source: vis_dev/glu-2.1/src/cmuBdd/bdd.3 @ 6

.\" BDD library man page
.TH BDD 3 "11 June 1993"
.SH NAME
bdd \- a binary decision diagram (BDD) package
.SH SYNOPSIS
.B #include <bdduser.h>
.SH DESCRIPTION
The
.B libbdd
library provides a set of routines for manipulating binary decision
diagrams (BDDs).  Some support is also provided for multi-terminal
BDDs (MTBDDs).  Programs designed to be used with the library should
use the
.B -lbdd -lmem
options to
.B cc
when linking.
.SH "LIST OF FUNCTIONS"
.nf
.ta 3in
\fIName\fP      \fIFunction\fP
bdd_init        Initialize the library
bdd_quit        Finish using the library
bdd_new_var_first       Create a variable first in the order
bdd_new_var_last        Create a variable last in the order
bdd_new_var_before      Create a variable before an existing one
bdd_new_var_after       Create a variable after an existing one
bdd_var_with_index      Obtain an existing variable
bdd_var_with_id Obtain an existing variable
bdd_one Constant TRUE
bdd_zero        Constant FALSE
bdd_and Logical AND
bdd_nand        Logical NAND
bdd_or  Logical OR
bdd_nor Logical NOR
bdd_xor Logical XOR
bdd_xnor        Logical XNOR
bdd_identity    Logical identity
bdd_not Logical NOT
bdd_ite Logical IF-THEN-ELSE
bdd_if  Get the variable of the top node in a BDD
bdd_then        Get the THEN branch of the top node in a BDD
bdd_else        Get the ELSE branch of the top node in a BDD
bdd_if_index    Get the index of the top variable in a BDD
bdd_if_id       Get a unique ID number for the top variable
bdd_intersects  Check intersection
bdd_implies     Check boolean implication
bdd_new_assoc   Make a new variable association
bdd_free_assoc  Free a variable association
bdd_temp_assoc  Set the temporary variable association
bdd_augment_temp_assoc  Set the temporary variable association
bdd_assoc       Set the current variable association
bdd_exists      Existential quantification
bdd_forall      Universal quantification
bdd_rel_prod    Relational product
bdd_compose     Substitute for a variable
bdd_substitute  Substitute for a set of variables
bdd_reduce      Simplify given a constraint
bdd_cofactor    Generalized cofactor
bdd_depends_on  Determine if a BDD depends on a variable
bdd_support     Find the support of a BDD
bdd_satisfy     Find a satisfying assignment
bdd_satisfy_support     Find a satisfying assignment
bdd_satisfying_fraction Fraction of valuations satisfying a BDD
bdd_swap_vars   Swap two variables in a BDD
bdd_apply2      Generic apply routine
bdd_apply1      Generic apply routine
bdd_size        Number of nodes in a BDD
bdd_size_multiple       Number of nodes in multiple BDDs
bdd_profile     Node profile of a BDD
bdd_profile_multiple    Node profile of multiple BDDs
bdd_function_profile    Function profile of a BDD
bdd_function_profile_multiple   Function profile of multiple BDDs
bdd_print_bdd   Print a BDD in human-readable form
bdd_print_profile       Print a node profile of a BDD
bdd_print_profile_multiple      Print a profile of multiple BDDs
bdd_print_function_profile      Print a function profile of a BDD
bdd_dump_bdd    Write a BDD to a file
bdd_undump_bdd  Load a BDD from a file
bdd_type        Classify a BDD
bdd_free        Decrease the reference count of a BDD
bdd_unfree      Increase the reference count of a BDD
bdd_clear_refs  Set all BDD reference counts to zero
bdd_gc  Garbage collect unused BDD nodes
bdd_total_size  Total number of BDD nodes in use
bdd_vars        Total number of variables in existence
bdd_cache_ratio Get/set operation result cache size
bdd_node_limit  Get/set the number of BDD nodes allowed
bdd_overflow    Get/clear overflow flag
bdd_overflow_closure    Set a closure to invoke on overflow
bdd_abort_closure       Used to abort operations in progress
bdd_stats       Print statistics
bdd_dynamic_reordering  Specify dynamic reordering technique
bdd_reorder     Invoke dynamic reordering
bdd_new_var_block       Create variable block
bdd_var_block_reorderable       Set block reorderability
mtbdd_free_terminal_closure     Called when freeing an MTBDD terminal
mtbdd_get_terminal      Get an MTBDD terminal node
mtbdd_terminal_value    Get the value of an MTBDD terminal node
mtbdd_ite       IF-THEN-ELSE operation for MTBDDs
mtbdd_equal     Equality operation for MTBDDs
mtbdd_transform Applies the current transform to an MTBDD
mtbdd_transform_closure Called to set the MTBDD transform
mtbdd_one_data  Sets the MTBDD data value for TRUE
.fi
.SH "BASIC CONCEPTS"
For a general overview of BDDs, see the original article by Bryant
[1].

