You can do this:
\documentclass[border=5pt]{standalone}
\usepackage{tikz}
\usetikzlibrary{positioning,decorations.markings,automata,angles,quotes}
\tikzset{
myneg/.style={
decoration={
markings,
mark=at position 0.5 with
{\draw (0,0) arc (0:180:#1) (0,0) arc (180:360:#1);}
},
postaction=decorate
},
myneg/.default=6pt,
mynode/.style={
draw,
circle,
minimum size=20pt
}
}
\begin{document}
\begin{tikzpicture}
\node[mynode] (a) at (0,0) {$a$};
\node[mynode] (b) at (3,0) {$b$};
\draw
(a) -- (b) node[above=8pt,midway,font=\scshape] {identity};
\begin{scope}[xshift=5cm]
\node[mynode] (a) at (0,0) {$a$};
\node[mynode] (b) at (3,0) {$b$};
\draw[myneg]
(a) -- (b) node[above=8pt,midway,font=\scshape] {not};
\end{scope}
\begin{scope}[yshift=-4cm]
\node[mynode] (a) at (0,2) {$a$};
\node[mynode] (b) at (0,0) {$b$};
\node[mynode] (c) at (0,-2) {$c$};
\node[mynode] (d) at (3,0) {$d$};
\coordinate (aux) at (d.west);
\foreach \Value in {a,b,c}
\draw (\Value) -- (aux);
\path
pic[draw,angle radius=35pt,"\raisebox{20pt}{$\lor$}",
angle eccentricity=1.25]
{angle=a--aux--c};
\path (a) -- (d) node[above=12pt,midway,font=\scshape] {or};
\end{scope}
\begin{scope}[xshift=5cm,yshift=-4cm]
\node[mynode] (a) at (0,2) {$a$};
\node[mynode] (b) at (0,-2) {$b$};
\node[mynode] (c) at (3,0) {$c$};
\foreach \Value in {a,b}
\draw (\Value) -- (c.west);
\path
(a) -- (c)
node[above=12pt,midway,font=\scshape] {and};
\node[left=of c,font=\scshape] {$\land$};
\end{scope}
\end{tikzpicture}
\end{document}

The negation was obtained using a decoration (with the help of the decorations.markings library) and the arc for the "OR" was produced with the help of the angles library.
Of course, depending on the actual kind of application, you could use additional libraries for the nodes, such as automata or chains.
An additional example showing the negation and or paths in action:
\documentclass[border=5pt]{standalone}
\usepackage{tikz}
\usetikzlibrary{positioning,decorations.markings,automata,angles,quotes}
\tikzset{
myneg/.style={
decoration={
markings,
mark=at position 0.5 with
{\draw (0,0) arc (0:180:#1) (0,0) arc (180:360:#1);}
},
postaction=decorate
},
myneg/.default=6pt,
mynode/.style={
draw,
circle,
minimum size=20pt
}
}
\begin{document}
\begin{tikzpicture}[node distance=1cm and 2cm]
\node[mynode] (b) at (0,0) {$b$};
\node[mynode,above=of b] (a) {$a$};
\node[mynode,below=of b] (c) {$c$};
\node[mynode,right=of b] (d) {$d$};
\node[mynode,above right=of d] (e) {$e$};
\node[mynode,below right=of d] (f) {$f$};
\coordinate (aux) at (d.west);
\foreach \Value in {a,b,c}
\draw (\Value) -- (aux);
\path
pic[draw,angle radius=35pt,"\raisebox{20pt}{$\lor$}",
angle eccentricity=1.25]
{angle=a--aux--c};
\draw[myneg]
(d) -- (e);
\draw[myneg]
(d) -- (f);
\end{tikzpicture}
\end{document}

automatalibrary?pics?graphslibrary and declare some new standard structures? – cfr Nov 08 '15 at 22:03pics orgraphs. It depends partly on the details of how you are going to need to combine these 'atomic' elements. – cfr Nov 08 '15 at 22:54