2,757 research outputs found
Church-Rosser Systems, Codes with Bounded Synchronization Delay and Local Rees Extensions
What is the common link, if there is any, between Church-Rosser systems,
prefix codes with bounded synchronization delay, and local Rees extensions? The
first obvious answer is that each of these notions relates to topics of
interest for WORDS: Church-Rosser systems are certain rewriting systems over
words, codes are given by sets of words which form a basis of a free submonoid
in the free monoid of all words (over a given alphabet) and local Rees
extensions provide structural insight into regular languages over words. So, it
seems to be a legitimate title for an extended abstract presented at the
conference WORDS 2017. However, this work is more ambitious, it outlines some
less obvious but much more interesting link between these topics. This link is
based on a structure theory of finite monoids with varieties of groups and the
concept of local divisors playing a prominent role. Parts of this work appeared
in a similar form in conference proceedings where proofs and further material
can be found.Comment: Extended abstract of an invited talk given at WORDS 201
A Survey on the Local Divisor Technique
Local divisors allow a powerful induction scheme on the size of a monoid. We
survey this technique by giving several examples of this proof method. These
applications include linear temporal logic, rational expressions with Kleene
stars restricted to prefix codes with bounded synchronization delay,
Church-Rosser congruential languages, and Simon's Factorization Forest Theorem.
We also introduce the notion of localizable language class as a new abstract
concept which unifies some of the proofs for the results above
A survey on the local divisor technique
© 2015 Elsevier B.V. Local divisors allow a powerful induction scheme on the size of a monoid. We survey this technique by giving several examples of this proof method. These applications include linear temporal logic, rational expressions with Kleene stars restricted to prefix codes with bounded synchronization delay, Church-Rosser congruential languages, and Simon's Factorization Forest Theorem. We also introduce the notion of a localizable language class as a new abstract concept which unifies some of the proofs for the results above
Cyclic Datatypes modulo Bisimulation based on Second-Order Algebraic Theories
Cyclic data structures, such as cyclic lists, in functional programming are
tricky to handle because of their cyclicity. This paper presents an
investigation of categorical, algebraic, and computational foundations of
cyclic datatypes. Our framework of cyclic datatypes is based on second-order
algebraic theories of Fiore et al., which give a uniform setting for syntax,
types, and computation rules for describing and reasoning about cyclic
datatypes. We extract the "fold" computation rules from the categorical
semantics based on iteration categories of Bloom and Esik. Thereby, the rules
are correct by construction. We prove strong normalisation using the General
Schema criterion for second-order computation rules. Rather than the fixed
point law, we particularly choose Bekic law for computation, which is a key to
obtaining strong normalisation. We also prove the property of "Church-Rosser
modulo bisimulation" for the computation rules. Combining these results, we
have a remarkable decidability result of the equational theory of cyclic data
and fold.Comment: 38 page
Continuity, Discontinuity and Dynamics in Mathematics & Economics - Reconsidering Rosser's Visions
Barkley Rosser has been a pioneer in arguing the case for the mathematics of discontinuity, broadly conceived, to be placed at the foundations of modelling economic dynamics. In this paper we reconsider this vision from the broad perspective of a variety of different kinds of mathematics and suggest a broadening of Rosser’s methodology to the study of economic dynamicsContinuity, Discontinuity, Economic Dynamics, Relaxation Oscillations
Godel's Incompleteness Phenomenon - Computationally
We argue that Godel's completeness theorem is equivalent to completability of
consistent theories, and Godel's incompleteness theorem is equivalent to the
fact that this completion is not constructive, in the sense that there are some
consistent and recursively enumerable theories which cannot be extended to any
complete and consistent and recursively enumerable theory. Though any
consistent and decidable theory can be extended to a complete and consistent
and decidable theory. Thus deduction and consistency are not decidable in
logic, and an analogue of Rice's Theorem holds for recursively enumerable
theories: all the non-trivial properties of such theories are undecidable
The Glasgow Parallel Reduction Machine: Programming Shared-memory Many-core Systems using Parallel Task Composition
We present the Glasgow Parallel Reduction Machine (GPRM), a novel, flexible
framework for parallel task-composition based many-core programming. We allow
the programmer to structure programs into task code, written as C++ classes,
and communication code, written in a restricted subset of C++ with functional
semantics and parallel evaluation. In this paper we discuss the GPRM, the
virtual machine framework that enables the parallel task composition approach.
We focus the discussion on GPIR, the functional language used as the
intermediate representation of the bytecode running on the GPRM. Using examples
in this language we show the flexibility and power of our task composition
framework. We demonstrate the potential using an implementation of a merge sort
algorithm on a 64-core Tilera processor, as well as on a conventional Intel
quad-core processor and an AMD 48-core processor system. We also compare our
framework with OpenMP tasks in a parallel pointer chasing algorithm running on
the Tilera processor. Our results show that the GPRM programs outperform the
corresponding OpenMP codes on all test platforms, and can greatly facilitate
writing of parallel programs, in particular non-data parallel algorithms such
as reductions.Comment: In Proceedings PLACES 2013, arXiv:1312.221
Algebraic and Combinatorial Methods in Computational Complexity
At its core, much of Computational Complexity is concerned with combinatorial objects and structures. But it has often proven true that the best way to prove things about these combinatorial objects is by establishing a connection (perhaps approximate) to a more well-behaved algebraic setting. Indeed, many of the deepest and most powerful results in Computational Complexity rely on algebraic proof techniques. The PCP characterization of NP and the Agrawal-Kayal-Saxena polynomial-time primality test are two prominent examples. Recently, there have been some works going in the opposite direction, giving alternative combinatorial proofs for results that were originally proved algebraically. These alternative proofs can yield important improvements because they are closer to the underlying problems and avoid the losses in passing to the algebraic setting. A prominent example is Dinur's proof of the PCP Theorem via gap amplification which yielded short PCPs with only a polylogarithmic length blowup (which had been the focus of significant research effort up to that point). We see here (and in a number of recent works) an exciting interplay between algebraic and combinatorial techniques. This seminar aims to capitalize on recent progress and bring together researchers who are using a diverse array of algebraic and combinatorial methods in a variety of settings
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