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\section*{Computability and Logic Homework 10}

\noindent
\textbf{Due: } {\bf Tuesday}, November 29, 2005 

\paragraph{1.}
For each of the following, indicate which variable occurrences are
bound and which are free.  For bound occurrences, indicate which
quantifier binds them.
\begin{itemize}
\item[a.] $P(z,y) \supset \forall x. (P(x,z) \supset \exists y. P(x,y))$.
\item[b.] $(\exists y. Q(x,y)) \supset (\exists y. \forall x. Q(x,y))$.
\item[c.] $P(f(x),f(y)) \supset \exists x.(P(x,y))$.
\end{itemize}

\paragraph{2.}
In class, I defined the notation $A[t/x]$ to stand for the
``capture-avoiding'' substitution of $t$ for $x$ in $A$.  This
substitution is well-defined no matter what variables are free in $t$
or bound in $A$.  Another way to define substitution is to stipulate
that the substitution of $t$ for $x$ in $A$ is {\em undefined} if it
would incur a capture.  To keep from getting confused, let $A\{t/x\}$
stand for this ``partial'' substitution operation.
Give a formal definition for $A\{t/x\}$ by recursion on $A$,
identifying all cases where it is not defined.

\paragraph{3.} For each formula below, give both
({\it i}) a model and ({\it ii}) a {\bf\em countermodel}, that is, a
structure that is not a model.
\begin{itemize}
\item[a.] $\forall x. \forall y. \forall z. ((P(x,y) \wedge P(y,z))
  \supset P(x,z))$.
\item[b.] $(\forall x.P(x,x)) \supset \exists y. \forall x. P(x,y)$.
\item[c.] $(\forall x.P(x,s(x))) \wedge (\forall x.\neg P(x,x))$.
\end{itemize}

\paragraph{4.}  
\begin{itemize}
\item[a.] Prove that if $x \notin FV(A)$, then $\models A \supset \forall x . B$
iff $\models A \supset B$.
\item[b.] Prove that if $x \notin FV(B)$, then $\models (\exists x
  . A) \supset B$ iff $\models A \supset B$.
\item[c.] Give an example of an $A$ for which $\not\models A \supset
  \forall x . P(x)$; prove your answer correct by giving an
  interpretation that makes this formula false.
\end{itemize}

\paragraph{5.}
Let's define a {\bf\em verification} of a formula $F$ to be a finite
set $V$, each of whose elements is either an atom or the negation of
an atom, such that $V \models F$.  For example, the set
\begin{quote}
$\{\neg Q(c,a)\}$ \quad is a verification of \quad $\exists x.\forall
y.(P(x,y) \wedge Q(c,x)) \supset R(y,y)$.
\end{quote}
Let's say a formula $F$ is {\bf\em affirmative} if, for any interpretation $(\mathcal{A},\eta)$ such
that $\mathcal{A} \models_\eta F$, there is a
verification $V$ of $F$ such that $\mathcal{A} \models_\eta V$.
$F$ is {\bf\em refutative} if
$\neg F$ is affirmative. 
\begin{itemize}
\item[a.] Prove that every atomic formula is both affirmative and refutative.
\item[b.] Suppose that $A$ and $B$ are affirmative.  {\bf Either}
  prove that $A \vee B$ is affirmative {\bf or} prove that $A \wedge
  B$ is affirmative (your choice).
\item[c.] Prove that if $A$ is {\em refutative} and $B$
  is affirmative, then $A \supset B$ is affirmative.
\item[d.] Prove that if $A$ is affirmative, then so is
  $\exists x . A$.
\item[e.] {\bf (2 points extra credit)} \\ Find an example of a formula
  that is {\em not} affirmative.  (Proof required for credit.)
\end{itemize}

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