Almost all of the BDD library routines require a BDD manager as one of
their arguments.  A BDD manager is a structure which holds various
variables used by the BDD routines.  The type
.B bdd_manager
is a pointer to this structure.  BDDs themselves are also represented
internally as structures.  The type
.B bdd
is a pointer to one of these structures.

There is a global ordering on the boolean variables which may appear
in a BDD.  The variable at the root of a BDD is earlier in the
ordering than all other variables in the BDD.  Each variable has an
index which represents its position in the ordering;
.I v1
appears before
.I v2
in the ordering if and only if the index for
.I v1
is less than the ordering for \fIv2\fR.  Each variable is also
assigned a unique ID number that is invariant.  Since variables can be
created at any position within the order, this is not true for the
index.  Also, the library supports dynamic variable reordering.  With
dynamic variable reordering, variables may be shuffled around in the
middle of an operation in order to reduce the number of BDD nodes in
use.

Some routines such as
.B bdd_substitute
require a mapping from variables to BDDs to operate.  This mapping is
supplied in the form of a variable association which is a set of
pairs.  The first element of each pair is the variable, and the second
element is the BDD that the variable is associated with.  Multiple
associations may exist at any one time.  Other routines such as
.B bdd_exists
require sets of variables.  Sets of variables are represented by
variable associations where only the fact that a variable is
associated with some BDD is significant.  There is one association,
called the temporary variable association, which is special in two
ways.  First, this association always exists.  Second, results are not
cached across calls when this association is used.  The temporary
association is intended for when an association will not be reused.
The advantage of using it is that setting the temporary association
does not require scanning the result cache to flush out-of-date
results.

The results returned by the library represent canonical forms and may
be checked for equivalence using the standard C comparison operators.
For example:

.nf
{
  bdd_manager bddm;
  bdd f;
  ...
  if (f == bdd_one(bddm))  /* Tautology check */
    ...
}
.fi

For checking for relations such as boolean implication, use
.B bdd_intersects
and \fBbdd_implies\fR.

Multi-terminal BDDs are like BDDs, except an MTBDD may have more than
just the constants TRUE and FALSE at the leaves.  Passing an MTBDD to
a routine expecting a BDD will give undefined results, except where
noted below.  MTBDDs are built up using
.B mtbdd_get_terminal
and \fBmtbdd_ite\fR.
.SH "STORAGE MANAGEMENT"
Each BDD node has an associated reference count which records the
number of references to the BDD (internal and external).  Whenever a
BDD is returned from a function, the reference count for its top node
is incremented.  (If the BDD did not exist before, the reference count
will be 1.)  Each time a garbage collection occurs, either internally
or because of a call to \fBbdd_gc\fR, all nodes which are not
referenced are reclaimed.  The reference count of a BDD may be
decremented by calling \fBbdd_free\fR.  This should be done whenever
possible for maximum space efficiency.  You may also specify a limit
for the total number of BDD nodes using \fBbdd_node_limit\fR.  If it
is not possible to complete an operation without exceeding this limit,
the operation is aborted and (by default) a null pointer is returned.
Whenever this happens, the reference counts of all nodes are restored
to what they were before the operation.  If a null pointer is passed
to a routine, the routine simply returns null.  Thus, it is not
necessary to check for overflows after each operation.  There is also
an internal flag that indicates whether any operation has caused an
overflow.  It may be read and reset by \fBbdd_overflow\fR.
Optionally, a user-defined closure may be invoked when an overflow
occurs; see \fBbdd_overflow_closure\fR.  Also see \fBbdd_free\fR,
\fBbdd_unfree\fR, \fBbdd_clear_refs\fR, \fBbdd_node_limit\fR and
\fBbdd_gc\fR.  The library also includes high-performance replacements
for
.B malloc
and \fBfree\fR.  See the discussion at the end of the section on
adding new routines.
