\chapter{The unwind Library}

This document describes the facilities provided by the \ml{unwind} library
for the HOL system~\cite{description}. The library provides conversions and
rules for unfolding, unwinding and pruning device implementations (logical
representations of hardware). For a detailed description of these techniques,
see~\cite{HVusingHOL}.

Most of the functions fall into one of five groups. The first group consists
of conversions and inference rules for moving universal quantifiers up and
down through conjunctions; they have names beginning with either
\ml{CONJ\_FORALL} or \ml{FORALL\_CONJ}. The second group of functions are for
unfolding, that is expanding sub-components using their definitions. The names
of these begin with \ml{UNFOLD}. The functions in the third group perform
unwinding and have names beginning with \ml{UNWIND}. The fourth group of
functions prune internal lines that have been unwound. Their names begin with
\ml{PRUNE}. The final group of functions combine unfolding, unwinding and
pruning. They have names beginning with \ml{EXPAND}.

I have tried to make the behaviour of the functions uniform. The conversions
apply to the smallest term possible, to provide maximum flexibility. The
inference rules, on the other hand, are designed to apply to the definition
of a hardware component. They expect to be given a theorem of the form:

\begin{small}\begin{verbatim}
   |- !x1 ... xn. DEVICE (x1,...,xn) = ?l1 ... lm. t1 /\ ... /\ tp
\end{verbatim}\end{small}


\section{Using the library}

The \ml{unwind} library can be loaded into a \HOL\ session using the function
\ml{load\_library}\index{load\_library@{\ptt load\_library}} (see the \HOL\
manual for a general description of library loading). The first action in the
load sequence initiated by \ml{load\_library} is to update the \HOL\
help\index{help!updating search path} search path. The help search path is
updated with a pathname to online help files for the \ML\ functions in the
library. After updating the help search path, the \ML\ functions in the
library are loaded into \HOL.

The following session shows how the \ml{unwind} library may be loaded using
\ml{load\_library}:

\setcounter{sessioncount}{1}
\begin{session}\begin{verbatim}
#load_library `unwind`;;
Loading library `unwind` ...
Updating help search path
..................................
Library `unwind` loaded.
() : void
\end{verbatim}\end{session}

We now illustrate the use of the library on the parity-checker example.
Firstly, we begin a new theory:

\begin{session}\begin{verbatim}
#new_theory `PARITY`;;
() : void
\end{verbatim}\end{session}

\vfill

\noindent
We define the sub-components used:

\vfill

\begin{session}\begin{verbatim}
#let ONE_DEF =
# new_definition
#  (`ONE_DEF`, "ONE(out:num->bool) = !t. out t = T");;
ONE_DEF = |- !out. ONE out = (!t. out t = T)
\end{verbatim}\end{session}

\vfill

\begin{session}\begin{verbatim}
#let NOT_DEF =
# new_definition
#  (`NOT_DEF`, "NOT(in,out:num->bool) = !t. out t = ~(in t)");;
NOT_DEF = |- !in out. NOT(in,out) = (!t. out t = ~in t)
\end{verbatim}\end{session}

\vfill

\begin{session}\begin{verbatim}
#let MUX_DEF =
# new_definition
#  (`MUX_DEF`,
#   "MUX(sw,in1,in2,out:num->bool) =
#     !t. out t = (sw t => in1 t | in2 t)");;
MUX_DEF =
|- !sw in1 in2 out.
    MUX(sw,in1,in2,out) = (!t. out t = (sw t => in1 t | in2 t))
\end{verbatim}\end{session}

\vfill

\begin{session}\begin{verbatim}
#let REG_DEF =
# new_definition
# (`REG_DEF`, "REG(in,out:num->bool) =
#              !t. out t = ((t=0) => F | in(t-1))");;
REG_DEF =
|- !in out. REG(in,out) = (!t. out t = ((t = 0) => F | in(t - 1)))
\end{verbatim}\end{session}

