8 research outputs found
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
Polynomial Path Orders: A Maximal Model
This paper is concerned with the automated complexity analysis of term
rewrite systems (TRSs for short) and the ramification of these in implicit
computational complexity theory (ICC for short). We introduce a novel path
order with multiset status, the polynomial path order POP*. Essentially relying
on the principle of predicative recursion as proposed by Bellantoni and Cook,
its distinct feature is the tight control of resources on compatible TRSs: The
(innermost) runtime complexity of compatible TRSs is polynomially bounded. We
have implemented the technique, as underpinned by our experimental evidence our
approach to the automated runtime complexity analysis is not only feasible, but
compared to existing methods incredibly fast. As an application in the context
of ICC we provide an order-theoretic characterisation of the polytime
computable functions. To be precise, the polytime computable functions are
exactly the functions computable by an orthogonal constructor TRS compatible
with POP*
Polynomial Path Orders
This paper is concerned with the complexity analysis of constructor term
rewrite systems and its ramification in implicit computational complexity. We
introduce a path order with multiset status, the polynomial path order POP*,
that is applicable in two related, but distinct contexts. On the one hand POP*
induces polynomial innermost runtime complexity and hence may serve as a
syntactic, and fully automatable, method to analyse the innermost runtime
complexity of term rewrite systems. On the other hand POP* provides an
order-theoretic characterisation of the polytime computable functions: the
polytime computable functions are exactly the functions computable by an
orthogonal constructor TRS compatible with POP*.Comment: LMCS version. This article supersedes arXiv:1209.379
A TERM-REWRITING CHARACTERIZATION OF PSPACE
Isabel Oitavem has introduced a term rewriting system (TRS) which captures the class FPS of polynomial-space computable functions. We propose an alternative TRS for FPS. As a consequence, it is obtained that FPS is the smallest class containing certain initial functions and closed under specific operations. It turns out that our characterization is relatively simple and suggests an uniform approach to the space-complexity
Proof Theory at Work: Complexity Analysis of Term Rewrite Systems
This thesis is concerned with investigations into the "complexity of term
rewriting systems". Moreover the majority of the presented work deals with the
"automation" of such a complexity analysis. The aim of this introduction is to
present the main ideas in an easily accessible fashion to make the result
presented accessible to the general public. Necessarily some technical points
are stated in an over-simplified way.Comment: Cumulative Habilitation Thesis, submitted to the University of
Innsbruc
Connecting the two worlds: well-partial-orders and ordinal notation systems
Kruskal claims in his now-classical 1972 paper [47] that well-partial-orders are among the most frequently rediscovered mathematical objects. Well partial-orders have applications in many fields outside the theory of orders: computer science, proof theory, reverse mathematics, algebra, combinatorics, etc.
The maximal order type of a well-partial-order characterizes that order’s strength. Moreover, in many natural cases, a well-partial-order’s maximal order type can be represented by an ordinal notation system. However, there are a number of natural well-partial-orders whose maximal order types and corresponding ordinal notation systems remain unknown. Prominent examples are Friedman’s well-partial-orders of trees with the gap-embeddability relation [76].
The main goal of this dissertation is to investigate a conjecture of Weiermann [86], thereby addressing the problem of the unknown maximal order types and corresponding ordinal notation systems for Friedman’s well-partial orders [76]. Weiermann’s conjecture concerns a class of structures, a typical member of which is denoted by T (W ), each are ordered by a certain gapembeddability relation. The conjecture indicates a possible approach towards determining the maximal order types of the structures T (W ). Specifically, Weiermann conjectures that the collapsing functions #i correspond to maximal linear extensions of these well-partial-orders T (W ), hence also that these collapsing functions correspond to maximal linear extensions of Friedman’s famous well-partial-orders