.SH "DETAILED DESCRIPTION"
.B bdd_manager
.br
.B bdd_init()
.in +4
Creates and initializes a new BDD manager.  Multiple BDD managers may
exist at any time.
.LP
.B void
.br
.B bdd_quit(bddm)
.br
.B bdd_manager bddm;
.in +4
Deallocates the BDD manager given by
.B bddm
and all the storage associated with it.
.LP
.B bdd
.br
.B bdd_new_var_first(bddm)
.br
.B bdd_manager bddm;
.in +4
Creates a new variable at the start of the BDD variable ordering and
returns the BDD for it.
.LP
.B bdd
.br
.B bdd_new_var_last(bddm)
.br
.B bdd_manager bddm;
.in +4
Creates a new variable at the end of the BDD variable ordering and
returns the BDD for it.
.LP
.B bdd
.br
.B bdd_new_var_before(bddm, var)
.br
.B bdd_manager bddm;
.br
.B bdd var;
.in +4
Creates a new variable which is before
.B var
in the BDD variable ordering and returns the BDD for the new variable.
.LP
.B bdd
.br
.B bdd_new_var_after(bddm, var)
.br
.B bdd_manager bddm;
.br
.B bdd var;
.in +4
Creates a new variable which is after
.B var
in the BDD variable ordering and returns the BDD for the new variable.
.LP
.B bdd
.br
.B bdd_var_with_index(bddm, i)
.br
.B bdd_manager bddm;
.br
.B long i;
.in +4
If a variable with index
.B i
has been created, returns the BDD for the variable.  If no such
variable exists, returns null.  See also \fBbdd_if_index\fR.
.LP
.B bdd
.br
.B bdd_var_with_id(bddm, i)
.br
.B bdd_manager bddm;
.br
.B long i;
.in +4
If a variable with ID
.B i
has been created, returns the BDD for the variable.  If no such
variable has been created, returns null.  See also \fBbdd_if_id\fR.
.LP
.B bdd
.br
.B bdd_one(bddm)
.br
.B bdd_manager bddm;
.in +4
Returns the BDD for the constant TRUE.
.LP
.B bdd
.br
.B bdd_zero(bddm)
.br
.B bdd_manager bddm;
.in +4
Returns the BDD for the constant FALSE.
.LP
.B bdd
.br
.B bdd_and(bddm, f, g)
.br
.B bdd_manager bddm;
.br
.B bdd f, g;
.in +4
Returns the BDD for the logical AND of
.B f
and \fBg\fR.
.LP
.B bdd
.br
.B bdd_nand(bddm, f, g)
.br
.B bdd_manager bddm;
.br
.B bdd f, g;
.in +4
Returns the BDD for the logical NAND of
.B f
and \fBg\fR.
.LP
.B bdd
.br
.B bdd_or(bddm, f, g)
.br
.B bdd_manager bddm;
.br
.B bdd f, g;
.in +4
Returns the BDD for the logical OR of
.B f
and \fBg\fR.
.LP
.B bdd
.br
.B bdd_nor(bddm, f, g)
.br
.B bdd_manager bddm;
.br
.B bdd f, g;
.in +4
Returns the BDD for the logical NOR of
.B f
and \fBg\fR.
.LP
.B bdd
.br
.B bdd_xor(bddm, f, g)
.br
.B bdd_manager bddm;
.br
.B bdd f, g;
.in +4
Returns the BDD for the logical XOR of
.B f
and \fBg\fR.
.LP
.B bdd
.br
.B bdd_xnor(bddm, f, g)
.br
.B bdd_manager bddm;
.br
.B bdd f, g;
.in +4
Returns the BDD for the logical XNOR of
.B f
and \fBg\fR.
.LP
.B bdd
.br
.B bdd_identity(bddm, f)
.br
.B bdd_manager bddm;
.br
.B bdd f;
.in +4
Returns the BDD for \fBf\fR.  The only real effect of this function is
to increase the reference count of \fBf\fR.  Also works with MTBDDs.
.LP
.B bdd
.br
.B bdd_not(bddm, f)
.br
.B bdd_manager bddm;
.br
.B bdd f;
.in +4
Returns the BDD for the logical NOT of \fBf\fR.
.LP
.B bdd
.br
.B bdd_ite(bddm, f, g, h)
.br
.B bdd_manager bddm;
.br
.B bdd f, g, h;
.in +4
Returns the BDD for the logical operation IF
.B f
THEN
.B g
ELSE \fBh\fR.