\vfill

\noindent
Now we define the parity-checker implementation:

\begin{session}\begin{verbatim}
#let PARITY_IMP_DEF =
# new_definition
#  (`PARITY_IMP_DEF`,
#   "PARITY_IMP(in,out) =
#    ?l1 l2 l3 l4 l5.
#     NOT(l2,l1) /\ MUX(in,l1,l2,l3) /\ REG(out,l2) /\
#     ONE l4     /\ REG(l4,l5)       /\ MUX(l5,l3,l4,out)");;
PARITY_IMP_DEF =
|- !in out.
    PARITY_IMP(in,out) =
    (?l1 l2 l3 l4 l5.
      NOT(l2,l1) /\
      MUX(in,l1,l2,l3) /\
      REG(out,l2) /\
      ONE l4 /\
      REG(l4,l5) /\
      MUX(l5,l3,l4,out))
\end{verbatim}\end{session}

\noindent
The function \ml{EXPAND\_AUTO\_RIGHT\_RULE} can be used to unfold, unwind and
prune the body of this definition:

\begin{session}\begin{verbatim}
#EXPAND_AUTO_RIGHT_RULE [ONE_DEF;NOT_DEF;MUX_DEF;REG_DEF] PARITY_IMP_DEF;;
|- !in out.
    PARITY_IMP(in,out) =
    (!t.
      out t =
      (((t = 0) => F | T) =>
       (in t =>
        ~((t = 0) => F | out(t - 1)) |
        ((t = 0) => F | out(t - 1))) |
       T))
\end{verbatim}\end{session}


\section{Automatic unwinding}

\def\putbox(#1,#2){\put(#1,#2){\framebox(2,2){}}}

Hardware implementations often contain feedbacks. This presents a problem
when trying to unwind and prune the internal lines in the logical
representation. The mutual dependencies between lines can cause a brute-force
unwind to loop indefinitely. To avoid this one has to be selective about
which lines to unwind. The tools in the \ml{unwind} library allow the user
to be selective in this way. However, it is possible for the machine itself to
be selective. The function \ml{UNWIND\_AUTO\_CONV} attempts to analyze the
dependencies between lines and unwind as far as possible without looping.

Consider the following term which arises in the parity-checker example:

\begin{small}\begin{verbatim}
   "?l1 l2 l3 l4 l5.
     (!t. l1 (t:num) = ~l2 t) /\
     (!t. l3 t = (in t => l1 t | l2 t)) /\
     (!t. l2 t = ((t = 0) => F | out (t - 1))) /\
     (!t. l4 t = T) /\
     (!t. l5 t = ((t = 0) => F | l4 (t - 1))) /\
     (!t. out t = (l5 t => l3 t | l4 t))"
\end{verbatim}\end{small}

\noindent
We can represent the dependencies of the lines using a directed graph:

{\setlength{\unitlength}{4mm}
\begin{center}
\begin{picture}(14,10)(0,0)
\put(0,2){\makebox(2,2){\small{\tt l2}}}
\put(4,2){\makebox(2,2){\small{\tt l1}}}
\put(8,4){\makebox(2,2){\small{\tt l3}}}
\put(12,6){\makebox(2,2){\small{\tt out}}}
\put(2,8){\makebox(2,2){\small{\tt l4}}}
\put(6,8){\makebox(2,2){\small{\tt l5}}}

\put(2,3){\vector(1,0){2}}
\put(2,3){\vector(3,1){6}}
\put(6,3){\vector(1,1){2}}
\put(10,5){\vector(1,1){2}}
\put(4,9){\vector(1,0){2}}
\put(4,9){\vector(4,-1){8}}
\put(8,9){\vector(2,-1){4}}

\put(13,6){\line(0,-1){6}}
\put(13,0){\line(-1,0){12}}
\put(1,0){\vector(0,1){2}}
\end{picture}
\end{center}}

\noindent
which can in turn be represented by the following list:

\begin{small}\begin{verbatim}
   l1, [l2]
   l3, [l1;l2]
   l2, [out]
   l4, []
   l5, [l4]
   out,[l5;l3;l4]
\end{verbatim}\end{small}

Since we wish to eliminate the internal lines, we want to be left with a
recursive equation for {\small\verb%out%} in terms of itself. We can do this
be `breaking the loop' at {\small\verb%out%}, giving the following structure:

\begin{small}\begin{verbatim}
   l1, [l2]
   l3, [l1;l2]
   l2, []
   l4, []
   l5, [l4]
\end{verbatim}\end{small}

\noindent
Note that {\small\verb%out%} has been removed from the structure. From the
graph we can see that {\small\verb%l2%} and {\small\verb%l4%} do not depend
on any internal lines. They can therefore be used to unwind without any risk
of looping. They can be recognized in the datastructure by the fact that their
corresponding dependency lists are empty.

