% This file is part of the Stanford GraphBase (c) Stanford University 1992
\def\title{GB\_\thinspace GATES}
@i boilerplate.w %<< legal stuff: PLEASE READ IT BEFORE MAKING ANY CHANGES!
\prerequisite{GB\_\thinspace GRAPH}
@* Introduction. This GraphBase module provides six external subroutines:
$$\vbox{\hsize=.8\hsize \everypar{\hangindent3em}
\noindent|risc|, a routine that creates a directed acyclic graph based on the
logic of a simple RISC computer;\par
\noindent|prod|, a routine that creates a directed acyclic graph based on the
logic of parallel multiplication circuits;\par
\noindent|print_gates|, a routine that outputs a symbolic representation of
such directed acyclic graphs;\par
\noindent|gate_eval|, a routine that evaluates such directed acyclic graphs by
assigning boolean values to each gate;\par
\noindent|partial_gates|, a routine that extracts a subgraph by assigning
random values to some of the input gates;\par
\noindent|run_risc|, a routine that can be used to play with the output
of |risc|.}$$
Examples of the use of these routines can be found in the demo programs
|take_risc| and |multiply|.
@(gb_gates.h@>=
extern Graph *risc(); /* make a network for a microprocessor */
extern Graph *prod(); /* make a network for high-speed multiplication */
extern void print_gates(); /* write a network to standard output file */
extern int gate_eval(); /* evaluate a network */
extern Graph *partial_gates(); /* reduce network size */
extern int run_risc(); /* simulate the microprocessor */
extern unsigned risc_state[]; /* the output of |run_risc| */
@ The directed acyclic graphs produced by |gb_gates| are GraphBase
graphs with special conventions related to logical networks. Each vertex
represents a gate of a network, and utility field |val| is a boolean
value associated with that gate. Utility field |typ| is an ASCII code
that tells what kind of gate is present:
{\advance\parindent 2em
\smallskip
\item{|'I'|} denotes an input gate, whose value is specified externally.
\smallskip
\item{|'&'|} denotes an \.{AND} gate, whose value is the logical {\sc AND} of
two or more previous gates (namely, 1 if all those gates are~1, otherwise~0).
\smallskip
\item{|'|'|} denotes an \.{OR} gate, whose value is the logical {\sc OR} of
two or more previous gates (namely, 0 if all those gates are~0, otherwise~1).
\smallskip
\item{|'^'|} denotes an \.{XOR} gate, whose value is the logical {\sc
EXCLUSIVE-OR} of two or more previous gates (namely, their sum modulo~2).
\smallskip
\item{|'~'|} denotes an inverter, whose value is the logical complement of
the value of a single previous gate.
\smallskip
\item{|'L'|} denotes a latch, whose value depends on past history; it is
the value that was assigned to a subsequent gate when the network was most
recently evaluated. Utility field |alt| points to that subsequent gate.
\smallskip}\noindent
Latches can be used to include ``state'' information in a circuit; for example,
they correspond to registers of the RISC machine constructed by |risc|.
The |prod| procedure does not use latches.
The vertices of the directed acyclic graph appear in a special ``topological''
order convenient for evaluation: All the input gates come first, followed
by all the latches; then come the other types of gates, whose values are
computed from their predecessors. The arcs of the graph run from each gate
to its arguments, and all arguments to a gate precede that gate.
If |g| points to such a graph of gates, the utility field |g->outs| points to
a list of |Arc| records, denoting ``outputs'' that might be used in
certain applications. For example, the outputs of the graphs
created by |prod| correspond to the bits of the product of the numbers
represented in the input gates.
A special convention is used so that the routines will support partial
evaluation: The |tip| fields in the output list either point to a
vertex or hold one of the constant values 0 or~1 when regarded as an
unsigned long integer.
@d val x.i /* the field containing a boolean value */
@d typ y.i /* the field containing the gate type */
@d alt z.v /* the field pointing to another related gate */
@d outs z.a /* the field pointing to the list of output gates */
@d is_boolean(v) ((unsigned long)(v)<=1) /* is a |tip| field constant? */
@d the_boolean(v) ((long)(v)) /* if so, this is its value */
@d tip_value(v) (is_boolean(v)? the_boolean(v): (v)->val)
@d AND '&'
@d OR '|'
@d NOT '~'
@d XOR '^'
@#
@f Graph int /* |gb_graph| defines the |Graph| type and a few others */
@f Vertex int
@f Arc int
@f Area int
@(gb_gates.h@>=
#define val @t\quad@> x.i /* the definitions are repeated in the header file */
#define typ @t\quad@> y.i
#define alt @t\quad@> z.v
#define outs @t\quad@> z.a
#define is_boolean(v) @t\quad@> ((unsigned long)(v)<=1)
#define the_boolean(v) @t\quad@> ((long)(v))
#define tip_value(v) @t\quad@> (is_boolean(v)? the_boolean(v): (v)->val)
#define AND @t\quad@> '&'
#define OR @t\quad@> '|'
#define NOT @t\quad@> '~'
#define XOR @t\quad@> '^'
@ Let's begin with the |gate_eval| procedure, because it is quite simple
and because it illustrates the conventions just explained. Given a gate
graph |g| and optional pointers |in_vec| and |out_vec|, the procedure
|gate_eval| will assign values to each gate of~|g|. If |in_vec| is
non-null, it should point to a string of characters, each |'0'| or~|'1'|,
that will be assigned to the first gates of the network, in order;
otherwise |gate_eval| assumes that all input gates have already received
appropriate values and it will not change them. New values are computed for
each gate after the bits of |in_vec| have been consumed.
If |out_vec| is non-null, it should point to a memory area capable of
receiving |m+1| characters, where |m| is the number of outputs of~|g|;
a string containing the respective output values will be deposited there.
If |gate_eval| encounters an unknown gate type, it terminates execution
prematurely and returns the value |-1|. Otherwise it returns~0.
@<The |gate_eval| routine@>=
int gate_eval(g,in_vec,out_vec)
Graph *g; /* graph with gates as vertices */
char *in_vec; /* string for input values, or |NULL| */
char *out_vec; /* string for output values, or |NULL| */
{@+register Vertex *v; /* the current vertex of interest */
Vertex *u, *uu; /* additional vertices being examined */
register Arc *a; /* the current arc of interest */
register char t; /* boolean value being computed */
if (!g) return -2; /* no graph supplied! */
v=g->vertices;
if (in_vec) @<Read a sequence of input values from |in_vec|@>;
for (; v<g->vertices+g->n; v++) {
switch (v->typ) { /* branch on type of gate */
case 'I': continue; /* this input gate's value should be externally set */
case 'L': t=v->alt->val;@+break;
@t\4\4@>@;@+@<Compute the value |t| of a classical logic gate@>;
default: return -1; /* unknown gate type! */
}
v->val=t; /* assign the computed value */
}
if (out_vec) @<Store the sequence of output values in |out_vec|@>;
return 0;
}
@ @<Read a sequence...@>=
while (*in_vec && v<g->vertices+g->n)
(v++)->val = *in_vec++ - '0';
@ @<Store the sequence of output values in |out_vec|@>=
{
for (a=g->outs; a; a=a->next)
*out_vec++='0'+tip_value(a->tip);
*out_vec=0; /* terminate the string */
}
@ @<Compute the value |t| of a classical logic gate@>=
case AND: t=1;
for (a=v->arcs; a; a=a->next)
t &= a->tip->val;
break;
case OR: t=0;
for (a=v->arcs; a; a=a->next)
t |= a->tip->val;
break;
case XOR: t=0;
for (a=v->arcs; a; a=a->next)
t ^= a->tip->val;
break;
case NOT: t=1-v->arcs->tip->val;
break;
@ Here now is an outline of the entire |gb_gates| module, as seen by
the \Cee\ compiler:
@p
#include "gb_flip.h" /* we will use the |gb_flip| routines for random numbers */
#include "gb_graph.h" /* and we will use the |gb_graph| data structures */
@#
@<Private variables@>@;
@<Global variables@>@;
@<Internal subroutines@>@;
@<The |gate_eval| routine@>@;
@<The |print_gates| routine@>@>;
@<The |risc| routine@>@;
@<The |run_risc| routine@>@;
@<The |prod| routine@>@;
@<The |partial_gates| routine@>@;
@* The RISC netlist. The subroutine call `|risc(regs)|' creates a
gate graph having |regs| registers; the value of |regs| must be
between 2 and~16, inclusive, otherwise |regs| is set to~16.
This gate graph describes the circuitry for a small RISC computer, defined
below. The total number of gates turns out to be |1400+115*regs|;
thus it lies between 1630 (when |regs=2|) and 3240 (when |regs=16|).
Exclusive-or gates are not used; the effect of xoring is obtained where
needed by means of {\sc AND}s, {\sc OR}s, and inverters.
If |risc| cannot do its thing, it returns |NULL| (\.{NULL})
and sets |panic_code|
to indicate the problem. Otherwise |risc| returns a pointer to the graph.
@d panic(c) @+{@+panic_code=c;@+gb_alloc_trouble=0;@+return NULL;@+}
@<The |risc| routine@>=
Graph *risc(regs)
unsigned regs; /* number of registers supported */
{@+@<Local variables for |risc|@>@;
@#
@<Initialize |new_graph| to an empty graph of the appropriate size@>;
@<Add the RISC data to |new_graph|@>;
if (gb_alloc_trouble) {
gb_recycle(new_graph);
panic(alloc_fault); /* oops, we ran out of memory somewhere back there */
}
return new_graph;
}
@ @<Local variables for |risc|@>=
Graph *new_graph; /* the graph constructed by |risc| */
register int k,r; /* all-purpose indices */
@ This RISC machine works with 16-bit registers and 16-bit data words.
It cannot write into memory, but it assumes the existence of an
external read-only memory. The circuit has 16 outputs, representing
the 16 bits of a memory address register; it also has 17 inputs, the
last 16 of which are supposed to be set to the contents of the memory
address computed on the previous cycle. Thus, we can run the machine
by accessing memory between calls of |gate_eval|. The first input
bit, called \.{RUN}, is normally set to~1; if it is~0, the other
inputs are effectively ignored and all registers and outputs will be
cleared to~0. Input bits for the memory appear in ``little-endian
order,'' i.e., least significant bit first; but the output bits for
the memory address register appear in ``big-endian order,'' i.e., most
significant bit first.
Words read from memory are interpreted as instructions having the following
format:
$$\vbox{\offinterlineskip
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&&\multispan3\hfill\.{OP}\hfill&&\multispan3\hfill\.{A}\hfill
&&\multispan7\hfill\.{SRC}\hfill&\cr
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\\15&\\14&\\13&\\12&\\11&\\10&\\9&\\8&\\7&\\6&\\5&\\4&\\3&\\2&\\1&%
\\0&\omit\cr}}$$
The \.{SRC} and \.A fields specify a ``source'' value.
If $\.A=0$, the source is \.{SRC}, treated as a 16-bit signed
number between $-8$ and $+7$ inclusive.