.LP
.B bdd
.br
.B bdd_if(bddm, f)
.br
.B bdd_manager bddm;
.br
.B bdd f;
.in +4
Returns the BDD for the variable which labels the root of the BDD
given by \fBf\fR.  Also works with MTBDDs.  The result is undefined if
.B f
is one of the constants TRUE or FALSE or an MTBDD terminal node.
.LP
.B bdd
.br
.B bdd_then(bddm, f)
.br
.B bdd_manager bddm;
.br
.B bdd f;
.in +4
Returns the BDD for the THEN branch of the root of the BDD given by
\fBf\fR.  Also works with MTBDDs.  The result is undefined if
.B f
is one of the constants TRUE or FALSE or an MTBDD terminal node.
.LP
.B bdd
.br
.B bdd_else(bddm, f)
.br
.B bdd_manager bddm;
.br
.B bdd f;
.in +4
Returns the BDD for the ELSE branch of the root of the BDD given by
\fBf\fR.  Also works with MTBDDs.  The result is undefined if
.B f
is one of the constants TRUE or FALSE or an MTBDD terminal node.
.LP
.B long
.br
.B bdd_if_index(bddm, f)
.br
.B bdd_manager bddm;
.br
.B bdd f;
.in +4
Returns the index of the variable which labels the root of the BDD
given by \fBf\fR.  Also works with MTBDDs.  The result is undefined if
.B f
is one of the constants TRUE or FALSE or an MTBDD terminal node.  The
variable at the start of variable ordering has index 0, the next has
index 1, etc.  Note that creating new variables may change the index
of existing variables.  Dynamic reordering may also change the index
of variables.
.LP
.B long
.br
.B bdd_if_id(bddm, f)
.br
.B bdd_manager bddm;
.br
.B bdd f;
.in +4
Returns a unique ID number for the variable which labels the root of
the BDD given by \fBf\fR.  Also works with MTBDDs.  The result is
undefined if
.B f
is one of the constants TRUE or FALSE or an MTBDD terminal node.  The
ID for a variable is fixed at the time the variable is created and
never changes after that.
.LP
.B bdd
.br
.B bdd_intersects(bddm, f, g)
.br
.B bdd_manager bddm;
.br
.B bdd f, g;
.in +4
Computes a BDD that implies the conjunction of
.B f
and \fBg\fR.  If the conjunction is not FALSE, then the BDD returned
will not be FALSE.  Also, the function tries to construct as few new
nodes as possible.  This routine is intended for cases where you need
to test for a FALSE conjunction, and, when it the conjunction is not
FALSE, to obtain just one valuation satisfying both
.B f
and \fBg\fR.  A non-FALSE result from
.B bdd_intersects
can be passed directly to a routine like \fBbdd_satisfy_support\fR.
.LP
.B bdd
.br
.B bdd_implies(bddm, f, g)
.br
.B bdd_manager bddm;
.br
.B bdd f, g;
.in +4
This is equivalent to calling
.B bdd_intersects
with
.B f
and NOT \fBg\fR.
.LP
.B int
.br
.B bdd_new_assoc(bddm, assoc, pairs)
.br
.B bdd_manager bddm;
.br
.B bdd *assoc;
.br
.B int pairs;
.in +4
Creates or finds a variable association.  The association is specified
by
.B assoc
and should be a null-terminated array of BDDs.  If
.B pairs
is 0, the array is assumed to be an array of variables.  In this case,
each variable is paired with the BDD for TRUE.  Such an association
may essentially be viewed as specifying a set of variables for use
with routines such as \fBbdd_exists\fR.  If
.B pairs
is nonzero, then the even numbered array elements should be variables
and the odd numbered elements should be the BDDs which they are mapped
to.  In both cases, the return value is an integer identifier for this
association.  Note: if the given association is equivalent to one
which already exists, the same identifier is used for both, and the
reference count of the association is increased by one.
.LP
.B void
.br
.B bdd_free_assoc(bddm, id)
.br
.B bdd_manager bddm;
.br
.B int id;
.in +4
Decrements the reference count of the variable association with
identifier \fBid\fR, and frees it if the reference count becomes zero.
