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\renewcommand{\not}{\neg}
\renewcommand{\and}{\wedge}
\renewcommand{\or}{\vee}
\newcommand{\impl}{\Rightarrow}
\newcommand{\lequiv}{\Leftrightarrow} %equivalence as logical connective
\newcommand{\all}{\forall}
\newcommand{\exi}{\exists}
\newcommand{\elc}{\models}  % semantic logical consequence
\newcommand{\ylc}{\vdash}  % syntactic logical consequence
%\equiv is already defined for === semantic equivalence
\newcommand{\union}{\cup}
\newcommand{\intersect}{\cap}
\newcommand{\true}{{\mathbb{T}}}
\newcommand{\false}{{\mathbb{F}}}
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\begin{document}

{\bf Computational Logic, WS 2018.
Homework 8, given Dec. 10, due Jan. 7.}

\bigskip

\noindent
1.
Construct inductively the function which gives the list of free variables of a formula in first order predicate logic.

\bigskip

\noindent
2.
Compute the truth value:
$\langle P[x] \impl P[f[x]] \rangle_I^{\{x \arrow 1\}}$
with respect to the interpretation

$I:\ \{
D_I = \{0, 1\},\ 
a_I = 0,\ 
f_I = \{0 \arrow 1, 1 \arrow 0\},\ 
P_I = \{0 \arrow \false, 1 \arrow \true\}\}$.

\bigskip

\noindent
3.
Investigate the interaction between the existential quantifier and the logical
connectives $\and, \or$. Find a counterexample for the non-equivalence.

\bigskip

\noindent
4.
Prove that if the formula $P[a]$ is satisfiable, then the formula
$\exi_x P[x]$ is also satisfiable.

\bigskip

\noindent
5.
Prove that if the formula $\all_x \exi_y P[x,y]$ is satisfiable, then the formula
$\all_x P[x,f[x]]$ is also satisfiable.


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