If $\.A=1$, the source is the contents of register \.{DST} plus the
(signed) value of \.{SRC}. If $\.A=2$, the source is the contents of register
\.{SRC}. And if $\.A=3$, the source is the contents of the memory location
whose address is the contents of register \.{SRC}. Thus, for example,
if $\.{DST}=3$ and $\.{SRC}=10$, and if \.{r3} contains 17 while \.{r10}
contains 1009, the source value will be $-6$ if $\.A=0$,
or $17-6=11$ if $\.A=1$, or 1009 if $\.A=2$, or the contents of memory location
1009 if $\.A=3$.
The \.{DST} field specifies the number of the destination register. This
register receives a new value based on its previous value and the source
value, as prescribed by the operation defined in the \.{OP} and \.{MOD}
fields. For example, when $\.{OP}=0$, a general logical operation is
performed: Suppose the bits of \.{MOD} are called $\mu_{11}\mu_{10}\mu_{01}
\mu_{00}$ from left to right; then if the $k$th bit of the destination register
currently is equal to~$i$ and the $k$th bit of the source value is
equal to~$j$, the general logical operator changes the $k$th bit of
the destination register to~$\mu_{ij}$. If the \.{MOD} bits are,
for example, $1010$, the source value is simply copied to the
destination register; if $\.{MOD}=0110$, an exclusive or is done;
if $\.{MOD}=0011$, the destination register is complemented and the
source value is effectively ignored.
The machine contains four status bits called \.S (sign), \.N (nonzero),
\.K (carry), and \.V (overflow). Every general logical operation sets
\.S equal to the sign of the new result transferred to the destination
register; this is bit~15, the most significant bit. A general logical
operation also sets \.N to~1 if any of the other 15 bits are~1, to~0
if all of the other bits are~0. Thus, \.S and \.N both become zero if and
only if the new result is entirely zero. Logical operations do not change
the values of \.K and~\.V; the latter are affected only by the arithmetic
operations described below.
The status of the \.S and \.N bits can be tested by using the
conditional load operator, $\.{OP}=2$: This operation loads the source
value into the destination register if and only if \.{MOD} bit
$\mu_{ij}=1$, where $i$ and~$j$ are the current values of \.S and~\.N,
respectively. For example, if $\.{MOD}=0011$, the source value is
loaded if and only if $\.S=0$, which means that the last value
affecting \.S and~\.N was greater than or equal to zero. If
$\.{MOD}=1111$, loading is always done; this is a way to move source
to destination without affecting \.S or~\.N.
A second conditional load operator, $\.{OP}=3$, is similar but
it is used for testing the status of \.K and~\.V instead of
\.S and~\.N. For example, a command having $\.{MOD}=1010$,
$\.{OP}=3$, $\.A=1$, and $\.{SRC}=1$ adds the current overflow bit to the
destination register. (Please take a moment to understand why
this is true.)
We have now described all the operations except those that
are performed when $\.{OP}=1$.
As you might expect, our machine is able to do rudimentary arithmetic,
and the general addition and subtraction operators can be found here,
together with various shift operators, depending on the value of \.{MOD}.
Eight of the $\.{OP}=1$ operations set the destination register to a shifted
version of the source value: $\.{MOD}=0$ means ``shift left~1,''
which is equivalent to multiplying the source by~2; $\.{MOD}=1$ means
``cyclic shift left~1,'' which is the same but also adding the
previous sign bit to the result; $\.{MOD}=2$ means ``shift left~4,''
which is equivalent to multiplying by~16; $\.{MOD}=3$ means ``cyclic
shift left~4''; $\.{MOD}=4$ means ``shift right~1,'' which is
equivalent to dividing the source by~2 and rounding down to the
next lower integer if there was a remainder; $\.{MOD}=5$ means
``unsigned shift right~1,'' which is the same except that the
most significant bit is always set to zero instead of retaining the
previous sign; $\.{MOD}=6$ means ``shift right~4,'' which is equivalent
to dividing the source by~16 and rounding down; $\.{MOD}=7$ means
``unsigned shift right~4.'' Each of these shift operations affects
\.S and~\.N, as in the case of logical operations. They also affect
\.K and~\.V, as follows: Shifting left sets \.K to~1 if and
only if at least one of the bits shifted off the left was nonzero,
and sets \.V to~1 if and only if the corresponding multiplication
would cause overflow.
Shifting right~1 sets \.K to the value of the bit
shifted out, and sets \.V to~0;
shifting right~4 sets \.K to the value of the last
bit shifted out, and sets \.V to the logical {\sc OR} of the other three
lost bits. The same values of \.K and \.V arise from cyclic or unsigned
shifts as from ordinary shifts.
When $\.{OP}=1$ and $\.{MOD}=8$, the source value is added to the
destination register. This sets \.S, \.N, and \.V as you would expect;
and it sets \.K to the carry you would get if treating the values as
16-bit unsigned integers. Another addition operation, having
$\.{MOD}=9$, is similar, but the current value of \.K is also added to
the result; in this case, the new value of \.N will be zero if and only if
the 15 non-sign bits of the result are zero and the previous values of
\.S and~\.N were also zero. This means
that you can use the first addition operation on the lower
halves of a 32-bit number and the second operation on the upper halves,
thereby obtaining a correct 32-bit result, with appropriate sign,
nonzero, carry, and overflow bits set.
Higher precision (48 bits, 64 bits, etc.)~can be obtained in a similar way.
When $\.{OP}=1$ and $\.{MOD}=10$, the source value is subtracted
from the destination register. Again, \.S, \.N, \.K, and \.V are set;
the \.K value in this case represents the ``borrow'' bit.
An auxiliary subtraction operation, having $\.{MOD}=11$, subtracts
also the current value of \.K, thereby allowing for correct 32-bit subtraction.
The operations for $\.{OP}=1$ and $\.{MOD}=12$, 13, and~14 are
``reserved for future expansion.'' Actually they will never change,
since this RISC chip is merely academic; if you check out the logic
below you will find that they simply set the destination register and
the four status bits all to zero.
There is one further operation, having $\.{OP}=1$ and $\.{MOD}=15$;
this is the special \.{JUMP} operation described below. It does not
affect \.S, \.N, \.K, or~\.V.
If the RISC is made with fewer than 16 registers, the higher-numbered ones
will effectively contain zero whenever their values are fetched.
But if you use them as destination registers, you will set
\.S, \.N, \.K, and~\.V as if actual numbers were being stored.
Register 0 is different from the other 15 registers: It is the location
of the current instruction. Therefore if you change the contents of
register~0, you are changing the control flow of the program. If you
do not change register~0, it automatically increases by~1.
Special treatment occurs when $\.A=3$ and $\.{SRC}=0$:
In such a case, the normal rules given above say that the source value
should be the contents of the memory location specified by register~0. But
that memory location holds the current instruction; so the machine
uses the {\it following\/} location instead, as a 16-bit source
operand. If the contents of register~0 are not changed by such a
two-word instruction, register~0 will increase by~2 instead of~1.
We have now learned everything about the machine except the operation
of the \.{JUMP} command. This command moves the source value to
register~0, thereby changing the flow of control; furthermore, if $\.{DST}\ne0$,
it also sets register \.{DST} to the location of the instruction
following the \.{JUMP}. Assembly language programmers will recognize
this as a convenient way to jump to a subroutine.
Example programs can be found in the |take_risc| module, which includes
a simple subroutine for multiplication and division.
@ A few auxiliary functions will ameliorate the task of constructing
the RISC logic. First comes a routine that ``christens'' a new gate,
assigning it a name and a type. The name is constructed from a prefix
and a serial number, where the prefix indicates the current portion of
logic being created.
@<Internal...@>=
static Vertex* new_vert(t)
char t; /* the type of the new gate */
{@+register Vertex *v;
v=next_vert++;
if (count<0) v->name=gb_save_string(prefix);
else {
sprintf(name_buf,"%s%d",prefix,count);
v->name=gb_save_string(name_buf);
count++;
}
v->typ=t;
return v;
}
@ @d start_prefix(s) strcpy(prefix,s);@+count=0
@d numeric_prefix(a,b) sprintf(prefix,"%c%d:",a,b);@+count=0;
@<Private...@>=
static Vertex* next_vert; /* the first vertex not yet assigned a name */
static char prefix[5]; /* prefix string for vertex names */
static int count; /* serial number for vertex names */
static char name_buf[100]; /* place to form vertex names */
@ Here are some trivial routines to create gates with 2, 3, or more
arguments. The arcs from a gate to its inputs are assigned length 100;
below we will assign length~1 to the arcs between an inverter and its
unique input. This makes the lengths of shortest paths in the resulting
network a bit more interesting than they would otherwise be.
@d DELAY 100
@<Internal...@>=
static Vertex* make2(t,v1,v2)
char t; /* the type of the new gate */
Vertex *v1,*v2;
{@+register Vertex *v=new_vert(t);
gb_new_arc(v,v1,DELAY);
gb_new_arc(v,v2,DELAY);
return v;
}
@#
static Vertex* make3(t,v1,v2,v3)
char t; /* the type of the new gate */
Vertex *v1,*v2,*v3;
{@+register Vertex *v=new_vert(t);
gb_new_arc(v,v1,DELAY);
gb_new_arc(v,v2,DELAY);
gb_new_arc(v,v3,DELAY);
return v;
}
@#
static Vertex* make4(t,v1,v2,v3,v4)
char t; /* the type of the new gate */
Vertex *v1,*v2,*v3,*v4;
{@+register Vertex *v=new_vert(t);
gb_new_arc(v,v1,DELAY);
gb_new_arc(v,v2,DELAY);
gb_new_arc(v,v3,DELAY);
gb_new_arc(v,v4,DELAY);
return v;
}
@#
static Vertex* make5(t,v1,v2,v3,v4,v5)
char t; /* the type of the new gate */
Vertex *v1,*v2,*v3,*v4,*v5;
{@+register Vertex *v=new_vert(t);
gb_new_arc(v,v1,DELAY);
gb_new_arc(v,v2,DELAY);
gb_new_arc(v,v3,DELAY);
gb_new_arc(v,v4,DELAY);
gb_new_arc(v,v5,DELAY);
return v;
}
@ We will use utility field |w.v| to store a pointer to the complement
of a gate, if that complement has been formed; this will prevent the creation
of a lot of gates that are equivalent to each other. The following subroutine
returns a pointer to the complement of a given gate.
@d bar w.v /* field pointing to complement, if known to exist */
@d signed(s,v) ((s)&1? v: comp(v))
@<Internal...@>=
static Vertex* comp(v)
Vertex *v;
{@+register Vertex *u;
if (v->bar) return v->bar;
u=next_vert++;
u->bar=v;@+v->bar=u;
sprintf(name_buf,"%s~",v->name);
u->name=gb_save_string(name_buf);
u->typ=NOT;
gb_new_arc(u,v,1);
return u;
}
@ To create a gate for the {\sc EXCLUSIVE-OR} of two arguments, we can
either construct the {\sc OR} of two {\sc AND}s, or the {\sc AND} of two
{\sc OR}s. We choose the former alternative:
@<Internal...@>=
static Vertex* make_xor(u,v)
Vertex *u,*v;
{@+register Vertex *t1,*t2;
t1=make2(AND,u,comp(v));
t2=make2(AND,comp(u),v);
return make2(OR,t1,t2);
}
@ OK, let's get going.