Once we have unwound with {\small\verb%l2%} and {\small\verb%l4%} they can be
removed from the datastructure:

\begin{small}\begin{verbatim}
   l1, []
   l3, [l1]
   l5, []
\end{verbatim}\end{small}

\noindent
We now see that {\small\verb%l1%} and {\small\verb%l5%} can be unwound to give:

\begin{small}\begin{verbatim}
   l3, []
\end{verbatim}\end{small}

\noindent
Unwinding {\small\verb%l3%} then leaves us with the required recursive
equation for {\small\verb%out%}.

The problem with the approach just described is that it only unwinds fully if
there is at most one loop in the circuit, and the output is in that loop. We
can be a bit more general. Consider the circuit:

\vfill

{\setlength{\unitlength}{4mm}
\begin{center}
\begin{picture}(4,18)(0,0)
\put(1,0){\makebox(2,2){\small{\tt out}}}
\putbox(1,4)
\put(0,6){\makebox(2,2){\small{\tt l2}}}
\putbox(1,8)
\put(0,10){\makebox(2,2){\small{\tt l1}}}
\putbox(1,12)
\put(1,16){\makebox(2,2){\small{\tt in}}}

\put(2,16){\vector(0,-1){2}}
\put(2,12){\vector(0,-1){2}}
\put(2,8){\vector(0,-1){2}}
\put(2,4){\vector(0,-1){2}}

\put(2,7){\circle*{0.2}}
\put(2,7){\line(1,0){2}}
\put(4,7){\line(0,1){8}}
\put(4,15){\line(-1,0){1}}
\put(3,15){\vector(0,-1){1}}
\end{picture}
\end{center}}

\vfill

\noindent
represented by the graph:

\vfill

\begin{small}\begin{verbatim}
   l1, [l2]
   l2, [l1]
   out,[l2]
\end{verbatim}\end{small}

\vfill

\noindent
There are no lines with an empty dependency list, and eliminating
{\small\verb%out%} will not help because it is not in the loop. However, if we
break the loop at {\small\verb%l2%} we can unwind {\small\verb%l1%}. This will
leave us with a recursive equation for {\small\verb%l2%} and an equation for
{\small\verb%out%} in terms of {\small\verb%l2%}. This is the best that we can
do, and it is now up to the user to deal with the recursive equation.

Now let's consider an example with more than one loop:

{\setlength{\unitlength}{4mm}
\begin{center}
\begin{picture}(12,30)(0,0)
\put(1,0){\makebox(2,2){\small{\tt out}}}
\putbox(1,4)
\put(0,6){\makebox(2,2){\small{\tt l5}}}
\putbox(1,8)
\put(0,10){\makebox(2,2){\small{\tt l4}}}
\putbox(1,12)
\put(0,14){\makebox(2,2){\small{\tt l3}}}
\putbox(1,16)
\putbox(5,16)
\put(0,18){\makebox(2,2){\small{\tt l2}}}
\put(6,18){\makebox(2,2){\small{\tt l6}}}
\putbox(1,20)
\putbox(9,20)
\put(0,22){\makebox(2,2){\small{\tt l1}}}
\put(10,22){\makebox(2,2){\small{\tt l7}}}
\putbox(1,24)
\put(1,28){\makebox(2,2){\small{\tt in}}}

\put(2,28){\vector(0,-1){2}}
\put(2,24){\vector(0,-1){2}}
\put(2,20){\vector(0,-1){2}}
\put(2,16){\vector(0,-1){2}}
\put(2,12){\vector(0,-1){2}}
\put(2,8){\vector(0,-1){2}}
\put(2,4){\vector(0,-1){2}}

