1,452 research outputs found
Polynomial Path Orders
This paper is concerned with the complexity analysis of constructor term
rewrite systems and its ramification in implicit computational complexity. We
introduce a path order with multiset status, the polynomial path order POP*,
that is applicable in two related, but distinct contexts. On the one hand POP*
induces polynomial innermost runtime complexity and hence may serve as a
syntactic, and fully automatable, method to analyse the innermost runtime
complexity of term rewrite systems. On the other hand POP* provides an
order-theoretic characterisation of the polytime computable functions: the
polytime computable functions are exactly the functions computable by an
orthogonal constructor TRS compatible with POP*.Comment: LMCS version. This article supersedes arXiv:1209.379
Polynomial Path Orders: A Maximal Model
This paper is concerned with the automated complexity analysis of term
rewrite systems (TRSs for short) and the ramification of these in implicit
computational complexity theory (ICC for short). We introduce a novel path
order with multiset status, the polynomial path order POP*. Essentially relying
on the principle of predicative recursion as proposed by Bellantoni and Cook,
its distinct feature is the tight control of resources on compatible TRSs: The
(innermost) runtime complexity of compatible TRSs is polynomially bounded. We
have implemented the technique, as underpinned by our experimental evidence our
approach to the automated runtime complexity analysis is not only feasible, but
compared to existing methods incredibly fast. As an application in the context
of ICC we provide an order-theoretic characterisation of the polytime
computable functions. To be precise, the polytime computable functions are
exactly the functions computable by an orthogonal constructor TRS compatible
with POP*
Definitions by Rewriting in the Calculus of Constructions
The main novelty of this paper is to consider an extension of the Calculus of
Constructions where predicates can be defined with a general form of rewrite
rules. We prove the strong normalization of the reduction relation generated by
the beta-rule and the user-defined rules under some general syntactic
conditions including confluence. As examples, we show that two important
systems satisfy these conditions: a sub-system of the Calculus of Inductive
Constructions which is the basis of the proof assistant Coq, and the Natural
Deduction Modulo a large class of equational theories.Comment: Best student paper (Kleene Award
On Sharing, Memoization, and Polynomial Time (Long Version)
We study how the adoption of an evaluation mechanism with sharing and
memoization impacts the class of functions which can be computed in polynomial
time. We first show how a natural cost model in which lookup for an already
computed value has no cost is indeed invariant. As a corollary, we then prove
that the most general notion of ramified recurrence is sound for polynomial
time, this way settling an open problem in implicit computational complexity
12th International Workshop on Termination (WST 2012) : WST 2012, February 19–23, 2012, Obergurgl, Austria / ed. by Georg Moser
This volume contains the proceedings of the 12th International Workshop on Termination (WST 2012), to be held February 19–23, 2012 in Obergurgl, Austria. The goal of the Workshop on Termination is to be a venue for presentation and discussion of all topics in and around termination. In this way, the workshop tries to bridge the gaps between different communities interested and active in research in and around termination. The 12th International Workshop on Termination in Obergurgl continues the successful workshops held in St. Andrews (1993), La Bresse (1995), Ede (1997), Dagstuhl (1999), Utrecht (2001), Valencia (2003), Aachen (2004), Seattle (2006), Paris (2007), Leipzig (2009), and Edinburgh (2010). The 12th International Workshop on Termination did welcome contributions on all aspects of termination and complexity analysis. Contributions from the imperative, constraint, functional, and logic programming communities, and papers investigating applications of complexity or termination (for example in program transformation or theorem proving) were particularly welcome. We did receive 18 submissions which all were accepted. Each paper was assigned two reviewers. In addition to these 18 contributed talks, WST 2012, hosts three invited talks by Alexander Krauss, Martin Hofmann, and Fausto Spoto
A combinatorial approach to scattering diagrams
Scattering diagrams arose in the context of mirror symmetry, but a special
class of scattering diagrams (the cluster scattering diagrams) were recently
developed to prove key structural results on cluster algebras. We use the
connection to cluster algebras to calculate the function attached to the
limiting wall of a rank-2 cluster scattering diagram of affine type. In the
skew-symmetric rank-2 affine case, this recovers a formula due to Reineke. In
the same case, we show that the generating function for signed Narayana numbers
appears in a role analogous to a cluster variable. In acyclic finite type, we
construct cluster scattering diagrams of acyclic finite type from Cambrian fans
and sortable elements, with a simple direct proof.Comment: This is the second half of arXiv:1712.06968, which was originally
titled "Scattering diagrams and scattering fans". The contents of this paper
will be removed from arXiv:1712.06968, which will be re-titled "Scattering
fans." Version 2: Minor expository changes. (We thank some anonymous referees
for helpful comments.
Distributed Coupled Multi-Agent Stochastic Optimization
This work develops effective distributed strategies for the solution of
constrained multi-agent stochastic optimization problems with coupled
parameters across the agents. In this formulation, each agent is influenced by
only a subset of the entries of a global parameter vector or model, and is
subject to convex constraints that are only known locally. Problems of this
type arise in several applications, most notably in disease propagation models,
minimum-cost flow problems, distributed control formulations, and distributed
power system monitoring. This work focuses on stochastic settings, where a
stochastic risk function is associated with each agent and the objective is to
seek the minimizer of the aggregate sum of all risks subject to a set of
constraints. Agents are not aware of the statistical distribution of the data
and, therefore, can only rely on stochastic approximations in their learning
strategies. We derive an effective distributed learning strategy that is able
to track drifts in the underlying parameter model. A detailed performance and
stability analysis is carried out showing that the resulting coupled diffusion
strategy converges at a linear rate to an neighborhood of the true
penalized optimizer
Inductive types in the Calculus of Algebraic Constructions
In a previous work, we proved that an important part of the Calculus of
Inductive Constructions (CIC), the basis of the Coq proof assistant, can be
seen as a Calculus of Algebraic Constructions (CAC), an extension of the
Calculus of Constructions with functions and predicates defined by higher-order
rewrite rules. In this paper, we prove that almost all CIC can be seen as a
CAC, and that it can be further extended with non-strictly positive types and
inductive-recursive types together with non-free constructors and
pattern-matching on defined symbols.Comment: Journal version of TLCA'0
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