@<Initialize |new_graph|...@>=
if (regs<2 || regs>16) regs=16;
new_graph=gb_new_graph(1400+115*regs);
if (new_graph==NULL)
panic(no_room); /* out of memory before we're even started */
sprintf(new_graph->id,"risc(%u)",regs);
strcpy(new_graph->format,"ZZZIIVZZZZZZZA");
next_vert=new_graph->vertices;
@ @<Add the RISC data to |new_graph|@>=
@<Create the inputs and latches@>;
@<Create gates for instruction decoding@>;
@<Create gates for fetching the source value@>;
@<Create gates for the general logic operation@>;
@<Create gates for the conditional load operations@>;
@<Create gates for the arithmetic operations@>;
@<Create gates that bring everything together properly@>;
if (next_vert!=new_graph->vertices+new_graph->n)
panic(impossible); /* oops, we miscounted; this should be impossible */
@ We will want to assign internal names to many of the most important
gates. Here are the names of inputs and latches.
@<Local variables for |risc|@>=
Vertex *run_bit; /* the \.{RUN} input */
Vertex *mem[16]; /* 16 bits of input from read-only memory */
Vertex *prog; /* first of 10 bits in the program register */
Vertex *sign; /* the latched value of \.S */
Vertex *nonzero; /* the latched value of \.N */
Vertex *carry; /* the latched value of \.K */
Vertex *overflow; /* the latched value of \.V */
Vertex *extra; /* latched status bit: are we doing an extra memory cycle? */
Vertex *reg[16]; /* the least-significant bit of a given register */
@ @d first_of(n,t) new_vert(t);@+for (k=1;k<n;k++)@+new_vert(t);
@<Create the inputs and latches@>=
strcpy(prefix,"RUN");@+count=-1;@+run_bit=new_vert('I');
start_prefix("M");@+for (k=0;k<16;k++)@+mem[k]=new_vert('I');
start_prefix("P");@+prog=first_of(10,'L');
strcpy(prefix,"S");@+count=-1;@+sign=new_vert('L');
strcpy(prefix,"N");@+nonzero=new_vert('L');
strcpy(prefix,"K");@+carry=new_vert('L');
strcpy(prefix,"V");@+overflow=new_vert('L');
strcpy(prefix,"X");@+extra=new_vert('L');
for (r=0;r<regs;r++) {
numeric_prefix('R',r);
reg[r]=first_of(16,'L');
}
@ The order of evaluation of function arguments is not defined in \Cee,
so we introduce a few macros that force left-to-right order.
@d do2(result,t,v1,v2)
{@+t1=v1;@+t2=v2;
result=make2(t,t1,t2);@+}
@d do3(result,t,v1,v2,v3)
{@+t1=v1;@+t2=v2;@+t3=v3;
result=make3(t,t1,t2,t3);@+}
@d do4(result,t,v1,v2,v3,v4)
{@+t1=v1;@+t2=v2;@+t3=v3;@+t4=v4;
result=make4(t,t1,t2,t3,t4);@+}
@d do5(result,t,v1,v2,v3,v4,v5)
{@+t1=v1;@+t2=v2;@+t3=v3;@+t4=v4;@+t5=v5;
result=make5(t,t1,t2,t3,t4,t5);@+}
@<Local variables for |risc|@>=
Vertex *t1,*t2,*t3,*t4,*t5; /* temporary holds to force evaluation order */
Vertex *tmp[16]; /* additional holding places for partial results */
Vertex *imm; /* is the source value immediate (a given constant)? */
Vertex *rel; /* is the source value relative to the
current destination register? */
Vertex *dir; /* should the source value be fetched directly
from a source register? */
Vertex *ind; /* should the source value be fetched indirectly from memory? */
Vertex *op; /* least significant bit of \.{OP} */
Vertex *cond; /* most significant bit of \.{OP} */
Vertex *mod[4]; /* the \.{MOD} bits */
Vertex *dest[4]; /* the \.{DEST} bits */
@ The sixth line of the program here can be translated into the logic
equation
$$ |op|=(|extra|\land|prog|)\lor(\mskip1mu\overline{|extra|}\land|mem[6]|)\,.$$
Once you see why, you'll be able to read the rest of this curious code.
@<Create gates for instruction decoding@>=
start_prefix("D");
do3(imm,AND,comp(extra),comp(mem[4]),comp(mem[5])); /* $\.A=0$ */
do3(rel,AND,comp(extra),mem[4],comp(mem[5])); /* $\.A=1$ */
do3(dir,AND,comp(extra),comp(mem[4]),mem[5]); /* $\.A=2$ */
do3(ind,AND,comp(extra),mem[4],mem[5]); /* $\.A=3$ */
do2(op,OR,make2(AND,extra,prog),make2(AND,comp(extra),mem[6]));
do2(cond,OR,make2(AND,extra,prog+1),make2(AND,comp(extra),mem[7]));
for (k=0;k<4;k++) {
do2(mod[k],OR,make2(AND,extra,prog+2+k),make2(AND,comp(extra),mem[8+k]));
do2(dest[k],OR,make2(AND,extra,prog+6+k),make2(AND,comp(extra),mem[12+k]));
}
@ @<Create gates for fetching the source value@>=
start_prefix("F");
@<Set |old_dest| to the present value of the destination register@>;
@<Set |old_src| to the present value of the source register@>;
@<Set |inc_dest| to |old_dest| plus \.{SRC}@>;
for (k=0;k<16;k++)@/
do4(source[k],OR,
make2(AND,imm,mem[k<4?k:3]),
make2(AND,rel,inc_dest[k]),@|
make2(AND,dir,old_src[k]),
make2(AND,extra,mem[k]));
@ Here and in the immediately following section we create {\sc OR} gates
|old_dest[k]| and |old_src[k]| that might have as many as 16~inputs. (The actual
number of inputs is |regs|.) All of the
other gates in the network will have at most five inputs.
@<Set |old_dest| to the present value of the destination register@>=
for (r=0;r<regs;r++) @/
do4(dest_match[r],AND,signed(r,dest[0]),signed(r>>1,dest[1]),@|
signed(r>>2,dest[2]),signed(r>>3,dest[3]));
for (k=0;k<16;k++) {
for (r=0;r<regs;r++)@/
tmp[r]=make2(AND,dest_match[r],reg[r]+k);
old_dest[k]=new_vert(OR);
for (r=0;r<regs;r++) gb_new_arc(old_dest[k],tmp[r],DELAY);
}
@ @<Set |old_src| to the present value of the source register@>=
for (k=0;k<16;k++) {
for (r=0;r<regs;r++)@/
do5(tmp[r],AND,reg[r]+k,signed(r,mem[0]),signed(r>>1,mem[1]),
signed(r>>2,mem[2]),signed(r>>3,mem[3]));
old_src[k]=new_vert(OR);
for (r=0;r<regs;r++) gb_new_arc(old_src[k],tmp[r],DELAY);
}
@ @<Local variables for |risc|@>=
Vertex *dest_match[16]; /* |dest_match[r]==1| iff $\.{DST}=r$ */
Vertex *old_dest[16]; /* contents of destination register before operation */
Vertex *old_src[16]; /* contents of source register before operation */
Vertex *inc_dest[16]; /* |old_dest| plus the \.{SRC} field */
Vertex *source[16]; /* source value for the operation */
Vertex *log[16]; /* result of general logic operation */
Vertex *shift[18]; /* result of shift operation, with carry and overflow */
Vertex *sum[18]; /* |old_dest| plus |source| plus optional carry */
Vertex *diff[18]; /* |old_dest| minus |source| minus optional borrow */
Vertex *next_loc[16]; /* contents of register 0, plus 1 */
Vertex *next_next_loc[16]; /* contents of register 0, plus 2 */
Vertex *result[18]; /* result of operating on |old_dest| and |source| */
@ @<Create gates for the general logic operation@>=
start_prefix("L");
for (k=0;k<16;k++)@/
do4(log[k],OR,@t}\3{-5@>
make3(AND,mod[0],comp(old_dest[k]),comp(source[k])),@t}\3{-5@>
make3(AND,mod[1],comp(old_dest[k]),source[k]),@t}\3{-5@>
make3(AND,mod[2],old_dest[k],comp(source[k])),@t}\3{-5@>
make3(AND,mod[3],old_dest[k],source[k]));
@ @<Create gates for the conditional load operations@>=
start_prefix("C");
do4(tmp[0],OR,@t}\3{-5@>
make3(AND,mod[0],comp(sign),comp(nonzero)),@t}\3{-5@>
make3(AND,mod[1],comp(sign),nonzero),@t}\3{-5@>
make3(AND,mod[2],sign,comp(nonzero)),@t}\3{-5@>
make3(AND,mod[3],sign,nonzero));
do4(tmp[1],OR,@t}\3{-5@>
make3(AND,mod[0],comp(carry),comp(overflow)),@t}\3{-5@>
make3(AND,mod[1],comp(carry),overflow),@t}\3{-5@>
make3(AND,mod[2],carry,comp(overflow)),@t}\3{-5@>
make3(AND,mod[3],carry,overflow));
do3(change,OR,comp(cond),make2(AND,tmp[0],comp(op)),make2(AND,tmp[1],op));
@ @<Local variables for |risc|@>=
Vertex *change; /* is the destination register supposed to change? */
@ Hardware is like software except that it performs all the operations
all the time and then selects only the results it needs. (If you think about
it, this is a profound observation about economics, society, and nature.
Gosh.)
@<Create gates that bring everything together properly@>=
start_prefix("Z");
@<Create gates for the |next_loc| and |next_next_loc| bits@>;
@<Create gates for the |result| bits@>;
@<Create gates for the new values of registers 1 to |regs|@>;
@<Create gates for the new values of \.S, \.N, \.K, and \.V@>;
@<Create gates for the new values of the program register and |extra|@>;
@<Create gates for the new values of register 0
and the memory address register@>;
@ @<Create gates for the |next_loc|...@>=
next_loc[0]=comp(reg[0]);@+next_next_loc[0]=reg[0];
next_loc[1]=make_xor(reg[0]+1,reg[0]);@+next_next_loc[1]=comp(reg[0]+1);
for (t5=reg[0]+1,k=2;k<16;t5=make2(AND,t5,reg[0]+k++)) {
next_loc[k]=make_xor(reg[0]+k,make2(AND,reg[0],t5));
next_next_loc[k]=make_xor(reg[0]+k,t5);
}
@ @<Create gates for the |result| bits@>=
jump=make5(AND,op,mod[0],mod[1],mod[2],mod[3]); /* assume |cond=0| */
for (k=0;k<16;k++) {
do5(result[k],OR,@t}\3{-5@>
make2(AND,comp(op),log[k]),@t}\3{-5@>
make2(AND,jump,next_loc[k]),@t}\3{-5@>
make3(AND,op,comp(mod[3]),shift[k]),@t}\3{-5@>
make5(AND,op,mod[3],comp(mod[2]),comp(mod[1]),sum[k]),@t}\3{-5@>
make5(AND,op,mod[3],comp(mod[2]),mod[1],diff[k]));
do2(result[k],OR,@t}\3{-5@>
make3(AND,cond,change,source[k]),@t}\3{-5@>
make2(AND,comp(cond),result[k]));
}
for (k=16;k<18;k++) /* carry and overflow bits of the result */
do3(result[k],OR,@t}\3{-5@>
make3(AND,op,comp(mod[3]),shift[k]),@t}\3{-5@>
make5(AND,op,mod[3],comp(mod[2]),comp(mod[1]),sum[k]),@t}\3{-5@>
make5(AND,op,mod[3],comp(mod[2]),mod[1],diff[k]));
@ The program register |prog| and the |extra| bit are needed for
the case when we must spend an extra cycle to fetch a word from memory.