\put(2,11){\circle*{0.2}}
\put(2,11){\line(1,0){4}}
\put(6,11){\vector(0,1){5}}
\put(6,18){\line(0,1){1}}
\put(6,19){\line(-1,0){3}}
\put(3,19){\vector(0,-1){1}}
\put(2,7){\circle*{0.2}}
\put(2,7){\line(1,0){8}}
\put(10,7){\vector(0,1){13}}
\put(10,22){\line(0,1){1}}
\put(10,23){\line(-1,0){7}}
\put(3,23){\vector(0,-1){1}}
\end{picture}
\end{center}}

\noindent
We could unwind {\small\verb%l1%} but then we would get stuck. If we break at
{\small\verb%l2%} or {\small\verb%l7%} we will still get stuck because of the
inner loop. If we break at {\small\verb%l5%} we can unwind {\small\verb%l7%},
but then get stuck. If we break at {\small\verb%l6%} we get stuck because of
the outer loop. However, if we break at {\small\verb%l3%} or {\small\verb%l4%}
both loops are broken and we can unwind fully to leave a recursive equation
for either {\small\verb%l3%} or {\small\verb%l4%} and an equation for
{\small\verb%out%} in terms of that line. So, the choice of where to break a
loop may determine how far the unwinding can go.

\ml{UNWIND\_AUTO\_CONV} attempts to break every loop in the circuit using the
minimum number of breaks, so that there are as few equations left as possible.
The function also gives priority to non-internal lines when determining where
to break, so that if possible the recursive equations are in terms of these
lines.

The algorithm used determines from the term a list of line variables. Each
line variable has a right-hand side of an equation associated with it. The
free variables in each right-hand side are computed and those that are also
line variables are placed in the dependency list for the corresponding line.
From the dependency structure, the loops are determined. Lines are then
eliminated so that all loops are broken.

A study of the following circuit reveals why {\em all\/} loops have to be
broken. If not all loops are broken, then a remaining loop can make the
breaking of other loops fruitless.

{\setlength{\unitlength}{4mm}
\begin{center}
\begin{picture}(12,30)(0,0)
\put(1,0){\makebox(2,2){\small{\tt out}}}
\putbox(1,4)
\put(0,6){\makebox(2,2){\small{\tt l5}}}
\putbox(1,8)
\put(0,10){\makebox(2,2){\small{\tt l4}}}
\putbox(1,12)
\putbox(9,12)
\put(0,14){\makebox(2,2){\small{\tt l3}}}
\put(10,14){\makebox(2,2){\small{\tt l7}}}
\putbox(1,16)
\putbox(5,16)
\put(0,18){\makebox(2,2){\small{\tt l2}}}
\put(6,18){\makebox(2,2){\small{\tt l6}}}
\putbox(1,20)
\put(0,22){\makebox(2,2){\small{\tt l1}}}
\putbox(1,24)
\put(1,28){\makebox(2,2){\small{\tt in}}}

\put(2,28){\vector(0,-1){2}}
\put(2,24){\vector(0,-1){2}}
\put(2,20){\vector(0,-1){2}}
\put(2,16){\vector(0,-1){2}}
\put(2,12){\vector(0,-1){2}}
\put(2,8){\vector(0,-1){2}}
\put(2,4){\vector(0,-1){2}}

\put(2,11){\circle*{0.2}}
\put(2,11){\line(1,0){4}}
\put(6,11){\vector(0,1){5}}
\put(6,18){\line(0,1){1}}
\put(6,19){\line(-1,0){3}}
\put(3,19){\vector(0,-1){1}}
\put(2,7){\circle*{0.2}}
\put(2,7){\line(1,0){8}}
\put(10,7){\vector(0,1){5}}
\put(10,14){\line(0,1){1}}
\put(10,15){\line(-1,0){7}}
\put(3,15){\vector(0,-1){1}}
\end{picture}
\end{center}}