.LP
.B void
.br
.B bdd_temp_assoc(bddm, assoc, pairs)
.br
.B bdd_manager bddm;
.br
.B bdd *assoc;
.br
.B int pairs;
.in +4
Sets the temporary variable association.  The arguments
.B assoc
and
.B pairs
are as in \fBbdd_new_assoc\fR.
.LP
.B void
.br
.B bdd_augment_temp_assoc(bddm, assoc, pairs)
.br
.B bdd_manager bddm;
.br
.B bdd *assoc;
.br
.B int pairs;
.in +4
Add to the temporary variable association.  The arguments
.B assoc
and
.B pairs
are as in \fBbdd_new_assoc\fR.  Any existing associations are
overwritten.  This is mainly used when doing things like substituting
for all variables in a BDD.  It isn't necessary to clear out the
temporary association in such cases, so you can save a little time by
using this routine.
.LP
.B int
.br
.B bdd_assoc(bddm, id)
.br
.B bdd_manager bddm;
.br
.B int id;
.in +4
Sets the current variable association to the one identified by
\fBid\fR.  The identifier for the old current association is returned.
The temporary variable association has identifier -1.
.LP
.B bdd
.br
.B bdd_exists(bddm, f)
.br
.B bdd_manager bddm;
.br
.B bdd f;
.in +4
Returns the BDD for
.B f
with all the variables that are paired with something in the current
variable association existentially quantified out.
.LP
.B bdd
.br
.B bdd_forall(bddm, f)
.br
.B bdd_manager bddm;
.br
.B bdd f;
.in +4
Returns the BDD for
.B f
with all the variables that are paired with something in the current
variable association universally quantified out.
.LP
.B bdd
.br
.B bdd_rel_prod(bddm, f, g)
.br
.B bdd_manager bddm;
.br
.B bdd f, g;
.in +4
Returns the BDD for the logical AND of
.B f
and
.B g
with all the variables that are paired with something in the current
variable association existentially quantified out.  If
.B f
and
.B g
are viewed as boolean relations, this operation corresponds to
relational product.  This routine is generally much more efficient
than doing the operations separately.
.LP
.B bdd
.br
.B bdd_compose(bddm, f, g, h)
.br
.B bdd_manager bddm;
.br
.B bdd f, g, h;
.in +4
Returns the BDD for the substitution of
.B h
for the variable
.B g
in \fBf\fR.  When
.B h
does not depend on \fBg\fR, the operation may be viewed as composition
of boolean functions.  If
.B h
does depend on \fBg\fR, it corresponds to instantaneous substitution
in a boolean formula.
.LP
.B bdd
.br
.B bdd_substitute(bddm, f)
.br
.B bdd_manager bddm;
.br
.B bdd f;
.in +4
Returns the BDD for
.B f
under a substitution defined by the current variable association.
Each variable is replaced by its associated BDD.  The substitution is
effectively simultaneous.
.LP
.B bdd
.br
.B bdd_reduce(bddm, f, g)
.br
.B bdd_manager bddm;
.br
.B bdd f, g;
.in +4
Returns a BDD which agrees with
.B f
for all valuations which satisfy \fBg\fR.  The result is usually
smaller in terms of number of BDD nodes than \fBf\fR.  This operation
is typically used in state space searches to simplify the
representation for the set of states which will be expanded at each
step.
.LP
.B bdd
.br
.B bdd_cofactor(bddm, f, g)
.br
.B bdd_manager bddm;
.br
.B bdd f, g;
.in +4
Returns a BDD for the generalized cofactor of
.B f
by \fBg\fR.  The BDD indicated by
.B g
should not be the constant FALSE.  For some properties of this
operation, see Touati
.I et al.
[2].
.LP
.B int
.br
.B bdd_depends_on(bddm, f, g)
.br
.B bdd_manager bddm;
.br
.B bdd f;
.br
.B bdd g;
.in +4
Returns 1 if the BDD or MTBDD
.B f
depends on the variable given by the BDD \fBg\fR, and returns 0
otherwise.
.LP
.B void
.br
.B bdd_support(bddm, f, support)
.br
.B bdd_manager bddm;
.br
.B bdd f;
.br
.B bdd *support;
.in +4
Stores the support of
.B f
as a null-terminated sequence of variables in \fBsupport\fR.  Works
for MTBDDs also.
.LP
.B bdd
.br
.B bdd_satisfy(bddm, f)
.br
.B bdd_manager bddm;
.br
.B bdd f;
.in +4
Returns a BDD which is not false, implies \fBf\fR, and has at most one
BDD node at each level.  The BDD indicated by
.B f
should not be the constant FALSE.