On the first cycle, |ind| is true, so a ``result'' is calculated but not
actually used. On the second cycle, |extra| is true.
A slight optimization has been introduced in order to make the circuit
a bit more interesting: If a conditional load instruction occurs with
indirect addressing and a false condition, the extra cycle is not taken.
(The |next_next_loc| values were computed for this reason.)
@d latchit(u,@!latch)
(latch)->alt=make2(AND,u,run_bit) /* |u&run_bit| is new value for |latch| */
@<Create gates for the new values of the program reg...@>=
for (k=0;k<10;k++)
latchit(mem[k+6],prog+k);
do2(nextra,OR,make2(AND,ind,comp(cond)),make2(AND,ind,change));
latchit(nextra,extra);
nzs=make4(OR,mem[0],mem[1],mem[2],mem[3]);
nzd=make4(OR,dest[0],dest[1],dest[2],dest[3]);
@ @<Local variables for |risc|@>=
Vertex *jump; /* is this command a \.{JUMP}, assuming |cond| is false? */
Vertex *nextra; /* must we take an extra cycle? */
Vertex *nzs; /* is the \.{SRC} field nonzero? */
Vertex *nzd; /* is the \.{DST} field nonzero? */
@ @<Create gates for the new values of registers 1 to |regs|@>=
t5=make2(AND,change,comp(ind)); /* should destination register change? */
for (r=1;r<regs;r++) {
t4=make2(AND,t5,dest_match[r]); /* should register |r| change? */
for (k=0;k<16;k++) {
do2(t3,OR,make2(AND,t4,result[k]),make2(AND,comp(t4),reg[r]+k));
latchit(t3,reg[r]+k);
}
}
@ @<Create gates for the new values of \.S, \.N, \.K, and \.V@>=
do4(t5,OR,@t}\3{-5@>
make2(AND,sign,cond),@t}\3{-5@>
make2(AND,sign,jump),@t}\3{-5@>
make2(AND,sign,ind),@t}\3{-5@>
make4(AND,result[15],comp(cond),comp(jump),comp(ind)));
latchit(t5,sign);
do4(t5,OR,@t}\3{-5@>
make4(OR,result[0],result[1],result[2],result[3]),@t}\3{-5@>
make4(OR,result[4],result[5],result[6],result[7]),@t}\3{-5@>
make4(OR,result[8],result[9],result[10],result[11]),@t}\3{-5@>
make4(OR,result[12],result[13],result[14],@t}\3{-5@>
@t\hskip5em@>make5(AND,make2(OR,nonzero,sign),op,mod[0],comp(mod[2]),mod[3])));
do4(t5,OR,@t}\3{-5@>
make2(AND,nonzero,cond),@t}\3{-5@>
make2(AND,nonzero,jump),@t}\3{-5@>
make2(AND,nonzero,ind),@t}\3{-5@>
make4(AND,t5,comp(cond),comp(jump),comp(ind)));
latchit(t5,nonzero);
do5(t5,OR,@t}\3{-5@>
make2(AND,overflow,cond),@t}\3{-5@>
make2(AND,overflow,jump),@t}\3{-5@>
make2(AND,overflow,comp(op)),@t}\3{-5@>
make2(AND,overflow,ind),@t}\3{-5@>
make5(AND,result[17],comp(cond),comp(jump),comp(ind),op));
latchit(t5,overflow);
do5(t5,OR,@t}\3{-5@>
make2(AND,carry,cond),@t}\3{-5@>
make2(AND,carry,jump),@t}\3{-5@>
make2(AND,carry,comp(op)),@t}\3{-5@>
make2(AND,carry,ind),@t}\3{-5@>
make5(AND,result[16],comp(cond),comp(jump),comp(ind),op));
latchit(t5,carry);
@ As usual, we have left the hardest case for last, hoping that we will
have learned enough tricks to handle it when the time of reckoning
finally arrives. The most subtle part of the logic here
is perhaps the case of a \.{JUMP} command with $\.A=3$;
we want to increase register~0 by~1 during the first cycle of
such a command, if $\.{SRC}=0$, so that the |result| will be
correct on the next cycle.
@<Create gates for the new values of register 0...@>=
skip=make2(AND,cond,comp(change)); /* false conditional? */
hop=make2(AND,comp(cond),jump); /* \.{JUMP} command? */
do4(normal,OR,@t}\3{-5@>
make2(AND,skip,comp(ind)),@t}\3{-5@>
make2(AND,skip,nzs),@t}\3{-5@>
make3(AND,comp(skip),ind,comp(nzs)),@t}\3{-5@>
make3(AND,comp(skip),comp(hop),nzd));
special=make3(AND,comp(skip),ind,nzs);
for (k=0;k<16;k++) {
do4(t5,OR,@t}\3{-5@>
make2(AND,normal,next_loc[k]),@t}\3{-5@>
make4(AND,skip,ind,comp(nzs),next_next_loc[k]),@t}\3{-5@>
make3(AND,hop,comp(ind),source[k]),@t}\3{-5@>
make5(AND,comp(skip),comp(hop),comp(ind),comp(nzd),result[k]));
do2(t4,OR,@t}\3{-5@>
make2(AND,special,reg[0]+k),@t}\3{-5@>
make2(AND,comp(special),t5));
latchit(t4,reg[0]+k);
do2(t4,OR,@t}\3{-5@>
make2(AND,special,old_src[k]),@t}\3{-5@>
make2(AND,comp(special),t5));
{@+register Arc *a=gb_virgin_arc();
a->tip=make2(AND,t4,run_bit);
a->next=new_graph->outs;
new_graph->outs=a; /* pointer to memory address bit */
}
} /* arcs for output bits will appear in big-endian order */
@ @<Local variables for |risc|@>=
Vertex *skip; /* are we skipping a conditional load operation? */
Vertex *hop; /* are we doing a \.{JUMP}? */
Vertex *normal; /* is this a case where register 0 is simply incremented? */
Vertex *special; /* is this a case where register 0 and the memory address
register will not coincide? */
@* Serial addition. We haven't yet specified the parts of |risc| that
deal with addition and subtraction; somehow, those parts wanted to
be separate from the rest. To complete our mission, we will use
subroutine calls of the form `|make_adder(n,x,y,z,carry,add)|',
where |x| and |y| are |n|-bit arrays of input gates and
|z|~is an |(n+1)|-bit array of output gates. If |add!=0|, the subroutine
computes |x+y|, otherwise it computes |x-y|. If |carry!=0|, the |carry| gate
is effectively added to~|y| before the operation.
A simple |n|-stage serial scheme, which reduces the problem of |n|-bit
addition to |(n-1)|-bit addition, is adequate for our purposes here.
(A parallel adder, which gains efficiency by reducing the problem size
from |n| to~$n/\phi$, can be found in the |prod| routine below.)
The handy identity $x-y=\overline{\overline x+y}$ is used to reduce
subtraction to addition.
@<Internal...@>=
static make_adder(n,x,y,z,carry,add)
unsigned n; /* number of bits */
Vertex *x[],*y[]; /* input gates */
Vertex *z[]; /* output gates */
Vertex *carry; /* add this to |y|, unless it's null */
char add; /* should we add or subtract? */
{@+register int k;
Vertex *t1,*t2,*t3,*t4; /* temporary storage used by |do4| */
if (!carry) {
z[0]=make_xor(x[0],y[0]);
carry=make2(AND,signed(add,x[0]),y[0]);
k=1;
} else k=0;
for (;k<n;k++) {
comp(x[k]);@+comp(y[k]);@+comp(carry); /* generate inverse gates */
do4(z[k],OR,@t}\3{-5@>
make3(AND,x[k],comp(y[k]),comp(carry)),@t}\3{-5@>
make3(AND,comp(x[k]),y[k],comp(carry)),@t}\3{-5@>
make3(AND,comp(x[k]),comp(y[k]),carry),@t}\3{-5@>
make3(AND,x[k],y[k],carry));
do3(carry,OR,@t}\3{-5@>
make2(AND,signed(add,x[k]),y[k]),@t}\3{-5@>
make2(AND,signed(add,x[k]),carry),@t}\3{-5@>
make2(AND,y[k],carry));
}
z[n]=carry;
}
@ OK, now we can add. What good does that do us?
In the first place, we need a 4-bit adder to compute the least
significant bits of $|old_dest|+\.{SRC}$. The other 12 bits of that
sum are simpler.
@<Set |inc_dest| to |old_dest| plus \.{SRC}@>=
make_adder(4,old_dest,mem,inc_dest,NULL,1);
up=make2(AND,inc_dest[4],comp(mem[3])); /* remaining bits must increase */
down=make2(AND,comp(inc_dest[4]),mem[3]); /* remaining bits must decrease */
for (k=4;;k++) {
comp(up);@+comp(down);
do3(inc_dest[k],OR,@t}\3{-5@>
make2(AND,comp(old_dest[k]),up),@t}\3{-5@>
make2(AND,comp(old_dest[k]),down),@t}\3{-5@>
make3(AND,old_dest[k],comp(up),comp(down)));
if (k<15) {
up=make2(AND,up,old_dest[k]);
down=make2(AND,down,comp(old_dest[k]));
} else break;
}
@ @<Local variables for |risc|@>=
Vertex *up,*down; /* gates used when computing |inc_dest| */
@ In the second place, we need a 16-bit adder and a 16-bit subtracter
for the four addition/subtraction commands.
