5,057 research outputs found
Almost Every Simply Typed Lambda-Term Has a Long Beta-Reduction Sequence
It is well known that the length of a beta-reduction sequence of a simply
typed lambda-term of order k can be huge; it is as large as k-fold exponential
in the size of the lambda-term in the worst case. We consider the following
relevant question about quantitative properties, instead of the worst case: how
many simply typed lambda-terms have very long reduction sequences? We provide a
partial answer to this question, by showing that asymptotically almost every
simply typed lambda-term of order k has a reduction sequence as long as
(k-1)-fold exponential in the term size, under the assumption that the arity of
functions and the number of variables that may occur in every subterm are
bounded above by a constant. To prove it, we have extended the infinite monkey
theorem for strings to a parametrized one for regular tree languages, which may
be of independent interest. The work has been motivated by quantitative
analysis of the complexity of higher-order model checking
Formal Languages in Dynamical Systems
We treat here the interrelation between formal languages and those dynamical
systems that can be described by cellular automata (CA). There is a well-known
injective map which identifies any CA-invariant subshift with a central formal
language. However, in the special case of a symbolic dynamics, i.e. where the
CA is just the shift map, one gets a stronger result: the identification map
can be extended to a functor between the categories of symbolic dynamics and
formal languages. This functor additionally maps topological conjugacies
between subshifts to empty-string-limited generalized sequential machines
between languages. If the periodic points form a dense set, a case which arises
in a commonly used notion of chaotic dynamics, then an even more natural map to
assign a formal language to a subshift is offered. This map extends to a
functor, too. The Chomsky hierarchy measuring the complexity of formal
languages can be transferred via either of these functors from formal languages
to symbolic dynamics and proves to be a conjugacy invariant there. In this way
it acquires a dynamical meaning. After reviewing some results of the complexity
of CA-invariant subshifts, special attention is given to a new kind of
invariant subshift: the trapped set, which originates from the theory of
chaotic scattering and for which one can study complexity transitions.Comment: 23 pages, LaTe
Evaluating Datalog via Tree Automata and Cycluits
We investigate parameterizations of both database instances and queries that
make query evaluation fixed-parameter tractable in combined complexity. We show
that clique-frontier-guarded Datalog with stratified negation (CFG-Datalog)
enjoys bilinear-time evaluation on structures of bounded treewidth for programs
of bounded rule size. Such programs capture in particular conjunctive queries
with simplicial decompositions of bounded width, guarded negation fragment
queries of bounded CQ-rank, or two-way regular path queries. Our result is
shown by translating to alternating two-way automata, whose semantics is
defined via cyclic provenance circuits (cycluits) that can be tractably
evaluated.Comment: 56 pages, 63 references. Journal version of "Combined Tractability of
Query Evaluation via Tree Automata and Cycluits (Extended Version)" at
arXiv:1612.04203. Up to the stylesheet, page/environment numbering, and
possible minor publisher-induced changes, this is the exact content of the
journal paper that will appear in Theory of Computing Systems. Update wrt
version 1: latest reviewer feedbac
Maude: specification and programming in rewriting logic
Maude is a high-level language and a high-performance system supporting executable specification and declarative programming in rewriting logic. Since rewriting logic contains equational logic, Maude also supports equational specification and programming in its sublanguage of functional modules and theories. The underlying equational logic chosen for Maude is membership equational logic, that has sorts, subsorts, operator overloading, and partiality definable by membership and equality conditions. Rewriting logic is reflective, in the sense of being able to express its own metalevel at the object level. Reflection is systematically exploited in Maude endowing the language with powerful metaprogramming capabilities, including both user-definable module operations and declarative strategies to guide the deduction process. This paper explains and illustrates with examples the main concepts of Maude's language design, including its underlying logic, functional, system and object-oriented modules, as well as parameterized modules, theories, and views. We also explain how Maude supports reflection, metaprogramming and internal strategies. The paper outlines the principles underlying the Maude system implementation, including its semicompilation techniques. We conclude with some remarks about applications, work on a formal environment for Maude, and a mobile language extension of Maude
Software Engineering and Complexity in Effective Algebraic Geometry
We introduce the notion of a robust parameterized arithmetic circuit for the
evaluation of algebraic families of multivariate polynomials. Based on this
notion, we present a computation model, adapted to Scientific Computing, which
captures all known branching parsimonious symbolic algorithms in effective
Algebraic Geometry. We justify this model by arguments from Software