\noindent
The dependency structure for the circuit is:

\begin{small}\begin{verbatim}
   l1, []
   l2, [l1]
   l3, [l2;l6]
   l4, [l3;l7]
   l5, [l4]
   l6, [l4]
   l7, [l5]
   out,[l5]
\end{verbatim}\end{small}

\noindent
The loops for the circuit are:

\begin{small}\begin{verbatim}
   [l3;l4;l6]
   [l4;l5;l7]
\end{verbatim}\end{small}

\noindent
Both loops can be broken by eliminating {\small\verb%l4%}:

\begin{small}\begin{verbatim}
   l1, []
   l2, [l1]
   l3, [l2;l6]
   l5, []
   l6, []
   l7, [l5]
   out,[l5]
\end{verbatim}\end{small}

\noindent
We can now unwind {\small\verb%l1%}, {\small\verb%l5%} and {\small\verb%l6%}:

\begin{small}\begin{verbatim}
   l2, []
   l3, [l2]
   l7, []
   out,[]
\end{verbatim}\end{small}

\noindent
and then unwind {\small\verb%l2%}, {\small\verb%l7%} and {\small\verb%out%},
followed by unwinding with {\small\verb%l3%} to yield a recursive equation for
{\small\verb%l4%} and all other equations in terms of {\small\verb%l4%}. All
the internal lines except for {\small\verb%l4%} can be pruned. This leaves
equations for {\small\verb%l4%} and {\small\verb%out%} only.

The technique does not always yield a single recursive equation. Mutual
recursion is also possible. This is illustrated by the following example:

{\setlength{\unitlength}{4mm}
\begin{center}
\begin{picture}(10,30)(0,0)
\put(1,0){\makebox(2,2){\small{\tt out}}}
\putbox(1,4)
\put(0,6){\makebox(2,2){\small{\tt l5}}}
\putbox(1,8)
\put(0,10){\makebox(2,2){\small{\tt l4}}}
\putbox(1,12)
\putbox(5,10)
\put(0,14){\makebox(2,2){\small{\tt l3}}}
\put(6,12){\makebox(2,2){\small{\tt l7}}}
\putbox(1,16)
\put(0,18){\makebox(2,2){\small{\tt l2}}}
\putbox(1,20)
\putbox(5,18)
\put(0,22){\makebox(2,2){\small{\tt l1}}}
\put(6,20){\makebox(2,2){\small{\tt l6}}}
\putbox(1,24)
\put(1,28){\makebox(2,2){\small{\tt in}}}

\put(2,28){\vector(0,-1){2}}
\put(2,24){\vector(0,-1){2}}
\put(2,20){\vector(0,-1){2}}
\put(2,16){\vector(0,-1){2}}
\put(2,12){\vector(0,-1){2}}
\put(2,8){\vector(0,-1){2}}
\put(2,4){\vector(0,-1){2}}

\put(2,15){\circle*{0.2}}
\put(2,15){\line(1,0){4}}
\put(6,15){\vector(0,1){3}}
\put(6,20){\line(0,1){2}}
\put(6,22){\vector(-1,0){3}}
\put(2,7){\circle*{0.2}}
\put(2,7){\line(1,0){4}}
\put(6,7){\vector(0,1){3}}
\put(6,12){\line(0,1){2}}
\put(6,14){\vector(-1,0){3}}
\put(6,7){\circle*{0.2}}
\put(6,7){\line(1,0){4}}
\put(10,7){\line(0,1){16}}
\put(10,23){\line(-1,0){7}}
\put(3,23){\vector(0,-1){1}}
\end{picture}
\end{center}}

There are three loops, but breaking at {\small\verb%l2%} and {\small\verb%l4%}
is sufficient to break all the loops. The result is three equations: an
equation for {\small\verb%l2%} in terms of itself and {\small\verb%l4%}, an
equation for {\small\verb%l4%} in terms of itself and {\small\verb%l2%}, and
an equation for {\small\verb%out%} in terms of {\small\verb%l4%}. So, it can
be seen that the loop analysis technique used by \ml{UNWIND\_AUTO\_CONV} does
not eliminate loops; it simply `shrinks' them.