@<Create gates for the arithmetic operations@>=
start_prefix("A");
@<Create gates for the shift operations@>;
make_adder(16,old_dest,source,sum,make2(AND,carry,mod[0]),1); /* adder */
make_adder(16,old_dest,source,diff,make2(AND,carry,mod[0]),0); /* subtracter */
do2(sum[17],OR,@t}\3{-5@>
make3(AND,old_dest[15],source[15],comp(sum[15])),@t}\3{-5@>
make3(AND,comp(old_dest[15]),comp(source[15]),sum[15])); /* overflow */
do2(diff[17],OR,@t}\3{-5@>
make3(AND,old_dest[15],comp(source[15]),comp(diff[15])),@t}\3{-5@>
make3(AND,comp(old_dest[15]),source[15],diff[15])); /* overflow */
@ @<Create gates for the shift operations@>=
for (k=0;k<16;k++)@/
do4(shift[k],OR,@t}\3{-5@>
(k==0? make4(AND,source[15],mod[0],comp(mod[1]),comp(mod[2])):@t}\3{-5@>
@t\hskip5em@>make3(AND,source[k-1],comp(mod[1]),comp(mod[2]))),@t}\3{-5@>
(k<4? make4(AND,source[k+12],mod[0],mod[1],comp(mod[2])):@t}\3{-5@>
@t\hskip5em@>make3(AND,source[k-4],mod[1],comp(mod[2]))),@t}\3{-5@>
(k==15? make4(AND,source[15],comp(mod[0]),comp(mod[1]),mod[2]):@t}\3{-5@>
@t\hskip5em@>make3(AND,source[k+1],comp(mod[1]),mod[2])),@t}\3{-5@>
(k>11? make4(AND,source[15],comp(mod[0]),mod[1],mod[2]):@t}\3{-5@>
@t\hskip5em@>make3(AND,source[k+4],mod[1],mod[2])));
do4(shift[16],OR,@t}\3{-5@>
make2(AND,comp(mod[2]),source[15]),@t}\3{-5@>
make3(AND,comp(mod[2]),mod[1],
make3(OR,source[14],source[13],source[12])),@t}\3{-5@>
make3(AND,mod[2],comp(mod[1]),source[0]),@t}\3{-5@>
make3(AND,mod[2],mod[1],source[3])); /* ``carry'' */
do3(shift[17],OR,@t}\3{-5@>
make3(AND,comp(mod[2]),comp(mod[1]),
make_xor(source[15],source[14])),@t}\3{-5@>
make4(AND,comp(mod[2]),mod[1],@t}\3{-5@>
@t\hskip5em@>make5(OR,source[15],source[14],
source[13],source[12],source[11]),@t}\3{-5@>
@t\hskip5em@>make5(OR,comp(source[15]),comp(source[14]),
comp(source[13]),@t}\3{-5@>
@t\hskip10em@>comp(source[12]),comp(source[11]))),@t}\3{-5@>
make3(AND,mod[2],mod[1],
make3(OR,source[0],source[1],source[2]))); /* ``overflow'' */
@* RISC management. The |run_risc| procedure takes a gate graph output by |risc|
and simulates its behavior, given the contents of its read-only memory.
(See the demonstration program |take_risc|, which appears in a module
by itself, for a typical illustration of how |run_risc| might be used.)
This procedure clears the simulated machine and begins executing the program
that starts at address~0. It stops when it gets to an address greater
than the size of read-only memory supplied. One way to stop it
is therefore to execute a command such as |0x0f00|, which will transfer
control to location |0xffff|; even better is |0x0f8f|, which transfers
to location |0xffff| without changing the status of \.S and \.N.
However, if the given read-only memory
contains a full set of $2^{16}$ words, |run_risc| will never stop.
When |run_risc| does stop, it returns 0 and puts the final contents of the
simulated registers into the global array |risc_state|.
Or, if |g| was not a decent graph, |run_risc| returns a negative value and
leaves |risc_state| untouched.
@<The |run_risc|...@>=
int run_risc(g,rom,size,trace_regs)
Graph *g; /* graph output by |risc| */
unsigned rom[]; /* contents of read-only memory */
unsigned size; /* length of |rom| vector */
unsigned trace_regs; /* if nonzero, this many registers will be traced */
{@+register unsigned l; /* memory address */
register unsigned m; /* memory or register contents */
register Vertex *v; /* the current gate of interest */
register Arc *a; /* the current output list element of interest */
register int k,r; /* general-purpose indices */
int x,s,n,c,o; /* status bits */
if (trace_regs) @<Print a headline@>;
m=gate_eval(g,"0",NULL); /* reset the RISC by turning off the \.{RUN} bit */
if (m<0) return m; /* not a valid gate graph! */
g->vertices->val=1; /* turn the \.{RUN} bit on */
while (1) {
for (a=g->outs,l=0;a;a=a->next) l=2*l+a->tip->val;
/* set $l=\null$memory address */
if (trace_regs) @<Print register contents@>;
if (l>=size) break; /* stop if memory check occurs */
for (v=g->vertices+1,m=rom[l];v<=g->vertices+16;v++,m>>=1)
v->val=m&1; /* store bits of memory word in the input gates */
gate_eval(g,NULL,NULL); /* do another RISC cycle */
}
if (trace_regs) @<Print a footline@>;
@<Dump the register contents into |risc_state|@>;
return 0;
}
@ If tracing is requested, we write on the standard output file.
@<Print a headline@>=
{
for (r=0;r<trace_regs;r++) printf(" r%-2d ",r); /* register names */
printf(" P XSNKV MEM\n"); /* |prog|, |extra|, status bits, memory */
}
@ @<Print a footline@>=
printf("Execution terminated with memory address %04x.\n",l);
@ Here we peek inside the circuit to see what values are about to
be latched.
@<Print register contents@>=
{ for (r=0;r<trace_regs;r++) {
v=g->vertices+(16*r+47); /* most significant bit of register |r| */
m=0;
if (v->typ=='L')
for (k=0,m=0;k<16;k++,v--) m=2*m+v->alt->val;
printf("%04x ",m);
}
for (k=0,m=0,v=g->vertices+26;k<10;k++,v--) m=2*m+v->alt->val; /* |prog| */
x=(g->vertices+31)->alt->val; /* |extra| */
s=(g->vertices+27)->alt->val; /* |sign| */
n=(g->vertices+28)->alt->val; /* |nonzero| */
c=(g->vertices+29)->alt->val; /* |carry| */
o=(g->vertices+30)->alt->val; /* |overflow| */
printf("%03x%c%c%c%c%c ",m<<2,
x?'X':'.', s?'S':'.', n?'N':'.', c?'K':'.', o?'V':'.');
if (l>=size) printf("????\n");
else printf("%04x\n",rom[l]);
}
@ @<Dump...@>=
for (r=0;r<16;r++) {
v=g->vertices+(16*r+47); /* most significant bit of register |r| */
m=0;
if (v->typ=='L')
for (k=0,m=0;k<16;k++,v--) m=2*m+v->alt->val;
risc_state[r]=m;
}
for (k=0,m=0,v=g->vertices+26;k<10;k++,v--) m=2*m+v->alt->val; /* |prog| */
m=4*m+(g->vertices+31)->alt->val; /* |extra| */
m=2*m+(g->vertices+27)->alt->val; /* |sign| */
m=2*m+(g->vertices+28)->alt->val; /* |nonzero| */
m=2*m+(g->vertices+29)->alt->val; /* |carry| */
m=2*m+(g->vertices+30)->alt->val; /* |overflow| */
risc_state[16]=m; /* program register and status bits go here */
risc_state[17]=l; /* this is the out-of-range address that caused termination */
@ @<Global variables@>=
unsigned risc_state[18];
@*Generalized gate graphs. For intermediate computations it is
convenient to allow two additional types of gates:
{\advance\parindent 2em
\smallskip
\item{|'C'|} denotes a constant gate of value |z.i|.
\smallskip
\item{|'='|} denotes a copy of a previous gate; utility field |alt|
points to that previous gate.
\smallskip}\noindent
Such gates might appear anywhere in the graph, possibly interspersed with
the inputs and latches.
Here is a simple subroutine that prints a symbolic representation of
a generalized gate graph on the standard output file:
@d bit z.i /* field containing the constant value of a |'C'| gate */
@<The |print_gates| routine@>=
static print_gate(v)
Vertex *v;
{@+register int t;
register Arc *a;
printf("%s = ",v->name);
switch(v->typ) {
case 'I':printf("input");@+break;
case 'L':printf("latch");
if (v->alt) printf("ed %s",v->alt->name);
break;
case '~':printf("~ ");@+break;
case 'C':printf("constant %d",v->bit); break;
case '=':printf("copy of %s",v->alt->name);
}
for (a=v->arcs;a;a=a->next) {
if (a!=v->arcs) printf(" %c ",v->typ);
printf(a->tip->name);
}
printf("\n");
}
@#
print_gates(g)
Graph *g;
{@+register Vertex *v;
register Arc *a;
for (v=g->vertices;v<g->vertices+g->n;v++) print_gate(v);
for (a=g->outs;a;a=a->next)
if (is_boolean(a->tip)) printf("Output %d\n",the_boolean(a->tip));
else printf("Output %s\n",a->tip->name);
}
@ @(gb_gates.h@>=
#define bit @t\quad@> z.i
@ The |reduce| routine takes a generalized graph |g| and uses the identities
$$\openup1\jot
\vbox{\halign{\hfil$x#0=\null$&$#$,\hfil\quad
&\hfil$x#1=\null$&$#$,\hfil\quad
&\hfil$x#x=\null$&$#$,\hfil\quad
&\hfil$x#\overline x=\null$&$#$,\hfil\cr
\land&0&\land&x&\land&x&\land&0\cr
\lor&x&\lor&1&\lor&x&\lor&1\cr
\oplus&x&\oplus&\overline x&\oplus&0&\oplus&1\cr}}$$
and $\overline{\overline x}=x$ to create an equivalent graph having no
|'C'| or |'='| or obviously redundant gates. The reduced graph also excludes
any gates that are not used directly or indirectly in the computation of
the output values.
@<Internal...@>=
static Graph* reduce(g)
Graph *g;
{@+register Vertex *u, *v; /* the current vertices of interest */
register Arc *a, *b; /* the current arcs of interest */
Arc *aa, *bb; /* their predecessors */
Vertex *latch_ptr; /* top of the latch list */
long n=0; /* the number of marked gates */
Graph *new_graph; /* the reduced gate graph */
Vertex *next_vert=NULL, *max_next_vert=NULL; /* allocation of new vertices */
Arc *avail_arc=NULL; /* list of recycled arcs */
Vertex *sentinel; /* end of the vertices */
if (g==NULL) panic(missing_operand); /* where is |g|? */
sentinel=g->vertices+g->n;
while (1) {
latch_ptr=NULL;
for (v=g->vertices;v<sentinel;v++)
@<Reduce gate |v|, if possible, or put it on the latch list@>;
@<Check to see if any latch has become constant; if not, |break|@>;
}
@<Mark all gates that are used in some output@>;
@<Copy all marked gates to a new graph@>;
gb_recycle(g);
return new_graph;
}
@ We will link latches together via their |v.v| fields.
