12,759 research outputs found
Superconvergence of Topological Entropy in the Symbolic Dynamics of Substitution Sequences
We consider infinite sequences of superstable orbits (cascades) generated by
systematic substitutions of letters in the symbolic dynamics of one-dimensional
nonlinear systems in the logistic map universality class. We identify the
conditions under which the topological entropy of successive words converges as
a double exponential onto the accumulation point, and find the convergence
rates analytically for selected cascades. Numerical tests of the convergence of
the control parameter reveal a tendency to quantitatively universal
double-exponential convergence. Taking a specific physical example, we consider
cascades of stable orbits described by symbolic sequences with the symmetries
of quasilattices. We show that all quasilattices can be realised as stable
trajectories in nonlinear dynamical systems, extending previous results in
which two were identified.Comment: This version: updated figures and added discussion of generalised
time quasilattices. 17 pages, 4 figure
The variable containment problem
The essentially free variables of a term in some -calculus, FV , form the set ( FV}. This set is significant once we consider equivalence classes of -terms rather than -terms themselves, as for instance in higher-order rewriting. An important problem for (generalised) higher-order rewrite systems is the variable containment problem: given two terms and , do we have for all substitutions and contexts [] that FV FV?
This property is important when we want to consider as a rewrite rule and keep -step rewriting decidable. Variable containment is in general not implied by FV FV. We give a decision procedure for the variable containment problem of the second-order fragment of . For full we show the equivalence of variable containment to an open problem in the theory of PCF; this equivalence also shows that the problem is decidable in the third-order case
Bounded-analytic sequent calculi and embeddings for hypersequent logics
A sequent calculus with the subformula property has long been recognised as a highly favourable starting point for the proof theoretic investigation of a logic. However, most logics of interest cannot be presented using a sequent calculus with the subformula property. In response, many formalisms more intricate than the sequent calculus have been formulated. In this work we identify an alternative: retain the sequent calculus but generalise the subformula property to permit specific axiom substitutions and their subformulas. Our investigation leads to a classification of generalised subformula properties and is applied to infinitely many substructural, intermediate, and modal logics (specifically: those with a cut-free hypersequent calculus). We also develop a complementary perspective on the generalised subformula properties in terms of logical embeddings. This yields new complexity upper bounds for contractive-mingle substructural logics and situates isolated results on the so-called simple substitution property within a general theory
Separation and Renaming in Nominal Sets
Nominal sets provide a foundation for reasoning about names. They are used primarily in syntax with binders, but also, e.g., to model automata over infinite alphabets. In this paper, nominal sets are related to nominal renaming sets, which involve arbitrary substitutions rather than permutations, through a categorical adjunction. In particular, the left adjoint relates the separated product of nominal sets to the Cartesian product of nominal renaming sets. Based on these results, we define the new notion of separated nominal automata. We show that these automata can be exponentially smaller than classical nominal automata, if the semantics is closed under substitutions
An Abstract Approach to Consequence Relations
We generalise the Blok-J\'onsson account of structural consequence relations,
later developed by Galatos, Tsinakis and other authors, in such a way as to
naturally accommodate multiset consequence. While Blok and J\'onsson admit, in
place of sheer formulas, a wider range of syntactic units to be manipulated in
deductions (including sequents or equations), these objects are invariably
aggregated via set-theoretical union. Our approach is more general in that
non-idempotent forms of premiss and conclusion aggregation, including multiset
sum and fuzzy set union, are considered. In their abstract form, thus,
deductive relations are defined as additional compatible preorderings over
certain partially ordered monoids. We investigate these relations using
categorical methods, and provide analogues of the main results obtained in the
general theory of consequence relations. Then we focus on the driving example
of multiset deductive relations, providing variations of the methods of matrix
semantics and Hilbert systems in Abstract Algebraic Logic
Generalised Unitarity for Dimensionally Regulated Amplitudes
We present a novel set of Feynman rules and generalised unitarity
cut-conditions for computing one-loop amplitudes via d-dimensional integrand
reduction algorithm. Our algorithm is suited for analytic as well as numerical
result, because all ingredients turn out to have a four-dimensional
representation. We will apply this formalism to NLO QCD corrections.Comment: Presented at SILAFAE 2014, 24-28 Nov, Ruta N, Medellin, Colombi
Classes of Terminating Logic Programs
Termination of logic programs depends critically on the selection rule, i.e.
the rule that determines which atom is selected in each resolution step. In
this article, we classify programs (and queries) according to the selection
rules for which they terminate. This is a survey and unified view on different
approaches in the literature. For each class, we present a sufficient, for most
classes even necessary, criterion for determining that a program is in that
class. We study six classes: a program strongly terminates if it terminates for
all selection rules; a program input terminates if it terminates for selection
rules which only select atoms that are sufficiently instantiated in their input
positions, so that these arguments do not get instantiated any further by the
unification; a program local delay terminates if it terminates for local
selection rules which only select atoms that are bounded w.r.t. an appropriate
level mapping; a program left-terminates if it terminates for the usual
left-to-right selection rule; a program exists-terminates if there exists a
selection rule for which it terminates; finally, a program has bounded
nondeterminism if it only has finitely many refutations. We propose a
semantics-preserving transformation from programs with bounded nondeterminism
into strongly terminating programs. Moreover, by unifying different formalisms
and making appropriate assumptions, we are able to establish a formal hierarchy
between the different classes.Comment: 50 pages. The following mistake was corrected: In figure 5, the first
clause for insert was insert([],X,[X]
Undecidability of Equality in the Free Locally Cartesian Closed Category (Extended version)
We show that a version of Martin-L\"of type theory with an extensional
identity type former I, a unit type N1 , Sigma-types, Pi-types, and a base type
is a free category with families (supporting these type formers) both in a 1-
and a 2-categorical sense. It follows that the underlying category of contexts
is a free locally cartesian closed category in a 2-categorical sense because of
a previously proved biequivalence. We show that equality in this category is
undecidable by reducing it to the undecidability of convertibility in
combinatory logic. Essentially the same construction also shows a slightly
strengthened form of the result that equality in extensional Martin-L\"of type
theory with one universe is undecidable
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