27 research outputs found
A uniform approach to fundamental sequences and hierarchies
In this article we give a unifying approach to the theory of fundamental sequences and their related Hardy hierarchies of number-theoretic functions and we show the equivalence of the new approach with the classical one
Multiply-Recursive Upper Bounds with Higman's Lemma
We develop a new analysis for the length of controlled bad sequences in
well-quasi-orderings based on Higman's Lemma. This leads to tight
multiply-recursive upper bounds that readily apply to several verification
algorithms for well-structured systems
A Computation of the Maximal Order Type of the Term Ordering on Finite Multisets
We give a sharpening of a recent result of Aschenbrenner and Pong about the maximal order type of the term ordering on the finite multisets over a wpo. Moreover we discuss an approach to compute maximal order types of well-partial orders which are related to tree embeddings
Reachability in Vector Addition Systems is Primitive-Recursive in Fixed Dimension
The reachability problem in vector addition systems is a central question,
not only for the static verification of these systems, but also for many
inter-reducible decision problems occurring in various fields. The currently
best known upper bound on this problem is not primitive-recursive, even when
considering systems of fixed dimension. We provide significant refinements to
the classical decomposition algorithm of Mayr, Kosaraju, and Lambert and to its
termination proof, which yield an ACKERMANN upper bound in the general case,
and primitive-recursive upper bounds in fixed dimension. While this does not
match the currently best known TOWER lower bound for reachability, it is
optimal for related problems
Bisimulation Equivalence of First-Order Grammars is ACKERMANN-Complete
Checking whether two pushdown automata with restricted silent actions are
weakly bisimilar was shown decidable by S\'enizergues (1998, 2005). We provide
the first known complexity upper bound for this famous problem, in the
equivalent setting of first-order grammars. This ACKERMANN upper bound is
optimal, and we also show that strong bisimilarity is primitive-recursive when
the number of states of the automata is fixed
Derivation Lengths Classification of G\"odel's T Extending Howard's Assignment
Let T be Goedel's system of primitive recursive functionals of finite type in
the lambda formulation. We define by constructive means using recursion on
nested multisets a multivalued function I from the set of terms of T into the
set of natural numbers such that if a term a reduces to a term b and if a
natural number I(a) is assigned to a then a natural number I(b) can be assigned
to b such that I(a) is greater than I(b). The construction of I is based on
Howard's 1970 ordinal assignment for T and Weiermann's 1996 treatment of T in
the combinatory logic version. As a corollary we obtain an optimal derivation
length classification for the lambda formulation of T and its fragments.
Compared with Weiermann's 1996 exposition this article yields solutions to
several non-trivial problems arising from dealing with lambda terms instead of
combinatory logic terms. It is expected that the methods developed here can be
applied to other higher order rewrite systems resulting in new powerful
termination orderings since T is a paradigm for such systems
Infinitary -Calculi from a Linear Perspective (Long Version)
We introduce a linear infinitary -calculus, called
, in which two exponential modalities are available, the
first one being the usual, finitary one, the other being the only construct
interpreted coinductively. The obtained calculus embeds the infinitary
applicative -calculus and is universal for computations over infinite
strings. What is particularly interesting about , is that
the refinement induced by linear logic allows to restrict both modalities so as
to get calculi which are terminating inductively and productive coinductively.
We exemplify this idea by analysing a fragment of built around
the principles of and . Interestingly, it enjoys
confluence, contrarily to what happens in ordinary infinitary
-calculi
Foundations of Online Structure Theory II: The Operator Approach
We introduce a framework for online structure theory. Our approach
generalises notions arising independently in several areas of computability
theory and complexity theory. We suggest a unifying approach using operators
where we allow the input to be a countable object of an arbitrary complexity.
We give a new framework which (i) ties online algorithms with computable
analysis, (ii) shows how to use modifications of notions from computable
analysis, such as Weihrauch reducibility, to analyse finite but uniform
combinatorics, (iii) show how to finitize reverse mathematics to suggest a fine
structure of finite analogs of infinite combinatorial problems, and (iv) see
how similar ideas can be amalgamated from areas such as EX-learning, computable
analysis, distributed computing and the like. One of the key ideas is that
online algorithms can be viewed as a sub-area of computable analysis.
Conversely, we also get an enrichment of computable analysis from classical
online algorithms
Simplification orders in term rewriting
Thema der Arbeit ist die Anwendung von Methoden der Beweistheorie
auf Termersetzungssysteme, deren Termination mittels einer
Simplifikationsordnung beweisbar ist. Es werden optimale
Schranken für Herleitungslängen im allgemeinen Fall und im
Fall der Termination mittels einer Knuth-Bendix-Ordnung (KBO)
angegeben. Zudem werden die Ordnungstypen von KBOs vollständig
klassifiziert und die unter KBO berechenbaren Funktionen
vorgestellt. Einen weiteren Schwerpunkt bildet die Untersuchung
der Löngen von Reduktionsketten, die bei einfach terminierenden
Termersetzungssysteme auftreten und bestimmten Wachstumsbedingungen
genügen
Snowflake groups, Perron-Frobenius eigenvalues, and isoperimetric spectra
The k-dimensional Dehn (or isoperimetric) function of a group bounds the
volume of efficient ball-fillings of k-spheres mapped into k-connected spaces
on which the group acts properly and cocompactly; the bound is given as a
function of the volume of the sphere. We advance significantly the observed
range of behavior for such functions. First, to each non-negative integer
matrix P and positive rational number r, we associate a finite, aspherical
2-complex X_{r,P} and calculate the Dehn function of its fundamental group
G_{r,P} in terms of r and the Perron-Frobenius eigenvalue of P. The range of
functions obtained includes x^s, where s is an arbitrary rational number
greater than or equal to 2. By repeatedly forming multiple HNN extensions of
the groups G_{r,P} we exhibit a similar range of behavior among
higher-dimensional Dehn functions, proving in particular that for each positive
integer k and rational s greater than or equal to (k+1)/k there exists a group
with k-dimensional Dehn function x^s. Similar isoperimetric inequalities are
obtained for arbitrary manifold pairs (M,\partial M) in addition to
(B^{k+1},S^k).Comment: 42 pages, 8 figures. Version 2: 47 pages, 8 figures; minor revisions
and reformatting; to appear in Geom. Topo