\documentclass[10pt]{article}
\usepackage{enumerate}
\usepackage{amsmath}
\usepackage{amssymb}
\newcommand{\entails}{\vDash}
\newcommand{\infers}{\vdash}
\renewcommand{\implies}{\Rightarrow}
\title{Artificial Intelligence - Lab 4}
\author{your name\\
\small \texttt{your\_email@ru.is}
}
\date{\today}
\begin{document}
\maketitle
\section*{Inference rules}
\begin{eqnarray}
\{ \alpha \implies \beta, \alpha \} & \infers & \beta\\
\{ \alpha \implies \beta, \neg \beta \} & \infers & \neg \alpha\\
\{ \alpha \land \beta \} & \infers & \alpha\\
\{ \alpha , \beta \} & \infers & \alpha \land \beta\\
\{ \alpha \} & \infers & \alpha \lor \beta\\
\{ \alpha \lor \beta, \neg \alpha \} & \infers & \beta\\
\alpha \Leftrightarrow \beta & \equiv & \beta \Leftrightarrow \alpha\\
\alpha \Leftrightarrow \beta & \equiv & (\alpha \implies \beta) \land \beta \implies \alpha\\
\alpha \implies \beta & \equiv & \neg \alpha \lor \beta\\
\alpha \land \beta & \equiv & \beta \land \alpha\\
\alpha \lor \beta & \equiv & \beta \lor \alpha\\
\neg (\alpha \land \beta) & \equiv & \neg \alpha \lor \neg \beta\\
\neg (\alpha \lor \beta) & \equiv & \neg \alpha \land \neg \beta\\
\neg \neg \alpha & \equiv & \alpha
\end{eqnarray}
\section*{Task 1 - Propositional symbols}
\begin{tabular}{l|l}
% Inside a tabular environment like this, "&" brings you
% to the next column in the table and "\\" ends the line
% of the table.
% To use any mathematical (or logical) symbols you need to
% be in math mode, which you start and end with "$".
% Look at the inference rules above to see how to produce
% logical symbols for "and", "or", "not", etc.
symbol & meaning \\
\hline
% TODO: remove this example and add your answer here
$R$ & it rains today \\
... & ... \\
\end{tabular}
\section*{Task 2 - Knowledge base}
\begin{tabular}{r|l}
\# & sentence \\
\hline
% TODO: remove this example and add your answer here
1 & $P \implies Q$ \\
2 & $P$ \\
\end{tabular}
\section*{Task 3 - Inference}
\begin{tabular}{r|l|l}
\# & inference rule & inferred sentence \\
\hline
% TODO: remove this example and add your answer here
1-2 & from KB & see above \\
3 & rule 1 applied to 1+2 & $Q$ \\
\end{tabular}
\end{document}
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