scholarly journals Pac-Learning Recursive Logic Programs: Efficient Algorithms

1995 ◽  
Vol 2 ◽  
pp. 501-539 ◽  
Author(s):  
W. W. Cohen
Keyword(s):  
Polynomial Time ◽  
Companion Paper ◽  
Efficient Algorithms ◽  
Pac Learning ◽  
Logic Programs ◽  
Constant Depth ◽  
Learning Problem ◽  
Equivalence Queries ◽  
Recursive Programs ◽  
Natural Way

We present algorithms that learn certain classes of function-free recursive logic programs in polynomial time from equivalence queries. In particular, we show that a single k-ary recursive constant-depth determinate clause is learnable. Two-clause programs consisting of one learnable recursive clause and one constant-depth determinate non-recursive clause are also learnable, if an additional ``basecase'' oracle is assumed. These results immediately imply the pac-learnability of these classes. Although these classes of learnable recursive programs are very constrained, it is shown in a companion paper that they are maximally general, in that generalizing either class in any natural way leads to a computationally difficult learning problem. Thus, taken together with its companion paper, this paper establishes a boundary of efficient learnability for recursive logic programs.

scholarly journals Pac-learning Recursive Logic Programs: Negative Results

1995 ◽  
Vol 2 ◽  
pp. 541-573 ◽  
Author(s):  
W. W. Cohen
Keyword(s):  
General Class ◽  
Recursive Function ◽  
Companion Paper ◽  
Natural Generalization ◽  
Pac Learning ◽  
Logic Programs ◽  
Constant Depth ◽  
Negative Results ◽  
Valiant’S Model ◽  
Positive Results

In a companion paper it was shown that the class of constant-depth determinate k-ary recursive clauses is efficiently learnable. In this paper we present negative results showing that any natural generalization of this class is hard to learn in Valiant's model of pac-learnability. In particular, we show that the following program classes are cryptographically hard to learn: programs with an unbounded number of constant-depth linear recursive clauses; programs with one constant-depth determinate clause containing an unbounded number of recursive calls; and programs with one linear recursive clause of constant locality. These results immediately imply the non-learnability of any more general class of programs. We also show that learning a constant-depth determinate program with either two linear recursive clauses or one linear recursive clause and one non-recursive clause is as hard as learning boolean DNF. Together with positive results from the companion paper, these negative results establish a boundary of efficient learnability for recursive function-free clauses.

Polynomial-time MAT learning of multilinear logic programs

1993 ◽  
pp. 63-74
Author(s):  
Kimihito Ito ◽  
Akihiro Yamamoto
Keyword(s):  
Polynomial Time ◽  
Logic Programs

scholarly journals POLYLINE FITTING OF PLANAR POINTS UNDER MIN-SUM CRITERIA

2006 ◽  
Vol 16 (02n03) ◽  
pp. 97-116 ◽  
Author(s):  
BORIS ARONOV ◽  
TETSUO ASANO ◽  
NAOKI KATOH ◽  
KURT MEHLHORN ◽  
TAKESHI TOKUYAMA
Keyword(s):  
Polynomial Time ◽  
Basic Problem ◽  
Efficient Algorithms ◽  
Approximation Schemes ◽  
Time Approximation ◽  
Set Of Points ◽  
Polynomial Time Approximation

Fitting a curve of a certain type to a given set of points in the plane is a basic problem in statistics and has numerous applications. We consider fitting a polyline with k joints under the min-sum criteria with respect to L1- and L2-metrics, which are more appropriate measures than uniform and Hausdorff metrics in statistical context. We present efficient algorithms for the 1-joint versions of the problem and fully polynomial-time approximation schemes for the general k-joint versions.

ORACLES IN $\Sigma^p_2$ ARE SUFFFICIENT FOR EXACT LEARNING

2000 ◽  
Vol 11 (04) ◽  
pp. 613-632 ◽  
Author(s):  
Johannes Köbler ◽  
Wolfgang Lindner
Keyword(s):  
Polynomial Time ◽  
Learning Model ◽  
Query Complexity ◽  
Boolean Circuits ◽  
Membership Queries ◽  
Exact Learning ◽  
Equivalence Queries

We study the learnability of representation classes in Angluin's exact learning model. In particular, we consider the following three query types: equivalence queries, equivalence and membership queries, and membership queries only. We show in all three cases that polynomial query complexity implies already polynomial-time learnability, provided that the learner additionally has access to an oracle in [Formula: see text]. It follows that boolean circuits are polynomial-time learnable with equivalence queries and the help of an oracle in [Formula: see text].a

