21 research outputs found
Weak Cat-Operads
An operad (this paper deals with non-symmetric operads)may be conceived as a
partial algebra with a family of insertion operations, Gerstenhaber's circle-i
products, which satisfy two kinds of associativity, one of them involving
commutativity. A Cat-operad is an operad enriched over the category Cat of
small categories, as a 2-category with small hom-categories is a category
enriched over Cat. The notion of weak Cat-operad is to the notion of Cat-operad
what the notion of bicategory is to the notion of 2-category. The equations of
operads like associativity of insertions are replaced by isomorphisms in a
category. The goal of this paper is to formulate conditions concerning these
isomorphisms that ensure coherence, in the sense that all diagrams of canonical
arrows commute. This is the sense in which the notions of monoidal category and
bicategory are coherent. The coherence proof in the paper is much simplified by
indexing the insertion operations in a context-independent way, and not in the
usual manner. This proof, which is in the style of term rewriting, involves an
argument with normal forms that generalizes what is established with the
completeness proof for the standard presentation of symmetric groups. This
generalization may be of an independent interest, and related to matters other
than those studied in this paper. Some of the coherence conditions for weak
Cat-operads lead to the hemiassociahedron, which is a polyhedron related to,
but different from, the three-dimensional associahedron and permutohedron.Comment: 38 pages, version prepared for publication in Logical Methods in
Computer Science, the authors' last version is v
Semantics of Higher-Order Recursion Schemes
Higher-order recursion schemes are recursive equations defining new
operations from given ones called "terminals". Every such recursion scheme is
proved to have a least interpreted semantics in every Scott's model of
\lambda-calculus in which the terminals are interpreted as continuous
operations. For the uninterpreted semantics based on infinite \lambda-terms we
follow the idea of Fiore, Plotkin and Turi and work in the category of sets in
context, which are presheaves on the category of finite sets. Fiore et al
showed how to capture the type of variable binding in \lambda-calculus by an
endofunctor H\lambda and they explained simultaneous substitution of
\lambda-terms by proving that the presheaf of \lambda-terms is an initial
H\lambda-monoid. Here we work with the presheaf of rational infinite
\lambda-terms and prove that this is an initial iterative H\lambda-monoid. We
conclude that every guarded higher-order recursion scheme has a unique
uninterpreted solution in this monoid
A Model of Cooperative Threads
We develop a model of concurrent imperative programming with threads. We
focus on a small imperative language with cooperative threads which execute
without interruption until they terminate or explicitly yield control. We
define and study a trace-based denotational semantics for this language; this
semantics is fully abstract but mathematically elementary. We also give an
equational theory for the computational effects that underlie the language,
including thread spawning. We then analyze threads in terms of the free algebra
monad for this theory.Comment: 39 pages, 5 figure
B\"uchi Complementation and Size-Change Termination
We compare tools for complementing nondeterministic B\"uchi automata with a
recent termination-analysis algorithm. Complementation of B\"uchi automata is a
key step in program verification. Early constructions using a Ramsey-based
argument have been supplanted by rank-based constructions with exponentially
better bounds. In 2001 Lee et al. presented the size-change termination (SCT)
problem, along with both a reduction to B\"uchi automata and a Ramsey-based
algorithm. The Ramsey-based algorithm was presented as a more practical
alternative to the automata-theoretic approach, but strongly resembles the
initial complementation constructions for B\"uchi automata. We prove that the
SCT algorithm is a specialized realization of the Ramsey-based complementation
construction. To do so, we extend the Ramsey-based complementation construction
to provide a containment-testing algorithm. Surprisingly, empirical analysis
suggests that despite the massive gap in worst-case complexity, Ramsey-based
approaches are superior over the domain of SCT problems. Upon further analysis
we discover an interesting property of the problem space that both explains
this result and provides a chance to improve rank-based tools. With these
improvements, we show that theoretical gains in efficiency of the rank-based
approach are mirrored in empirical performance