@<Check to see if any latch has become constant; if not, |break|@>=
{@+char no_constants_yet=1;
for (v=latch_ptr;v;v=v->v.v) {
u=v->alt; /* the gate whose value will be latched */
if (u->typ=='=')
v->alt=u->alt;
else if (u->typ=='C') {
v->typ='C';@+v->bit=u->bit;@+no_constants_yet=0;
}
}
if (no_constants_yet) break;
}
@ @d foo x.v /* link field used to find all the gates later */
@<Reduce gate |v|, if possible, or put it on the latch list@>=
{
switch(v->typ) {
case 'L': v->v.v=latch_ptr;@+latch_ptr=v;@+break;
case 'I': case 'C': break;
case '=': u=v->alt;
if (u->typ=='=')
v->alt=u->alt;
else if (u->typ=='C') {
v->bit=u->bit;@+goto make_v_constant;
}
break;
case AND:@<Try to reduce an {\sc AND} gate@>;@+goto test_single_arg;
case OR:@<Try to reduce an {\sc OR} gate@>;@+goto test_single_arg;
case XOR:@<Try to reduce an {\sc EXCLUSIVE-OR} gate@>;
@+goto test_single_arg;
case NOT:@<Try to reduce an inverter@>;@+break;
test_single_arg: if (v->arcs->next) break;
v->alt=v->arcs->tip;
make_v_eq: v->typ='='; goto make_v_arcless;
make_v_1: v->bit=1;@+goto make_v_constant;
make_v_0: v->bit=0;
make_v_constant: v->typ='C';
make_v_arcless: v->arcs=NULL;
}
v->bar=NULL; /* this field will point to the complement, if computed later */
done: v->foo=v+1; /* this field will link all the vertices together */
}
@ @<Try to reduce an inverter@>=
u=v->arcs->tip;
if (u->typ=='=')
u=v->arcs->tip=u->alt;
if (u->typ=='C') {
v->bit=1-u->bit;@+goto make_v_constant;
} else if (u->bar) { /* this inverse already computed */
v->alt=u->bar;@+goto make_v_eq;
} else {
u->bar=v;@+v->bar=u;@+goto done;
}
@ @<Try to reduce an {\sc AND} gate@>=
for (a=v->arcs,aa=NULL;a;a=a->next) {
u=a->tip;
if (u->typ=='=')
u=a->tip=u->alt;
if (u->typ=='C') {
if (u->bit==0) goto make_v_0;
goto bypass_arg_of_and;
} else for (b=v->arcs;b!=a;b=b->next) {
if (b->tip==u) goto bypass_arg_of_and;
if (b->tip==u->bar) goto make_v_0;
}
aa=a;@+continue;
bypass_arg_of_and: if (aa) aa->next=a->next;
else v->arcs=a->next;
}
if (v->arcs==NULL) goto make_v_1;
@ @<Try to reduce an {\sc OR} gate@>=
for (a=v->arcs,aa=NULL;a;a=a->next) {
u=a->tip;
if (u->typ=='=')
u=a->tip=u->alt;
if (u->typ=='C') {
if (u->bit) goto make_v_1;
goto bypass_arg_of_or;
} else for (b=v->arcs;b!=a;b=b->next) {
if (b->tip==u) goto bypass_arg_of_or;
if (b->tip==u->bar) goto make_v_1;
}
aa=a;@+continue;
bypass_arg_of_or: if (aa) aa->next=a->next;
else v->arcs=a->next;
}
if (v->arcs==NULL) goto make_v_0;
@ @<Try to reduce an {\sc EXCLUSIVE-OR} gate@>=
{@+int cmp=0;
for (a=v->arcs,aa=NULL;a;a=a->next) {
u=a->tip;
if (u->typ=='=')
u=a->tip=u->alt;
if (u->typ=='C') {
if (u->bit) cmp=1-cmp;
goto bypass_arg_of_xor;
} else for (bb=NULL,b=v->arcs;b!=a;b=b->next) {
if (b->tip==u) goto double_bypass;
if (b->tip==u->bar) {
cmp=1-cmp;
goto double_bypass;
}
bb=b;@+ continue;
double_bypass: if (bb) bb->next=b->next;
else v->arcs=b->next;
goto bypass_arg_of_xor;
}
aa=a;@+ continue;
bypass_arg_of_xor: if (aa) aa->next=a->next;
else v->arcs=a->next;
a->a.a=avail_arc;
avail_arc=a;
}
if (v->arcs==NULL) {
v->bit=cmp;
goto make_v_constant;
}
if (cmp) @<Complement one argument of |v|@>;
}
@ @<Complement one argument of |v|@>=
{
for (a=v->arcs;;a=a->next) {
u=a->tip;
if (u->bar) break; /* good, the complement is already known */
if (a->next==NULL) { /* oops, this is our last chance */
@<Create a new vertex for complement of |u|@>;
break;
}
}
a->tip=u->bar;
}
@ Here we've come to a subtle point: The ``reduced'' graph might
actually be larger than the original, in the sense of having more
vertices (although fewer arcs), if there are a lot of |XOR| gates
involving an input that is set to the constant value~1. Therefore
we must have the ability to allocate new vertices during the
reduction phase of |reduce|. At least one arc has been added to
the |avail_arc| list whenever we reach this portion of the program.
@<Create a new vertex for complement of |u|@>=
if (next_vert==max_next_vert) {
next_vert=gb_alloc_type(7,@[Vertex@],g->aux_data);
if (next_vert==NULL) {
gb_recycle(g);
panic(no_room+1); /* can't get auxiliary storage! */
}
max_next_vert=next_vert+7;
}
next_vert->typ=NOT;
sprintf(name_buf,"%s~",u->name);
next_vert->name=gb_save_string(name_buf);
next_vert->arcs=avail_arc; /* this is known to be non-|NULL| */
avail_arc->tip=u;
avail_arc=avail_arc->a.a;
next_vert->arcs->next=NULL;
next_vert->bar=u;
next_vert->foo=u->foo;
u->foo=u->bar=next_vert++;
@ During the marking phase, we will use the |w.v| field to link the
list of nodes-to-be-marked. That field will turn out to be non-|NULL|
only in the marked nodes. (We no longer use its former meaning related
to complementation, so we call it |lnk| instead of |bar|.)
@d lnk w.v /* stack link for marking */
@<Mark all gates that are used in some output@>=
{
for (v=g->vertices;v!=sentinel;v=v->foo) v->lnk=NULL;
for (a=g->outs;a;a=a->next) {
v=a->tip;
if (is_boolean(v)) continue;
if (v->typ=='=')
v=a->tip=v->alt;
if (v->typ=='C') { /* this output is constant, so make it boolean */
a->tip=(Vertex*)v->bit;
continue;
}
@<Mark all gates that are used to compute |v|@>;
}
}
@ @<Mark all gates that are used to compute |v|@>=
if (v->lnk==NULL) {
v->lnk=sentinel; /* |v| will now be the top of stack of nodes to be marked */
do {
n++;
b=v->arcs;
if (v->typ=='L') {
u=v->alt; /* latch vertices have a ``hidden'' dependency */
if (u<v) n++; /* latched input value will get a special gate */
if (u->lnk==NULL) {
u->lnk=v->lnk;
v=u;
} else v=v->lnk;
} else v=v->lnk;
for (;b;b=b->next) {
u=b->tip;
if (u->lnk==NULL) {
u->lnk=v;
v=u;
}
}
} while (v!=sentinel);
}
@ It is easier to copy a directed acyclic graph than to copy a general graph,
but we do have to contend with the feedback in latches.
@d reverse_arc_list(@!alist)
{@+for (aa=alist,b=NULL;aa;b=aa,aa=a) {
a=aa->next;
aa->next=b;
}
alist=b;@+}
@<Copy all marked gates to a new graph@>=
new_graph=gb_new_graph(n);
if (new_graph==NULL) {
gb_recycle(g);
panic(no_room+2); /* out of memory */
}
strcpy(new_graph->id,g->id);
strcpy(new_graph->format,"ZZZIIVZZZZZZZA");
next_vert=new_graph->vertices;
for (v=g->vertices,latch_ptr=NULL;v!=sentinel;v=v->foo) {
if (v->lnk) { /* yes, |v| is marked */
u=v->lnk=next_vert++; /* make note of where we've copied it */
@<Make |u| a copy of |v|; put it on the latch list if it's a latch@>;
}
}
@<Fix up the |alt| fields of the newly copied latches@>;
reverse_arc_list(g->outs);
for (a=g->outs;a;a=a->next) {
b=gb_virgin_arc();
b->tip=is_boolean(a->tip)? a->tip: a->tip->lnk;
b->next=new_graph->outs;
new_graph->outs=b;
}
@ @<Make |u| a copy of |v|; put it on the latch list if it's a latch@>=
u->name=gb_save_string(v->name);
u->typ=v->typ;
if (v->typ=='L') {
u->alt=latch_ptr;@+latch_ptr=v;
}
reverse_arc_list(v->arcs);
for (a=v->arcs;a;a=a->next)
gb_new_arc(u,a->tip->lnk,a->len);
@ @<Fix up the |alt| fields of the newly copied latches@>=
while (latch_ptr) {
u=latch_ptr->lnk; /* the copy of a latch */
v=u->alt;
u->alt=latch_ptr->alt->lnk;
latch_ptr=v;
if (u->alt<u) @<Replace |u->alt| by a new gate that copies an input@>;
}
@ Suppose we had a latch whose value was originally the {\sc AND} of
two inputs, where one of those inputs has now been set to~1. Then the
latch should still refer to a subsequent gate, equal to value of the
other input on the previous cycle. We create such a gate here, making
it an {\sc OR} of two identical inputs, because we're not supposed to
leave any |'='| in the result of |reduce|, and because every {\sc OR}
is supposed to have at least two inputs.
@<Replace |u->alt| by a new gate that copies an input@>=
{
v=u->alt; /* the input gate that should be copied for latching */
u->alt=next_vert++;
sprintf(name_buf,"%s>%s",v->name,u->name);
u=u->alt;
u->name=gb_save_string(name_buf);
u->typ=OR;
gb_new_arc(u,v,DELAY);@+gb_new_arc(u,v,DELAY);
}
@* Parallel multiplication. Now comes the |prod| routine,
which constructs a rather different network of gates, based this time
on a divide-and-conquer paradigm. Let's take a breater before we tackle it.
(Deep breath.)
The subroutine call |prod(m,n)| creates
a network for the binary multiplication of unsigned
|m|-bit numbers by |n|-bit numbers, assuming that |m>=2| and |n>=2|.
There is no upper limit on the sizes of |m| and~|n|, except of course
the limits imposed by the size of memory in which this routine is run.
The overall strategy used by |prod| is to start with a generalized
gate graph for multiplication in which many of the gates are
identically zero or copies of other gates. Then the |reduce| routine
will perform local optimizations leading to the desired result. Since
there are no latches, some of the complexities of the general |reduce|
routine are avoided.
All of the |AND|, |OR|, and |XOR| gates of the network returned by
|prod| have exactly two inputs. The depth of the circuit (i.e., the
length of its longest path) is $3\log m/\!\log 1.5 + \log(m+n)/\!\log\phi
+O(1)$, where $\phi=(1+\sqrt5\,)/2$ is the golden ratio. The total number
of gates is $6mn+5m^2+O\bigl((m+n)\log(m+n)\bigr)$.
There is a demonstration program called |multiply| that uses |prod| to
compute products of large integers.
@<The |prod| routine@>=
Graph* prod(m,n)
unsigned m,n; /* lengths of the binary numbers to be multiplied */
{@+@<Local variables for |prod|@>@;
@#
if (m<2) m=2;
if (n<2) n=2;
@<Allocate space for a temporary graph |g| and for auxiliary tables@>;
@<Fill |g| with generalized gates that do parallel multiplication@>;
if (gb_alloc_trouble) {
gb_recycle(g);@+panic(alloc_fault); /* too big */
}
g=reduce(g);
return g; /* if |g==NULL|, the |panic_code| was set by |reduce| */
}
@ The divide-and-conquer recurrences used in this network lead to interesting
patterns. First we use a method for parallel column addition that reduces
the sum of three numbers to the sum of two numbers; repeated use of this
reduction makes it possible to reduce the sum of |m| numbers to a sum of
just two numbers, with a total circuit depth that satisfies the
recurrence $T(3N)=T(2N)+O(1)$. Secondly, when the result has been reduced
to a sum of two numbers, we use a parallel addition scheme based on
recursively ``golden sectioning the data''; in other words, the recursion
partitions the data into two parts such that the ratio of the larger part
to the smaller part is approximately $\phi$. This technique proves to be
slightly better than a binary partition would be, both asymptotically and
for small values of~$m+n$.