Recurrence with affine level mappings is P-time decidable for CLP

2007 ◽  
Vol 8 (01) ◽  
pp. 111-119 ◽  
Author(s):  
FRED MESNARD ◽  
ALEXANDER SEREBRENIK
Keyword(s):  
Polynomial Time ◽  
Logic Programs ◽  
Constraint Logic Programs

AbstractIn this paper we introduce a class of constraint logic programs such that their termination can be proved by using affine level mappings. We show that membership to this class is decidable in polynomial time.

scholarly journals Actively Learning Concepts and Conjunctive Queries under ELr-Ontologies

2021 ◽  
Author(s):  
Maurice Funk ◽  
Jean Christoph Jung ◽  
Carsten Lutz
Keyword(s):  
Active Learning ◽  
Polynomial Time ◽  
Description Logic ◽  
Learning Algorithm ◽  
Domain Expert ◽  
Conjunctive Queries ◽  
Membership Queries ◽  
Equivalence Queries

We consider the problem to learn a concept or a query in the presence of an ontology formulated in the description logic ELr, in Angluin's framework of active learning that allows the learning algorithm to interactively query an oracle (such as a domain expert). We show that the following can be learned in polynomial time: (1) EL-concepts, (2) symmetry-free ELI-concepts, and (3) conjunctive queries (CQs) that are chordal, symmetry-free, and of bounded arity. In all cases, the learner can pose to the oracle membership queries based on ABoxes and equivalence queries that ask whether a given concept/query from the considered class is equivalent to the target. The restriction to bounded arity in (3) can be removed when we admit unrestricted CQs in equivalence queries. We also show that EL-concepts are not polynomial query learnable in the presence of ELI-ontologies.

From equivalence queries to PAC learning: The case of implication theories

2020 ◽  
Vol 127 ◽  
pp. 1-16
Author(s):  
Ramil Yarullin ◽  
Sergei Obiedkov
Keyword(s):  
Pac Learning ◽  
Equivalence Queries

scholarly journals Efficient algorithms for solving the p-Laplacian in polynomial time

2020 ◽  
Vol 146 (2) ◽  
pp. 369-400
Author(s):  
Sébastien Loisel
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Polynomial Time ◽  
Numerical Experiments ◽  
Nonlinear Partial Differential Equation ◽  
Numerical Algorithms ◽  
Efficient Algorithms ◽  
Barrier Method ◽  
Newton Iterations ◽  
Grid Points

Abstract The p-Laplacian is a nonlinear partial differential equation, parametrized by $$p \in [1,\infty ]$$ p ∈ [ 1 , ∞ ] . We provide new numerical algorithms, based on the barrier method, for solving the p-Laplacian numerically in $$O(\sqrt{n}\log n)$$ O ( n log n ) Newton iterations for all $$p \in [1,\infty ]$$ p ∈ [ 1 , ∞ ] , where n is the number of grid points. We confirm our estimates with numerical experiments.

scholarly journals Relating polynomial time to constant depth

1998 ◽  
Vol 207 (1) ◽  
pp. 159-170 ◽  
Author(s):  
Heribert Vollmer
Keyword(s):  
Polynomial Time ◽  
Constant Depth

scholarly journals Sparse regular variation

2021 ◽  
Vol 53 (4) ◽  
pp. 1115-1148
Author(s):  
Nicolas Meyer ◽  
Olivier Wintenberger
Keyword(s):  
Weak Convergence ◽  
Extreme Events ◽  
Spectral Measure ◽  
Regular Variation ◽  
Theoretical Framework ◽  
Efficient Algorithms ◽  
Dependence Structure ◽  
Random Vectors ◽  
Sparsity Structure ◽  
Natural Way

AbstractRegular variation provides a convenient theoretical framework for studying large events. In the multivariate setting, the spectral measure characterizes the dependence structure of the extremes. This measure gathers information on the localization of extreme events and often has sparse support since severe events do not simultaneously occur in all directions. However, it is defined through weak convergence, which does not provide a natural way to capture this sparsity structure. In this paper, we introduce the notion of sparse regular variation, which makes it possible to better learn the dependence structure of extreme events. This concept is based on the Euclidean projection onto the simplex, for which efficient algorithms are known. We prove that under mild assumptions sparse regular variation and regular variation are equivalent notions, and we establish several results for sparsely regularly varying random vectors.

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