2,844 research outputs found
Session Types in Abelian Logic
There was a PhD student who says "I found a pair of wooden shoes. I put a
coin in the left and a key in the right. Next morning, I found those objects in
the opposite shoes." We do not claim existence of such shoes, but propose a
similar programming abstraction in the context of typed lambda calculi. The
result, which we call the Amida calculus, extends Abramsky's linear lambda
calculus LF and characterizes Abelian logic.Comment: In Proceedings PLACES 2013, arXiv:1312.221
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
Analytical learning and term-rewriting systems
Analytical learning is a set of machine learning techniques for revising the representation of a theory based on a small set of examples of that theory. When the representation of the theory is correct and complete but perhaps inefficient, an important objective of such analysis is to improve the computational efficiency of the representation. Several algorithms with this purpose have been suggested, most of which are closely tied to a first order logical language and are variants of goal regression, such as the familiar explanation based generalization (EBG) procedure. But because predicate calculus is a poor representation for some domains, these learning algorithms are extended to apply to other computational models. It is shown that the goal regression technique applies to a large family of programming languages, all based on a kind of term rewriting system. Included in this family are three language families of importance to artificial intelligence: logic programming, such as Prolog; lambda calculus, such as LISP; and combinatorial based languages, such as FP. A new analytical learning algorithm, AL-2, is exhibited that learns from success but is otherwise quite different from EBG. These results suggest that term rewriting systems are a good framework for analytical learning research in general, and that further research should be directed toward developing new techniques
The Grail theorem prover: Type theory for syntax and semantics
As the name suggests, type-logical grammars are a grammar formalism based on
logic and type theory. From the prespective of grammar design, type-logical
grammars develop the syntactic and semantic aspects of linguistic phenomena
hand-in-hand, letting the desired semantics of an expression inform the
syntactic type and vice versa. Prototypical examples of the successful
application of type-logical grammars to the syntax-semantics interface include
coordination, quantifier scope and extraction.This chapter describes the Grail
theorem prover, a series of tools for designing and testing grammars in various
modern type-logical grammars which functions as a tool . All tools described in
this chapter are freely available
Physics, Topology, Logic and Computation: A Rosetta Stone
In physics, Feynman diagrams are used to reason about quantum processes. In
the 1980s, it became clear that underlying these diagrams is a powerful analogy
between quantum physics and topology: namely, a linear operator behaves very
much like a "cobordism". Similar diagrams can be used to reason about logic,
where they represent proofs, and computation, where they represent programs.
With the rise of interest in quantum cryptography and quantum computation, it
became clear that there is extensive network of analogies between physics,
topology, logic and computation. In this expository paper, we make some of
these analogies precise using the concept of "closed symmetric monoidal
category". We assume no prior knowledge of category theory, proof theory or
computer science.Comment: 73 pages, 8 encapsulated postscript figure
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