\def\flog{\mathop{\rm flog}\nolimits}
We define $\flog N$, the Fibonacci logarithm of~$N$, to be the smallest
nonnegative integer~$k$ such that $N\le F_{k+1}$. Let $N=m+n$. Our parallel
adder for two numbers of $N$ bits will turn out to have depth at most
$2+\flog N$. The unreduced graph~|g| in our circuit for multiplication
will have fewer than $(6m+3\flog N)N$ gates.
@<Allocate space for a temporary graph |g| and for auxiliary tables@>=
m_plus_n=m+n;@+@<Compute $f=\flog(m+n)$@>;
g=gb_new_graph((6*m-7+3*f)*m_plus_n);
if (g==NULL) panic(no_room); /* out of memory before we're even started */
sprintf(g->id,"prod(%u,%u)",m,n);
strcpy(g->format,"ZZZIIVZZZZZZZA");
long_tables=gb_alloc_type(2*m_plus_n+f,@[long@],g->aux_data);
vert_tables=gb_alloc_type(f*m_plus_n,@[Vertex*@],g->aux_data);
if (gb_alloc_trouble) {
gb_recycle(g);
panic(no_room+1); /* out of memory trying to create auxiliary tables */
}
@ @<Local variables for |prod|@>=
unsigned m_plus_n; /* guess what this variable holds */
int f; /* initially $\flog(m+n)$, later flog of other things */
Graph *g; /* graph of generalized gates, to be reduced eventually */
long *long_tables; /* beginning of auxiliary array of |long| numbers */
Vertex **vert_tables; /* beginning of auxiliary array of gate pointers */
@ @<Compute $f=\flog(m+n)$@>=
f=4;@+j=3;@+k=5; /* $j=F_f$, $k=F_{f+1}$ */
while (k<m_plus_n) {
k=k+j;
j=k-j;
f++;
}
@ The well-known formulas for a ``full adder,''
$$ x+y+z=s+2c,\qquad
\hbox{where $s=x\oplus y\oplus z$ and $c=xy\lor yz\lor zx$},$$
can be applied to each bit of an $N$-bit number, thereby providing us
with a way to reduce the sum of three numbers to the sum of two.
The input gates of our network will be called $x_0$, $x_1$, \dots,~$x_{m-1}$,
$y_0$,~$y_1$, \dots,~$y_{n-1}$, and the outputs will be called
$z_0$, $z_1$, \dots,~$z_{m+n-1}$. The logic of the |prod| network will compute
$$(z_{m+n-1}\ldots z_1z_0)_2=(x_{m-1}\ldots x_1x_0)_2\cdot
(y_{n-1}\ldots y_1y_0)_2\,,$$
by first considering the product to be the $m$-fold sum
$A_0+A_1+\cdots+A_{m-1}$, where
$$A_j=2^jx_j\cdot(y_{n-1}\ldots y_1y_0)_2\,,\qquad 0\le j<m.$$
Then the three-to-two rule for addition is used to define further
numbers $A_m$, $A_{m+1}$, \dots,~$A_{3m-5}$ by the scheme
$$A_{m+2j}+A_{m+2j+1}=A_{3j}+A_{3j+1}+A_{3j+2}\,,\qquad 0\le j\le m-3.$$
[A similar but slightly less efficient scheme was used by Pratt and
Stockmeyer in {\sl Journal of Computer and System Sciences \bf12} (1976),
Proposition~5.3. The recurrence used here is related to the Josephus
problem with step-size~3; see {\sl Concrete Mathematics},
{\mathhexbox278}3.3.]
For this purpose we compute intermediate results $P_j$, $Q_j$, and~$R_j$
by the rules
$$\eqalign{P_j&=A_{3j}\oplus A_{3j+1}\,;\cr
Q_j&=A_{3j}\land A_{3j+1}\,;\cr
A_{m+2j}&=P_j\oplus A_{3j+2}\,;\cr
R_j&=P_j\land A_{3j+2}\,;\cr
A_{m+2j+1}&=2(Q_j\lor R_j)\,.\cr}$$
Finally we let
$$\eqalign{U&=A_{3m-6}\oplus A_{3m-5}\,,\cr
V&=A_{3m-6}\land A_{3m-5}\,;\cr}$$
these are the values that would be $P_{m-2}$ and $Q_{m-2}$ if the previous
formulas were allowed to run past $j=m-3$. The final result
$Z=(z_{m+n-1}\ldots z_1z_0)_2$ can now be expressed as
$$Z=U+2V\,.$$
The gates of the first part of the network are conveniently obtained
in groups of $N=m+n$, representing the bits of the quantities $A_j$,
$P_j$, $Q_j$, $R_j$, $U$, and~$V$. We will put the least significant bit
of $A_j$ in gate position |g->vertices+a(j)*N|, where $a(j)=j+1$ for
$0\le j<m$ and $a(m+2j+t)=m+5j+3+2t$ for $0\le j\le m-3$, $0\le t\le1$.
@<Fill |g| with generalized gates that do parallel multiplication@>=
next_vert=g->vertices;
start_prefix("X");@+x=first_of(m,'I');
start_prefix("Y");@+y=first_of(n,'I');
@<Define $A_j$ for $0\le j<m$@>;
@<Define $P_j$, $Q_j$, $A_{m+2j}$, $R_j$, and $A_{m+2j+1}$
for $0\le j\le m-3$@>;
@<Define $U$ and $V$@>;
@<Compute the final result $Z$ by parallel addition@>;
@ @<Local variables for |prod|@>=
register int i,j,k,l; /* all-purpose indices */
register Vertex *v; /* current vertex of interest */
Vertex *x,*y; /* least-significant bits of the input gates */
Vertex *alpha,*beta; /* least-significant bits of arguments */
@ @<Define $A_j$ for $0\le j<m$@>=
for (j=0; j<m; j++) {
numeric_prefix('A',j);
for (k=0; k<j; k++) {
v=new_vert('C');@+v->bit=0; /* this gate is the constant 0 */
}
for (k=0; k<n; k++)
make2(AND,x+j,y+k);
for (k=j+n; k<m_plus_n; k++) {
v=new_vert('C');@+v->bit=0; /* this gate is the constant 0 */
}
}
@ Since |m| is |unsigned|, it is necessary to say `|j<m-2|' here instead
of `|j<=m-3|'.
@d a_pos(j) (j<m? j+1: m+5*((j-m)>>1)+3+(((j-m)&1)<<1))
@<Define $P_j$, $Q_j$, $A_{m+2j}$, $R_j$, and $A_{m+2j+1}$...@>=
for (j=0; j<m-2; j++) {
alpha=g->vertices+(a_pos(3*j)*m_plus_n);
beta=g->vertices+(a_pos(3*j+1)*m_plus_n);
numeric_prefix('P',j);
for (k=0; k<m_plus_n; k++)
make2(XOR,alpha+k,beta+k);
numeric_prefix('Q',j);
for (k=0; k<m_plus_n; k++)
make2(AND,alpha+k,beta+k);
alpha=next_vert-2*m_plus_n;
beta=g->vertices+(a_pos(3*j+2)*m_plus_n);
numeric_prefix('A',m+2*j);
for (k=0; k<m_plus_n; k++)
make2(XOR,alpha+k,beta+k);
numeric_prefix('R',j);
for (k=0; k<m_plus_n; k++)
make2(AND,alpha+k,beta+k);
alpha=next_vert-3*m_plus_n;
beta=next_vert-m_plus_n;
numeric_prefix('A',m+2*j+1);
v=new_vert('C');@+v->bit=0; /* another 0, it multiplies $Q\lor R$ by 2 */
for (k=0; k<m_plus_n-1; k++)
make2(OR,alpha+k,beta+k);
}
@ Actually $v_{m+n-1}$ will never be used (it has to be zero); but we
compute it anyway. We don't have to worry about such nitty gritty details
because |reduce| will get rid of all the obvious redundancy.
@<Define $U$ and $V$@>=
alpha=g->vertices+(a_pos(3*m-6)*m_plus_n);
beta=g->vertices+(a_pos(3*m-5)*m_plus_n);
start_prefix("U");
for (k=0; k<m_plus_n; k++)
make2(XOR,alpha+k,beta+k);
start_prefix("V");
for (k=0; k<m_plus_n; k++)
make2(AND,alpha+k,beta+k);
@* Parallel addition. It's time now to take a deep breath; we have finished the
parallel multiplier except for one last step, the design of a parallel
adder.
The adder is based on the following theory:
We want to perform the binary addition
$$\vbox{\halign{\hfil$#$&&\ \hfil$#$\cr
u_{N-1}&\ldots&u_2&u_1&u_0\cr
v_{N-2}&\ldots&v_1&v_0\cr
\noalign{\kern2pt\hrule\kern4pt}
z_{N-1}&\ldots&z_2&z_1&z_0\cr}}$$
where we know that $u_k+v_k\le1$ for all~$k$. It follows that $z_k=u_k\oplus
w_k$, where $w_0=0$ and
$$ w_k\;=\;v_{k-1}\;\lor\;u_{k-1}v_{k-2}\;\lor\;u_{k-1}u_{k-2}v_{k-3}\;\lor
\;\cdots\;\lor\;u_{k-1}\ldots u_1v_0$$
for $k>0$. The problem has therefore been reduced to the evaluation
of $w_1$, $w_2$, \dots, $w_{N-1}$.
Let $c_k^{\,j}$ denote the {\sc OR} of the first $j$ terms in the formula
that defines $w_k$, and let $d_k^{\,j}$ denote the $j$-fold product
$u_{k-1}u_{k-2}\ldots u_{k-j}$.
Then $w_k=c_k^k$, and we can use a recursive scheme of the form
$$c_k^{\,j}=c_k^{\,i}\lor d_k^{\,i}c_{k-i}^{\,j-i}\,,\qquad
d_k^{\,j}=d_k^{\,i}d_{k-i}^{\,j-i}\,,\qquad j\ge2,$$
to do the evaluation.
\def\down{\mathop{\rm down}}
It turns out that this recursion behaves very nicely if we choose
$i=\down[j]$, where $\down[j]$ is defined for $j>1$ by the formula
$$\down[j]\;=\;j-F_{(\flog j)-1}\,.$$
For example, we have $\flog18=7$ because $F_7=13<18\le21=F_8$,
hence $\down[18]=18-F_6=10$.