.LP
.B bdd
.br
.B bdd_satisfy_support(bddm, f)
.br
.B bdd_manager bddm;
.br
.B bdd f;
.in +4
Returns a BDD which is not false, implies \fBf\fR, has at most one
BDD node at each level, and has a node labeled with each variable
which is paired with something in the current variable association.
If
.B f
is the constant FALSE, the result is undefined.
.LP
.B double
.br
.B bdd_satisfying_fraction(bddm, f)
.br
.B bdd_manager bddm;
.br
.B bdd f;
.in +4
Returns the fraction of valuations which satisfy \fBf\fR.  If
.B f
is a function of 
.I n
variables, then 2 to the power
.I n
times this fraction is the number of valuations which satisfy \fBf\fR.
.LP
.B bdd
.br
.B bdd_swap_vars(bddm, f, g, h)
.br
.B bdd_manager bddm;
.br
.B bdd f;
.br
.B bdd g;
.br
.B bdd h;
.in +4
Returns the BDD for
.B f
with
.B g
substituted for
.B h
and
.B h
substituted for \fBg\fR.  The substitution is effectively
simultaneous.
.LP
.B bdd
.br
.B bdd_apply2(bddm, terminal_fn, f, g, env)
.br
.B bdd_manager bddm;
.br
.B bdd (*terminal_fn)();
.br
.B bdd f;
.br
.B bdd g;
.br
.B pointer env;
.in +4
This is a generic two-argument operation.  The behavior of the
operation on terminal values is given by \fBterminal_fn\fR.  It should
take as arguments: the BDD manager, pointers to two BDDs (the
arguments for the call), and the pointer given by \fBenv\fR.  If the
value of the call can be determined immediately from the arguments, it
should return that value.  Otherwise, it should return a null pointer.
In this case, it may also use the BDD pointers that it received to
alter the arguments to the call.  A typical use for this ability is to
put the arguments in a canonical order for commutative operations.
The function should not alter the reference counts of either the
arguments or the returned value.  Also, the returned value (if
non-null) has its temporary reference count incremented once
automatically.  If your function always returns one of the arguments
or TRUE or FALSE, this is the right thing and you don't have to worry
about it.  If you want to call other routines to determine the return
value, you should read the section on adding new routines below.
Works with MTBDDs.
.LP
.B bdd
.br
.B bdd_apply1(bddm, terminal_fn, f, env)
.br
.B bdd_manager bddm;
.br
.B bdd (*terminal_fn)();
.br
.B bdd f;
.br
.B pointer env;
.in +4
This is a generic one-argument operation.  It is basically like
\fBbdd_apply2\fR, except that
.B terminal_fn
takes a single BDD pointer argument instead of the pair of pointers in
the two-argument case.  Works with MTBDDs.
.LP
.B long
.br
.B bdd_size(bddm, f, negout)
.br
.B bdd_manager bddm;
.br
.B bdd f;
.br
.B int negout;
.in +4
Returns the number of nodes in \fBf\fR.  The parameter
.B negout
is a flag indicating whether negative output pointers should be
considered.  The library uses this type of pointer flag internally,
so if the flag is nonzero, the actual number of nodes used is
returned.  If the flag is zero, the return value is the number of
nodes which would be needed to represent
.B f
using a basic BDD.  Works for MTBDDs too.
.LP
.B long
.br
.B bdd_size_multiple(bddm, fs, negout)
.br
.B bdd_manager bddm;
.br
.B bdd *fs;
.br
.B int negout;
.in +4
Returns the number of nodes in the set of BDDs or MTBDDs given by
\fBfs\fR, which should be a null-terminated array.  Nodes which are
shared among the BDDs are only counted once.  The parameter
.B negout
is as in \fBbdd_size\fR.
.LP
.B void
.br
.B bdd_profile(bddm, f, level_counts, negout)
.br
.B bdd_manager bddm;
.br
.B bdd f;
.br
.B long *level_counts;
.br
.B int negout;
.in +4
Returns the ``node profile'' of \fBf\fR, i.e., the number of nodes at
each level in \fBf\fR.  The parameter
.B level_counts
should be an array of longs of size one plus the number of variables
in existence (see \fBbdd_vars\fR).  On return, this array holds the
profile; the \fIi\fRth entry is the number of nodes labeled with the
variable of index \fIi\fR.  The last entry corresponds to the nodes
for TRUE and FALSE.  The parameter
.B negout
is as in \fBbdd_size\fR.  Works for MTBDDs too; in this case, the
last entry corresponds to the MTBDD terminal nodes.