Engineering. Finally we exhibit a class of simple elimination problems of
effective Algebraic Geometry which require exponential time to be solved by
branching parsimonious algorithms of our computation model.Comment: 70 pages. arXiv admin note: substantial text overlap with
arXiv:1201.434
Dagstuhl Reports : Volume 1, Issue 2, February 2011
Online Privacy: Towards Informational Self-Determination on the Internet (Dagstuhl Perspectives Workshop 11061) : Simone Fischer-Hübner, Chris Hoofnagle, Kai Rannenberg, Michael Waidner, Ioannis Krontiris and Michael Marhöfer Self-Repairing Programs (Dagstuhl Seminar 11062) : Mauro Pezzé, Martin C. Rinard, Westley Weimer and Andreas Zeller Theory and Applications of Graph Searching Problems (Dagstuhl Seminar 11071) : Fedor V. Fomin, Pierre Fraigniaud, Stephan Kreutzer and Dimitrios M. Thilikos Combinatorial and Algorithmic Aspects of Sequence Processing (Dagstuhl Seminar 11081) : Maxime Crochemore, Lila Kari, Mehryar Mohri and Dirk Nowotka Packing and Scheduling Algorithms for Information and Communication Services (Dagstuhl Seminar 11091) Klaus Jansen, Claire Mathieu, Hadas Shachnai and Neal E. Youn
Order-Related Problems Parameterized by Width
In the main body of this thesis, we study two different order theoretic problems. The first problem, called Completion of an Ordering, asks to extend a given finite partial order to a complete linear order while respecting some weight constraints. The second problem is an order reconfiguration problem under width constraints.
While the Completion of an Ordering problem is NP-complete, we show that it lies in FPT when parameterized by the interval width of ρ. This ordering problem can be used to model several ordering problems stemming from diverse application areas, such as graph drawing, computational social choice, and computer memory management. Each application yields a special partial order ρ. We also relate the interval width of ρ to parameterizations for these problems that have been studied earlier in the context of these applications, sometimes improving on parameterized algorithms that have been developed for these parameterizations before. This approach also gives some practical sub-exponential time algorithms for ordering problems.
In our second main result, we combine our parameterized approach with the paradigm of solution diversity. The idea of solution diversity is that instead of aiming at the development of algorithms that output a single optimal solution, the goal is to investigate algorithms that output a small set of sufficiently good solutions that are sufficiently diverse from one another. In this way, the user has the opportunity to choose the solution that is most appropriate to the context at hand. It also displays the richness of the solution space. There, we show that the considered diversity version of the Completion of an Ordering problem is fixed-parameter tractable with respect to natural paramaters that capture the notion of diversity and the notion of sufficiently good solutions. We apply this algorithm in the study of the Kemeny Rank Aggregation class of problems, a well-studied class of problems lying in the intersection of order theory and social choice theory.
Up to this point, we have been looking at problems where the goal is to find an optimal solution or a diverse set of good solutions. In the last part, we shift our focus from finding solutions to studying the solution space of a problem. There we consider the following order reconfiguration problem: Given a graph G together with linear orders τ and τ ′ of the vertices of G, can one transform τ into τ ′ by a sequence of swaps of adjacent elements in such a way that at each time step the resulting linear order has cutwidth (pathwidth) at most w? We show that this problem always has an affirmative answer when the input linear orders τ and τ ′ have cutwidth (pathwidth) at most w/2. Using this result, we establish a connection between two apparently unrelated problems: the reachability problem for two-letter string rewriting systems and the graph isomorphism problem for graphs of bounded cutwidth. This opens an avenue for the study of the famous graph isomorphism problem using techniques from term rewriting theory.
In addition to the main part of this work, we present results on two unrelated problems, namely on the Steiner Tree problem and on the Intersection Non-emptiness problem from automata theory.Doktorgradsavhandlin
Answering Regular Path Queries on Workflow Provenance
This paper proposes a novel approach for efficiently evaluating regular path
queries over provenance graphs of workflows that may include recursion. The
approach assumes that an execution g of a workflow G is labeled with
query-agnostic reachability labels using an existing technique. At query time,
given g, G and a regular path query R, the approach decomposes R into a set of
subqueries R1, ..., Rk that are safe for G. For each safe subquery Ri, G is
rewritten so that, using the reachability labels of nodes in g, whether or not
there is a path which matches Ri between two nodes can be decided in constant
time. The results of each safe subquery are then composed, possibly with some
small unsafe remainder, to produce an answer to R. The approach results in an
algorithm that significantly reduces the number of subqueries k over existing
techniques by increasing their size and complexity, and that evaluates each
subquery in time bounded by its input and output size. Experimental results
demonstrate the benefit of this approach
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