Let us write $j\to\down[j]$, and consider the oriented tree on the set
of all positive integers that is defined by this relation. One of the
paths in this tree is, for example, $18\to10\to5\to3\to2\to1$. Our
recurrence for $w_{18}=c_{18}^{18}$ involves $c_{18}^{10}$, which
involves $c_{18}^5$, which involves $c_{18}^3$, and so on; in general,
we will compute $c_k^{\,j}$ for all $j$ with $k\to^*j$, and we will
compute $d_k^{\,j}$ for all $j$ with $k\to^+j$. It is not difficult to
prove that $$k\;\to^*\;j\;\to\;i\qquad\hbox{implies}\qquad
k-i\;\to^*\;j-i\,;$$ therefore the auxiliary factors $c_{k-i}^{\,j-i}$
and $d_{k-i}^{\,j-i}$ needed in the recurrence scheme will already
have been evaluated. (Indeed, one can prove more: Let $l=\flog k$. If
the complete path from $k$ to~$1$ in the tree is $k=k_0\to
k_1\to\cdots\to k_t=1$, then the differences $k_0-k_1$, $k_1-k_2$,
\dots, $k_{t-2}-k_{t-1}$ will consist of precisely the Fibonacci
numbers $F_{l-1}$, $F_{l-2}$, \dots,~$F_2$ except for the numbers that
appear when $F_{l+1}-k$ is written as a sum of non-consecutive
Fibonacci numbers.)
It can also be shown that, when $k>1$, we have
$$\flog k=\min_{0<j<n}\,\max\bigl(1+\flog j,\,2+\flog(k-j)\bigr)\,,$$
and that $\down[k]$ is the smallest~$j$ such that the minimum is
achieved in this equation. Therefore the depth of the circuit for
computing $w_k$ from the $u$'s and~$v$'s is exactly $\flog k$.
In particular, we can be sure that at most $3\flog N$ gates will be
created when computing $z_k$, and that there will be at most $3N\flog N$
gates in the parallel addition portion of the circuit.
@<Compute the final result $Z$ by parallel addition@>=
@<Set up auxiliary tables to handle Fibonacci-based recurrences@>;
@<Create the gates for $W$, remembering intermediate results that
might be reused later@>;
@<Compute the last gates $Z=U\oplus W$, and record their locations
as outputs of the network@>;
g->n=next_vert-g->vertices; /* reduce to the actual number of gates used */
@ When we have created a gate for $w_k$, we will store its address as
the value of $w[k]$ in an auxiliary table. When we've created a gate
for $c_k^{\,i}$ where $i<k$ is a Fibonacci number~$F_{l+1}$ and $l=\flog i\ge2$,
we will store its address as the value of $c[k+(l-2)N]$; the gate
$d_k^{\,i}$ will immediately follow this one. Tables of $\flog j$ and $\down[j]$
will facilitate all these manipulations.
@<Set up auxiliary tables to handle Fibonacci-based recurrences@>=
w=vert_tables;
c=w+m_plus_n;
flog=long_tables;
down=flog+m_plus_n+1;
anc=down+m_plus_n;
flog[1]=0;@+flog[2]=2;
down[1]=0;@+down[2]=1;
for (i=3,j=2,k=3,l=3; l<=m_plus_n; l++) {
if (l>k) {
k=k+j;
j=k-j;
i++; /* $F_i=j<l\le k=F_{i+1}$ */
}
flog[l]=i;
down[l]=l-k+j;
}
@ @<Local variables for |prod|@>=
Vertex *uu, *vv; /* pointer to $u_0$ and $v_0$ */
Vertex **w; /* table of pointers to $w_k$ */
Vertex **c; /* table of pointers to potentially
important intermediate values $c_k^{\,i}$ */
Vertex *cc,*dd; /* pointers to $c_k^{\,i}$ and $d_k^{\,i}$ */
long *flog; /* table of flog values */
long *down; /* table of down values */
long *anc; /* table of ancestors of the current $k$ */
@ @<Create the gates for $W$, remembering intermediate results that
might be reused later@>=
vv=next_vert-m_plus_n;@+uu=vv-m_plus_n;
start_prefix("W");
v=new_vert('C');@+v->bit=0;@+w[0]=v; /* $w_0=0$ */
v=new_vert('=');@+v->alt=vv;@+w[1]=v; /* $w_1=v_0$ */
for (k=2;k<m_plus_n;k++) {
@<Set the |anc| table to a list of the ancestors of |k| in decreasing order,
stopping with |anc[l]=2|@>;
i=1;@+cc=vv+k-1;@+dd=uu+k-1;
while (1) {
j=anc[l]; /* now $i=\down[j]$ */
@#
@<Compute the gate $b_k^{\,j}=d_k^{\,i}\land c_{k-i}^{\,j-i}$@>;
@<Compute the gate $c_k^{\,j}=c_k^{\,i}\lor b_k^{\,j}$@>;
if (flog[j]<flog[j+1]) /* $j$ is a Fibonacci number */
c[k+(flog[j]-2)*m_plus_n]=v;
if (l==0) break;
cc=v;
@<Compute the gate $d_k^{\,j}=d_k^{\,i}\land d_{k-i}^{\,j-i}$@>;
dd=v;
i=j;
l--;
}
w[k]=v;
}
@ If $k\to j$ we call $j$ an ``ancestor'' of $k$ because we are thinking
of the tree defined by `$\to$'; this tree is rooted at $2\to1$.
@<Set the |anc| table to a list of the ancestors of |k| in decreasing order,
stopping with |anc[l]=2|@>=
for (l=0,j=k;;l++,j=down[j]) {
anc[l]=j;
if (j==2) break;
}
@ @d spec_gate(v,a,k,j,t)
v=next_vert++;
sprintf(name_buf,"%c%d:%d",a,k,j);
v->name=gb_save_string(name_buf);
v->typ=t;
@<Compute the gate $b_k^{\,j}=d_k^{\,i}\land c_{k-i}^{\,j-i}$@>=
spec_gate(v,'B',k,j,AND);
gb_new_arc(v,dd,DELAY); /* first argument is $d_k^{\,i}$ */
f=flog[j-i]; /* get ready to compute the second argument, $c_{k-i}^{\,j-i}$ */
gb_new_arc(v,f>0? c[k-i+(f-2)*m_plus_n]:vv+k-i-1,DELAY);
@ @<Compute the gate $c_k^{\,j}=c_k^{\,i}\lor b_k^{\,j}$@>=
if (l) {
spec_gate(v,'C',k,j,OR);
} else v=new_vert(OR); /* if $l$ is zero, this gate is $c_k^k=w_k$ */
gb_new_arc(v,cc,DELAY); /* first argument is $c_k^{\,i}$ */
gb_new_arc(v,next_vert-2); /* second argument is $b_k^{\,j}$ */
@ Here we reuse the value $f=\flog(j-i)$ computed a minute ago.
@<Compute the gate $d_k^{\,j}=d_k^{\,i}\land d_{k-i}^{\,j-i}$@>=
spec_gate(v,'D',k,j,AND);
gb_new_arc(v,dd,DELAY); /* first argument is $d_k^{\,i}$ */
gb_new_arc(v,f>0? c[k-i+(f-2)*m_plus_n]+1:uu+k-i-1,DELAY);
/* $d_{k-i}^{\,j-i}$ */
@ The output list will contain the gates in ``big-endian order''
$z_{m+n-1}$ \dots, $z_1$, $z_0$, because we insert them into the
|outs| list in little-endian order.
@<Compute the last gates $Z=U\oplus W$...@>=
start_prefix("Z");
for (k=0;k<m_plus_n;k++) {@+register Arc *a=gb_virgin_arc();
a->tip=make2(XOR,uu+k,w[k]);
a->next=g->outs;
g->outs=a;
}
@* Partial evaluation. The subroutine call |partial_gates(g,r,prob,seed,buf)|
creates a new gate graph from a given gate graph~|g| by ``partial evaluation,''
i.e., by setting some of the inputs to constant values and simplifying the
result. The new graph is usually smaller than |g|; it may, in fact, be a great
deal smaller. Graph~|g| is destroyed in the process.
The first |r| inputs of |g| are retained; each remaining input is
retained with probability |prob/65536|, and if not retained it is assigned
a random constant value. For example, about half of the inputs will become
constant if |prob=32768|.
The |seed| parameter defines a machine-independent source of random
numbers, and it may be given any value between $0$ and $2^{31}-1$.
If the |buf| parameter is non-null, it should be the address of a string.
In such a case, |partial_gates| will put a record of its partial evaluation
into that string; |buf| will contain one character for each input gate
after the first |r|, namely |'*'| if the input was
retained, |'0'| if it was set to~$0$, or |'1'| if it was set to~$1$.
The new graph will contain only gates that contribute to the computation
of at least one output value. Therefore some input gates may disappear
even though they were supposedly ``retained,'' i.e., even though their
value has not been set constant. The |name| field of a vertex can be
used to determine exactly which input gates have survived.
If graph |g| was created by |risc|, users will probably want to make
|r>=1|, since the whole RISC circuit collapses to zero whenever its
first input `\.{RUN}' is set to 0.
An interesting class of graphs is produced by
the function call |partial_gates(prod(m,n),m,0,seed,NULL)|, which
creates a graph corresponding to a circuit that multiplies a given |m|-bit
number by a fixed (but randomly selected) |n|-bit constant. If the constant
is not zero, all |m| of the ``retained'' input gates necessarily survive.
The demo program called |multiply| illustrates such circuits.
The graph |g| might be a generalized network; i.e., it might
have the |'C'| or |'='| gates described earlier. Notice that if |r| is
sufficiently large, |partial_gates| becomes equivalent to the |reduce|
routine. Therefore we need not make that private routine public.
As usual, the result will be |NULL|, and |panic_code| will be set,
if |partial_gates| is unable to complete its task.
@<The |partial_gates| routine@>=
Graph *partial_gates(g,r,prob,seed,buf)
Graph *g; /* generalized gate graph */
unsigned r; /* the number of initial gates to leave untouched */
unsigned long prob; /* scaled probability of touching subsequent input gates */
long seed; /* seed value for random number generation */
char *buf; /* optional parameter for information about partial assignment */
{@+register Vertex *v; /* the current gate of interest */
if (g==NULL) panic(missing_operand); /* where is |g|? */
gb_init_rand(seed); /* get them random numbers rolling */
for (v=g->vertices+r;v<g->vertices+g->n;v++)
switch (v->typ) {
case 'C': case '=': continue; /* input gates may still follow */
case 'I': if ((gb_next_rand()>>15)>=prob) {
v->typ='C';@+v->bit=gb_next_rand()>>30;
if (buf) *buf++=v->bit+'0';
} else if (buf) *buf++='*';
break;
default: goto done; /* no more input gates can follow */
}
done:if (buf) *buf=0; /* terminate the string */
g=reduce(g);
@<Give the reduced graph a suitable |id|@>;
return g; /* if |(g==NULL)|, a |panic_code| has been set by |reduce| */
}
@ The |buf| parameter is not recorded in the graph's |id| field, since it
has no effect on the graph itself.
@<Give the reduced graph a suitable |id|@>=
if (g) {
strcpy(name_buf,g->id);
if (strlen(name_buf)>54) strcpy(name_buf+51,"...");
sprintf(g->id,"partial_gates(%s,%u,%lu,%ld)",name_buf,r,prob,seed);
}
@* Index. Here is a list that shows where the identifiers of this program are
defined and used.