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\begin{document}

\centerline{\bf \large 6.824 Artificial Intelligence, Fall 1993}
\centerline{\bf \large Problem Set 1}
\medskip
\medskip

\centerline{Due:  Wednesday, September 23}


%% check confusion over attributes, attributes, f's, x's.

\noindent
{\bf Problem 1} 

Consider an instance space where each instance is
an object with $n$ attributes $x_1, x_2, \ldots x_n$ where each $x_i$
takes on real number values in the closed interval $[0,1]$.
In this problem we consider conjunctive concepts, i.e., concepts
defined by conjunctions of pure atoms for linear attributes where
a pure atom is a formula of the form $r_1 \leq x_i \leq r_2$,
$r_1 < x_i \leq r_2$, $r_1 \leq x_i < r_2$, or $r_1 < x_i < r_2$
where $r_1$ and $r_2$ are real numbers.

{\bf (a)} Describe the procedures needed to implement Mitchel's algorithm for this
concept space.  In other words, describe procedures for updating the most general
and most specific versions when a negative and positive example are given.

{\bf (b)} Show that, for this concept space, the set of most general versions
maintained by Mitchell's algorithm, i.e., the upper fringe of the
version space, can grow exponentially large.  More specifically, give
a set of negative examples such that the number of different minimal
concepts in this concept space that cover those examples is exponential in $n$,
the number of attributes in the instance space.

\vskip .5in
\noindent
{\bf Problem 2}

Now consider an instance space where each instance is
a structure with $n$ attributes $x_1, x_2, \ldots x_n$ where each
attribute ranges over the two values $T$ and $F$.
Each instance can be viewed as a truth assignment to the attributes
--- each instance $x$ assigns the feature $x_i$ either the value
$T$ or the value $F$.

For this instance space a concept can be defined by a Boolean formula
built from the attributes.  For example, the formula $x_1
\wedge \neg x_2$ is true of an instance $x$ if and only if $x$
assigns the feature $x_1$ the value $T$ and the feature $x_2$ the
value $F$.  Any Boolean combination of feature names determines a concept.

{\bf (a)} How many instances are there in the instance space
defined by $n$ Boolean attributes.

{\bf (b)} How many concepts are there for the instance space defined
by $n$ Boolean attributes.  In answering this question take an
{\em extensional} view of concepts, i.e., assume that two Boolean formulas
which define the same set of instances are different representation of the
same concept.

{\bf (c)} Suppose that we are trying to learn some fixed target
concept. We are given a set of instances, each
selected at random according to some fixed but unknown probability
distribution on the instance space. Each selected instance is
classified as either a positive or negative example according to the
fixed target concept.  Give the number of instances sufficient to
imply that an arbitrary Boolean concept consistent with these
instances will, with probability at least $1-\delta$, incorrectly
classify  at most a fraction $\epsilon$ of the instance space.  Your
answer should be an expression in terms of $\delta$, $\epsilon$, and
the number $n$ of attributes.  (This is called the {\em sample complexity}
required for PAC learning for this concept space.)

\vskip .5in
\noindent
{\bf Problem 3}

In this problem we again consider the instance space defined by $n$
Boolean attributes.  However we restrict concepts to those that can be
expressed as Boolean combinations of literals that involve at most $k$
Binary Boolean connectives each of which is either $\vee$ or $\wedge$.
For example, $x_1 \vee (x_2 \wedge \neg x_3)$ is a Boolean combination
of literals containing two binary Boolean connectives.

{\bf (a)} Show that for fixed $k$ the number of examples sufficient
for PAC-learning over the concept space of $k$-binary operators is
polynomial in $1/\epsilon, 1/\delta$, and $n$.  Don't worry about
getting tight bounds on the sample complexity required for PAC
learning --- just give some polynomial that is sufficient.  Show
the derivation of your polynomial.

{\bf (b)} Explain why the polynomial sample complexity you gave in part
{\bf a} does not immediately imply that this concept space is
PAC learnable in polynomial time.

%%
\vskip .5in
\noindent
{\bf Problem 4}

Consider an instance space in which each instance is an integer between
1 and $n$.  Consider a concept space in which each concept is a
disjunction of pure literals, i.e., a formula of the form
$p_1 \leq i \leq q_1 \vee p_2 \leq i \leq q_2 \vee \ldots \vee p_k \leq i \leq q_k$
where $i$ ranges over instances (integers from 1 to $n$) and each
$p_i$ and $q_i$ are integers between 1 and $n$.

{\bf (a)} Arguing only from the cardinality of the concept space
show that the number of samples sufficient for PAC-learning
this concept space is polynomial in $1/\epsilon$, $1/\delta$, $k$ and $\log n$.

{\bf (b)} Give a polynomial time algorithm for PAC learning for this
concept space (it should be polynomial in $1/\epsilon$, $1/\delta$, $k$ and $\log n$).

{\bf (c)}  Show that for a fixed $k$ this concept space has a bounded
VC-dimension that is independent of $n$.  Use this fact to give a
number of samples sufficient for PAC learning that is indendent of $n$.

\end{document}  
