66,667 research outputs found
Complexity Hierarchies and Higher-order Cons-free Term Rewriting
Constructor rewriting systems are said to be cons-free if, roughly,
constructor terms in the right-hand sides of rules are subterms of the
left-hand sides; the computational intuition is that rules cannot build new
data structures. In programming language research, cons-free languages have
been used to characterize hierarchies of computational complexity classes; in
term rewriting, cons-free first-order TRSs have been used to characterize the
class PTIME.
We investigate cons-free higher-order term rewriting systems, the complexity
classes they characterize, and how these depend on the type order of the
systems. We prove that, for every K 1, left-linear cons-free systems
with type order K characterize ETIME if unrestricted evaluation is used
(i.e., the system does not have a fixed reduction strategy).
The main difference with prior work in implicit complexity is that (i) our
results hold for non-orthogonal term rewriting systems with no assumptions on
reduction strategy, (ii) we consequently obtain much larger classes for each
type order (ETIME versus EXPTIME), and (iii) results for cons-free
term rewriting systems have previously only been obtained for K = 1, and with
additional syntactic restrictions besides cons-freeness and left-linearity.
Our results are among the first implicit characterizations of the hierarchy E
= ETIME ETIME ... Our work confirms prior
results that having full non-determinism (via overlapping rules) does not
directly allow for characterization of non-deterministic complexity classes
like NE. We also show that non-determinism makes the classes characterized
highly sensitive to minor syntactic changes like admitting product types or
non-left-linear rules.Comment: extended version of a paper submitted to FSCD 2016. arXiv admin note:
substantial text overlap with arXiv:1604.0893
Complexity Hierarchies and Higher-Order Cons-Free Rewriting
Constructor rewriting systems are said to be cons-free if, roughly,
constructor terms in the right-hand sides of rules are subterms of constructor
terms in the left-hand side; the computational intuition is that rules cannot
build new data structures. It is well-known that cons-free programming
languages can be used to characterize computational complexity classes, and
that cons-free first-order term rewriting can be used to characterize the set
of polynomial-time decidable sets.
We investigate cons-free higher-order term rewriting systems, the complexity
classes they characterize, and how these depend on the order of the types used
in the systems. We prove that, for every k 1, left-linear cons-free
systems with type order k characterize ETIME if arbitrary evaluation is
used (i.e., the system does not have a fixed reduction strategy).
The main difference with prior work in implicit complexity is that (i) our
results hold for non-orthogonal term rewriting systems with possible rule
overlaps with no assumptions about reduction strategy, (ii) results for such
term rewriting systems have previously only been obtained for k = 1, and with
additional syntactic restrictions on top of cons-freeness and left-linearity.
Our results are apparently among the first implicit characterizations of the
hierarchy E = ETIME ETIME .... Our work
confirms prior results that having full non-determinism (via overlaps of rules)
does not directly allow characterization of non-deterministic complexity
classes like NE. We also show that non-determinism makes the classes
characterized highly sensitive to minor syntactic changes such as admitting
product types or non-left-linear rules.Comment: Extended version (with appendices) of a paper published in FSCD 201
Trading inference effort versus size in CNF Knowledge Compilation
Knowledge Compilation (KC) studies compilation of boolean functions f into
some formalism F, which allows to answer all queries of a certain kind in
polynomial time. Due to its relevance for SAT solving, we concentrate on the
query type "clausal entailment" (CE), i.e., whether a clause C follows from f
or not, and we consider subclasses of CNF, i.e., clause-sets F with special
properties. In this report we do not allow auxiliary variables (except of the
Outlook), and thus F needs to be equivalent to f.
We consider the hierarchies UC_k <= WC_k, which were introduced by the
authors in 2012. Each level allows CE queries. The first two levels are
well-known classes for KC. Namely UC_0 = WC_0 is the same as PI as studied in
KC, that is, f is represented by the set of all prime implicates, while UC_1 =
WC_1 is the same as UC, the class of unit-refutation complete clause-sets
introduced by del Val 1994. We show that for each k there are (sequences of)
boolean functions with polysize representations in UC_{k+1}, but with an
exponential lower bound on representations in WC_k. Such a separation was
previously only know for k=0. We also consider PC < UC, the class of
propagation-complete clause-sets. We show that there are (sequences of) boolean
functions with polysize representations in UC, while there is an exponential
lower bound for representations in PC. These separations are steps towards a
general conjecture determining the representation power of the hierarchies PC_k
< UC_k <= WC_k. The strong form of this conjecture also allows auxiliary
variables, as discussed in depth in the Outlook.Comment: 43 pages, second version with literature updates. Proceeds with the
separation results from the discontinued arXiv:1302.442
On the construction of hierarchic models
One of the main problems in the field of model-based diagnosis of technical systems today is finding the most useful model or models of the system being diagnosed. Often, a model showing the physical components and the connections between them is all that is available. As systems grow larger and larger, the run-time performance of diagnostic algorithms decreases considerably when using these detailed models. A solution to this problem is using a hierarchic model. This allows us to first diagnose the system using an abstract model, and then use this solution to guide the diagnostic process using a more detailed model. The main problem with this approach is acquiring the hierarchic model. We give a generic hierarchic diagnostic algorithm and show how the use of certain classes of hierarchic models can increase the performance of this algorithm. We then present linear time algorithms for the automatic construction of these hierarchic models, using the detailed model and extra information about cost of probing points and invertibility of components
Biosynthetic potentials of metabolites and their hierarchical organization
Peer reviewedPublisher PD
Building and Refining Abstract Planning Cases by Change of Representation Language
ion is one of the most promising approaches to improve the performance of
problem solvers. In several domains abstraction by dropping sentences of a
domain description -- as used in most hierarchical planners -- has proven
useful. In this paper we present examples which illustrate significant
drawbacks of abstraction by dropping sentences. To overcome these drawbacks, we
propose a more general view of abstraction involving the change of
representation language. We have developed a new abstraction methodology and a
related sound and complete learning algorithm that allows the complete change
of representation language of planning cases from concrete to abstract.
However, to achieve a powerful change of the representation language, the
abstract language itself as well as rules which describe admissible ways of
abstracting states must be provided in the domain model. This new abstraction
approach is the core of Paris (Plan Abstraction and Refinement in an Integrated
System), a system in which abstract planning cases are automatically learned
from given concrete cases. An empirical study in the domain of process planning
in mechanical engineering shows significant advantages of the proposed
reasoning from abstract cases over classical hierarchical planning.Comment: See http://www.jair.org/ for an online appendix and other files
accompanying this articl
The algebraic and Hamiltonian structure of the dispersionless Benney and Toda hierarchies
The algebraic and Hamiltonian structures of the multicomponent dispersionless
Benney and Toda hierarchies are studied. This is achieved by using a modified
set of variables for which there is a symmetry between the basic fields. This
symmetry enables formulae normally given implicitly in terms of residues, such
as conserved charges and fluxes, to be calculated explicitly. As a corollary of
these results the equivalence of the Benney and Toda hierarchies is
established. It is further shown that such quantities may be expressed in terms
of generalized hypergeometric functions, the simplest example involving
Legendre polynomials. These results are then extended to systems derived from a
rational Lax function and a logarithmic function. Various reductions are also
studied.Comment: 29 pages, LaTe
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