LEARNING CLASSES OF LINEARLY SEPARABLE BOOLEAN FUNCTIONS FROM POSITIVE EXAMPLES

1992 ◽  
Vol 03 (01) ◽  
pp. 41-54
Author(s):  
PAOLA CAMPADELLI ◽  
ANNA MORPURGO
Keyword(s):  
Lower Bound ◽  
Sample Size ◽  
Polynomial Time ◽  
Boolean Functions ◽  
Learning Model ◽  
Probably Approximately Correct Learning ◽  
Probably Approximately Correct

This paper deals with learnability from positive examples of subclasses of linearly separable boolean functions in the framework of the probably approximately correct learning model. We prove that classes of functions defined by binary threshold neurons with n inputs and g(n) unknown weights are learnable in polynomial time iff g(n)=O(log n) and give an upper and a lower bound on the sample size.

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.

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

SIMULATIONS BY TIME-BOUNDED COUNTER MACHINES

2011 ◽  
Vol 22 (02) ◽  
pp. 395-409 ◽  
Author(s):  
HOLGER PETERSEN
Keyword(s):  
Lower Bound ◽  
Real Time ◽  
Polynomial Time ◽  
Time Complexity ◽  
Linear Time ◽  
Fixed Number ◽  
Exponential Time ◽  
New Family ◽  
Counter Machines ◽  
Time Bounds

We investigate the efficiency of simulations of storages by several counters. A simulation of a pushdown store is described which is optimal in the sense that reducing the number of counters of a simulator leads to an increase in time complexity. The lower bound also establishes a tight counter hierarchy in exponential time. Then we turn to simulations of a set of counters by a different number of counters. We improve and generalize a known simulation in polynomial time. Greibach has shown that adding s + 1 counters increases the power of machines working in time ns. Using a new family of languages we show here a tight hierarchy result for machines with the same polynomial time-bound. We also prove hierarchies for machines with a fixed number of counters and with growing polynomial time-bounds. For machines with one counter and an additional "store zero" instruction we establish the equivalence of real-time and linear time. If at least two counters are available, the classes of languages accepted in real-time and linear time can be separated.

Lower Bounds on the VC Dimension of Smoothly Parameterized Function Classes

1995 ◽  
Vol 7 (5) ◽  
pp. 1040-1053 ◽  
Author(s):  
Wee Sun Lee ◽  
Peter L. Bartlett ◽  
Robert C. Williamson
Keyword(s):  
Lower Bounds ◽  
Differentiable Manifold ◽  
Function Class ◽  
Basis Functions ◽  
Vc Dimension ◽  
Probably Approximately Correct Learning ◽  
Learning Framework ◽  
Function Classes ◽  
Probably Approximately Correct ◽  
The Relationship

We examine the relationship between the VC dimension and the number of parameters of a threshold smoothly parameterized function class. We show that the VC dimension of such a function class is at least k if there exists a k-dimensional differentiable manifold in the parameter space such that each member of the manifold corresponds to a different decision boundary. Using this result, we are able to obtain lower bounds on the VC dimension proportional to the number of parameters for several thresholded function classes including two-layer neural networks with certain smooth activation functions and radial basis functions with a gaussian basis. These lower bounds hold even if the magnitudes of the parameters are restricted to be arbitrarily small. In Valiant's probably approximately correct learning framework, this implies that the number of examples necessary for learning these function classes is at least linear in the number of parameters.

scholarly journals Augmenting the Power of (Partial) MaxSat Resolution with Extension

2020 ◽  
Vol 34 (02) ◽  
pp. 1561-1568 ◽  
Author(s):  
Javier Larrosa ◽  
Emma Rollon
Keyword(s):  
Lower Bound ◽  
Objective Function ◽  
Lower Bounds ◽  
Polynomial Time ◽  
Proof System ◽  
Extension Rule ◽  
Resolution Proof

The refutation power of SAT and MaxSAT resolution is challenged by problems like the soft and hard Pigeon Hole Problem PHP for which short refutations do not exist. In this paper we augment the MaxSAT resolution proof system with an extension rule. The new proof system MaxResE is sound and complete, and more powerful than plain MaxSAT resolution, since it can refute the soft and hard PHP in polynomial time. We show that MaxResE refutations actually subtract lower bounds from the objective function encoded by the formulas. The resulting formula is the residual after the lower bound extraction. We experimentally show that the residual of the soft PHP (once its necessary cost of 1 has been efficiently subtracted with MaxResE) is a concise, easy to solve, satisfiable problem.

scholarly journals New Constructions of Balanced Boolean Functions with Maximum Algebraic Immunity, High Nonlinearity and Optimal Algebraic Degree

2020 ◽  
Vol 17 (7) ◽  
pp. 639-654
Author(s):  
Dheeraj Kumar SHARMA ◽  
Rajoo PANDEY
Keyword(s):  
Lower Bound ◽  
Boolean Function ◽  
Boolean Functions ◽  
Primitive Element ◽  
Galois Field ◽  
Algebraic Degree ◽  
Algebraic Immunity ◽  
Greatest Common Divisor ◽  
Primitive Elements ◽  
High Nonlinearity

This paper consists of proposal of two new constructions of balanced Boolean function achieving a new lower bound of nonlinearity along with high algebraic degree and optimal or highest algebraic immunity. This construction has been made by using representation of Boolean function with primitive elements. Galois Field,  used in this representation has been constructed by using powers of primitive element such that greatest common divisor of power and  is 1. The constructed balanced  variable Boolean functions achieve higher nonlinearity, algebraic degree of , and algebraic immunity of   for odd ,  for even . The nonlinearity of Boolean function obtained in the proposed constructions is better as compared to existing Boolean functions available in the literature without adversely affecting other properties such as balancedness, algebraic degree and algebraic immunity.

Sign in / Sign up

Export Citation Format

Share Document