gromov-Witten
theory of . Combining this result with the similar looking
refined tropical correspondence theorem for log Gromov-Witten invariants, we
obtain some non-trivial relation between relative and log Gromov-Witten
invariants for and Hirzebruch surfaces. We also prove that the
Block-G\"ottsche invariants of and are related by
the Abramovich-Bertram formula.Comment: 44 pages, 8 figures, revised version, exposition greatly improved,
main results unchanged, published in Selecta Mathematic
Quantum Turing Machines Computations and Measurements
Contrary to the classical case, the relation between quantum programming
languages and quantum Turing Machines (QTM) has not being fully investigated.
In particular, there are features of QTMs that have not been exploited, a
notable example being the intrinsic infinite nature of any quantum computation.
In this paper we propose a definition of QTM, which extends and unifies the
notions of Deutsch and Bernstein and Vazirani. In particular, we allow both
arbitrary quantum input, and meaningful superpositions of computations, where
some of them are "terminated" with an "output", while others are not. For some
infinite computations an "output" is obtained as a limit of finite portions of
the computation. We propose a natural and robust observation protocol for our
QTMs, that does not modify the probability of the possible outcomes of the
machines. Finally, we use QTMs to define a class of quantum computable
functions---any such function is a mapping from a general quantum state to a
probability distribution of natural numbers. We expect that our class of
functions, when restricted to classical input-output, will be not different
from the set of the recursive functions.Comment: arXiv admin note: substantial text overlap with arXiv:1504.02817 To
appear on MDPI Applied Sciences, 202
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