288 research outputs found
String matching problems over free partially commutative monoids
AbstractThis paper studies two string matching problems over free partially commutative monoids. We analyze these two problems in detail, and present two efficient polynomial time algorithms for solving them
Frex: dependently-typed algebraic simplification
We present an extensible, mathematically-structured algebraic simplification
library design. We structure the library using universal algebraic concepts: a
free algebra -- fral -- and a free extension -- frex -- of an algebra by a set
of variables. The library's dependently-typed API guarantees simplification
modules, even user-defined ones, are terminating, sound, and complete with
respect to a well-specified class of equations. Completeness offers intangible
benefits in practice -- our main contribution is the novel design. Cleanly
separating between the interface and implementation of simplification modules
provides two new modularity axes. First, simplification modules share thousands
of lines of infrastructure code dealing with term-representation,
pretty-printing, certification, and macros/reflection. Second, new
simplification modules can reuse existing ones. We demonstrate this design by
developing simplification modules for monoid varieties: ordinary, commutative,
and involutive. We implemented this design in the new Idris2 dependently-typed
programming language, and in Agda
A Context-theoretic Framework for Compositionality in Distributional Semantics
Techniques in which words are represented as vectors have proved useful in
many applications in computational linguistics, however there is currently no
general semantic formalism for representing meaning in terms of vectors. We
present a framework for natural language semantics in which words, phrases and
sentences are all represented as vectors, based on a theoretical analysis which
assumes that meaning is determined by context.
In the theoretical analysis, we define a corpus model as a mathematical
abstraction of a text corpus. The meaning of a string of words is assumed to be
a vector representing the contexts in which it occurs in the corpus model.
Based on this assumption, we can show that the vector representations of words
can be considered as elements of an algebra over a field. We note that in
applications of vector spaces to representing meanings of words there is an
underlying lattice structure; we interpret the partial ordering of the lattice
as describing entailment between meanings. We also define the context-theoretic
probability of a string, and, based on this and the lattice structure, a degree
of entailment between strings.
We relate the framework to existing methods of composing vector-based
representations of meaning, and show that our approach generalises many of
these, including vector addition, component-wise multiplication, and the tensor
product.Comment: Submitted to Computational Linguistics on 20th January 2010 for
revie
Non-Deterministic Communication Complexity of Regular Languages
In this thesis, we study the place of regular languages within the
communication complexity setting. In particular, we are interested in the
non-deterministic communication complexity of regular languages.
We show that a regular language has either O(1) or Omega(log n)
non-deterministic complexity. We obtain several linear lower bound results
which cover a wide range of regular languages having linear non-deterministic
complexity. These lower bound results also imply a result in semigroup theory:
we obtain sufficient conditions for not being in the positive variety Pol(Com).
To obtain our results, we use algebraic techniques. In the study of regular
languages, the algebraic point of view pioneered by Eilenberg (\cite{Eil74})
has led to many interesting results. Viewing a semigroup as a computational
device that recognizes languages has proven to be prolific from both semigroup
theory and formal languages perspectives. In this thesis, we provide further
instances of such mutualism.Comment: Master's thesis, 93 page
Concurrent Kleene Algebra with Tests and Branching Automata
We introduce concurrent Kleene algebra with tests (CKAT) as a combination of Kleene algebra with tests (KAT) of Kozen and Smith with concurrent Kleene algebras (CKA), introduced by Hoare, Möller, Struth and Wehrman. CKAT provides a relatively simple algebraic model for reasoning about semantics of concurrent programs. We generalize guarded strings to guarded series-parallel strings , or gsp-strings, to give a concrete language model for CKAT. Combining nondeterministic guarded automata of Kozen with branching automata of Lodaya and Weil one obtains a model for processing gsp-strings in parallel. To ensure that the model satisfies the weak exchange law (x‖y)(z‖w)≤(xz)‖(yw) of CKA, we make use of the subsumption order of Gischer on the gsp-strings. We also define deterministic branching automata and investigate their relation to (nondeterministic) branching automata. To express basic concurrent algorithms, we define concurrent deterministic flowchart schemas and relate them to branching automata and to concurrent Kleene algebras with tests
The conjugacy problem in right-angled Artin groups and their subgroups
29 pages, 7 figuresInternational audienceWe prove that the conjugacy problem in right-angled Artin groups (RAAGs), as well as in a large and natural class of subgroups of RAAGs, can be solved in linear-time. This class of subgroups contains, for instance, all graph braid groups (i.e. fundamental groups of configuration spaces of points in graphs), many hyperbolic groups, and it coincides with the class of fundamental groups of ``special cube complexes'' studied independently by Haglund and Wise
Knapsack in Graph Groups, HNN-Extensions and Amalgamated Products
It is shown that the knapsack problem, which was introduced by Myasnikov et al. for arbitrary finitely generated groups, can be solved in NP for graph groups. This result even holds if the group elements are represented in a compressed form by SLPs, which generalizes the classical NP-completeness result of the integer knapsack problem. We also prove general transfer results: NP-membership of the knapsack problem is passed on to finite extensions, HNN-extensions over finite associated subgroups, and amalgamated products with finite identified subgroups
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