25,762 research outputs found
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
On the Number of Variables in Special Classes of Random Lambda-Terms
We investigate the number of variables in two special subclasses of lambda-terms that are restricted by a bound of the number of abstractions between a variable and its binding lambda, and by a bound of the nesting levels of abstractions, respectively. These restrictions are on the one hand very natural from a practical point of view, and on the other hand they simplify the counting problem compared to that of unrestricted lambda-terms in such a way that the common methods of analytic combinatorics are applicable.
We will show that the total number of variables is asymptotically normally distributed for both subclasses of lambda-terms with mean and variance asymptotically equal to C_1 n and C_2 n, respectively, where the constants C_1 and C_2 depend on the bound that has been imposed. So far we just derived closed formulas for the constants in case of the class of lambda-terms with a bounded number of abstractions between each variable and its binding lambda. However, for the other class of lambda-terms that we consider, namely lambda-terms with a bounded number of nesting levels of abstractions, we investigate the number of variables in the different abstraction levels and thereby exhibit very interesting results concerning the distribution of the variables within those lambda-terms
Composition operators and generalized primes
We study composition operators on the Hardy space of
Dirichlet series with square summable coefficients. Our main result is a
necessary condition, in terms of a Nevanlinna-type counting function, for a
certain class of composition operators to be compact on . To do
that we extend our notions to a Hardy space of
generalized Dirichlet series, induced in a natural way by a sequence of
Beurling's primes.Comment: This paper has been accepted for publication in Proceedings of the
AM
On the number of lambda terms with prescribed size of their De Bruijn representation
John Tromp introduced the so-called 'binary lambda calculus' as a way to
encode lambda terms in terms of binary words. Later, Grygiel and Lescanne
conjectured that the number of binary lambda terms with free indices and of
size (encoded as binary words of length ) is for
. We generalize the proposed notion of size and
show that for several classes of lambda terms, including binary lambda terms
with free indices, the number of terms of size is with some class dependent constant , which in particular
disproves the above mentioned conjecture. A way to obtain lower and upper
bounds for the constant near the leading term is presented and numerical
results for a few previously introduced classes of lambda terms are given
Boltzmann samplers for random generation of lambda terms
Randomly generating structured objects is important in testing and optimizing
functional programs, whereas generating random -terms is more specifically
needed for testing and optimizing compilers. For that a tool called QuickCheck
has been proposed, but in this tool the control of the random generation is
left to the programmer. Ten years ago, a method called Boltzmann samplers has
been proposed to generate combinatorial structures. In this paper, we show how
Boltzmann samplers can be developed to generate lambda-terms, but also other
data structures like trees. These samplers rely on a critical value which
parameters the main random selector and which is exhibited here with
explanations on how it is computed. Haskell programs are proposed to show how
samplers are actually implemented
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
On Role Logic
We present role logic, a notation for describing properties of relational
structures in shape analysis, databases, and knowledge bases. We construct role
logic using the ideas of de Bruijn's notation for lambda calculus, an encoding
of first-order logic in lambda calculus, and a simple rule for implicit
arguments of unary and binary predicates. The unrestricted version of role
logic has the expressive power of first-order logic with transitive closure.
Using a syntactic restriction on role logic formulas, we identify a natural
fragment RL^2 of role logic. We show that the RL^2 fragment has the same
expressive power as two-variable logic with counting C^2 and is therefore
decidable. We present a translation of an imperative language into the
decidable fragment RL^2, which allows compositional verification of programs
that manipulate relational structures. In addition, we show how RL^2 encodes
boolean shape analysis constraints and an expressive description logic.Comment: 20 pages. Our later SAS 2004 result builds on this wor
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