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

scholarly journals On the Learnability of Possibilistic Theories

2020 ◽  
Author(s):  
Cosimo Persia ◽  
Ana Ozaki
Keyword(s):  
Large Class ◽  
Polynomial Time ◽  
Classical Logic ◽  
Learning Model ◽  
Membership Queries ◽  
Exact Model ◽  
Exact Learning ◽  
Probably Approximately Correct ◽  
Horn Theories ◽  
Pac Model

We investigate learnability of possibilistic theories from entailments in light of Angluin’s exact learning model. We consider cases in which only membership, only equivalence, and both kinds of queries can be posed by the learner. We then show that, for a large class of problems, polynomial time learnability results for classical logic can be transferred to the respective possibilistic extension. In particular, it follows from our results that the possibilistic extension of propositional Horn theories is exactly learnable in polynomial time. As polynomial time learnability in the exact model is transferable to the classical probably approximately correct (PAC) model extended with membership queries, our work also establishes such results in this model.

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.

scholarly journals Additive-error fine-grained quantum supremacy

Quantum ◽  
2020 ◽  
Vol 4 ◽  
pp. 329
Author(s):  
Tomoyuki Morimae ◽  
Suguru Tamaki
Keyword(s):  
Quantum Computing ◽  
Complexity Theory ◽  
Polynomial Time ◽  
Boolean Circuits ◽  
Exponential Time ◽  
Fine Grained ◽  
Additive Error ◽  
Universal Quantum ◽  
The One ◽  
Computing Models

It is known that several sub-universal quantum computing models, such as the IQP model, the Boson sampling model, the one-clean qubit model, and the random circuit model, cannot be classically simulated in polynomial time under certain conjectures in classical complexity theory. Recently, these results have been improved to ``fine-grained" versions where even exponential-time classical simulations are excluded assuming certain classical fine-grained complexity conjectures. All these fine-grained results are, however, about the hardness of strong simulations or multiplicative-error sampling. It was open whether any fine-grained quantum supremacy result can be shown for a more realistic setup, namely, additive-error sampling. In this paper, we show the additive-error fine-grained quantum supremacy (under certain complexity assumptions). As examples, we consider the IQP model, a mixture of the IQP model and log-depth Boolean circuits, and Clifford+T circuits. Similar results should hold for other sub-universal models.

DERANDOMIZED LEARNING OF BOOLEAN FUNCTIONS OVER FINITE ABELIAN GROUPS

2001 ◽  
Vol 12 (04) ◽  
pp. 491-516
Author(s):  
M. SITHARAM ◽  
TIMOTHY STRANEY
Keyword(s):  
Boolean Functions ◽  
Partial Information ◽  
Abelian Groups ◽  
Query Complexity ◽  
Character Theory ◽  
Finite Abelian Groups ◽  
Training Set ◽  
Membership Queries ◽  
Approximation Classes ◽  
Query Model

We employ the Always Approximately Correct or AAC model defined in [35], to prove learnability results for classes of Boolean functions over arbitrary finite Abelian groups. This model is an extension of Angluin's Query model of exact learning. The Boolean functions we consider belong to approximation classes, i.e. functions that are approximable (in various norms) by few Fourier basis functions, or irreducible characters of the domain Abelian group. We contrast our learnability results to previous results for similar classes in the PAC model of learning with and without membership queries. In addition, we discuss new, natural issues and questions that arise when the AAC model is used. One such question is whether a uniform training set is available for learning any function in a given approximation class. No analogous question seems to have been studied in the context of Angluin's Query model. Another question is whether the training set can be found quickly if the approximation class of the function is completely unknown to the learner, or only partial information about the approximation class is given to the learner (in addition to the answers to membership queries). In order to prove the learnability results in this paper we require new techniques for efficiently sampling Boolean functions using the character theory of finite Abelian groups, as well as the development of algebraic algorithms. The techniques result in other natural applications closely related to learning, for example, query complexity of deterministic algorithms for testing linearity, efficient pseudorandom generators, and estimating VC dimensions for classes of Boolean functions over finite Abelian groups.

scholarly journals Exact learning of juntas from membership queries

2018 ◽  
Vol 742 ◽  
pp. 82-97
Author(s):  
Nader H. Bshouty ◽  
Areej Costa
Keyword(s):  
Membership Queries ◽  
Exact Learning
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