.LP
.B void
.br
.B bdd_profile_multiple(bddm, fs, level_counts, negout)
.br
.B bdd_manager bddm;
.br
.B bdd* fs;
.br
.B long *level_counts;
.br
.B int negout;
.in +4
Returns the ``node profile'' of the set of BDDs or MTBDDs given by
\fBfs\fR, which should be a null-terminated array.  The parameters
\fBlevel_counts\fR and
.B negout
are as in \fBbdd_profile\fR.
.LP
.B void
.br
.B bdd_function_profile(bddm, f, func_counts)
.br
.B bdd_manager bddm;
.br
.B bdd f;
.br
.B long *func_counts;
.in +4
Returns the ``function profile'' of \fBf\fR, i.e., the number of
functions at or below each level in \fBf\fR.  The parameter
.B func_counts
should be an array of longs of size one plus the number of variables
in existence (see \fBbdd_vars\fR).  On return, this array holds the
profile.  The \fIi\fRth entry corresponds to the number of functions
which can be obtained by restricting those variables of index less
than \fIi\fR, provided that
.B f
has at least one node labeled with the variable of index \fIi\fR.  If
.B f
has no nodes labeled with the variable of index \fIi\fR, then the
\fIi\fRth entry of the profile is 0.  Works for MTBDDs also.
.LP
.B void
.br
.B bdd_function_profile_multiple(bddm, fs, func_counts)
.br
.B bdd_manager bddm;
.br
.B bdd *fs;
.br
.B long *func_counts;
.in +4
Returns the ``function profile'' of the set of BDDs or MTBDDs given by
\fBfs\fR, which should be a null-terminated array.  The parameter
.B func_counts
is as in \fBbdd_function_profile\fR.
.LP
.B void
.br
.B bdd_print_bdd(bddm, f, naming_fn, terminal_id_fn, env, fp)
.br
.B bdd_manager bddm;
.br
.B bdd f;
.br
.B char *(*naming_fn)();
.br
.B char *(*terminal_id_fn)();
.br
.B pointer env;
.br
.B FILE *fp;
.in +4
Prints a human-readable representation of the BDD or MTBDD
.B f
to the file given by \fBfp\fR.  The
.B naming_fn
should be a pointer to a function taking a \fBbdd_manager\fR, a
.B bdd
and the pointer given by \fBenv\fR.  This function should return
either a null pointer or a string that is the name of the supplied
variable.  If it returns a null pointer, a default name is generated
based on the index of the variable.  It is also legal for
.B naming_fn
to be null; in this case, default names are generated for all variables.
The macro
.B bdd_naming_fn_none
is a null pointer of suitable type.
.B terminal_id_fn
should be a pointer to a function taking a
.B bdd_manager
and two longs, plus the pointer given by \fBenv\fR.  It should
return either a null pointer or a string representing the MTBDD
terminal represented by the given value.  If it returns a null
pointer, or if
.B terminal_id_fn
is null, then default names are generated for the terminals.
The macro
.B bdd_terminal_id_fn_none
is a null pointer of suitable type.
.LP
.B void
.br
.B bdd_print_profile(bddm, f, naming_fn, env, width, fp)
.br
.B bdd_manager bddm;
.br
.B bdd f;
.br
.B char *(*naming_fn)();
.br
.B pointer env;
.br
.B int width;
.br
.B FILE *fp;
.in +4
Prints a node profile of a BDD in histogram form.  The argument
.B naming_fn
should be as described in \fBbdd_print_bdd\fR.  The width of the
output stream is specified by \fBwidth\fR.  This is used to determine
how to scale the histogram.
.LP
.B void
.br
.B bdd_print_profile_multiple(bddm, fs, naming_fn, env, width, fp)
.br
.B bdd_manager bddm;
.br
.B bdd *fs;
.br
.B char *(*naming_fn)();
.br
.B pointer env;
.br
.B int width;
.br
.B FILE *fp;
.in +4
Prints a node profile of a set of BDDs, which should be given as a
null-terminated array.  The other arguments are as in
\fBbdd_print_profile\fR.
.LP
.B void
.br
.B bdd_print_function_profile(bddm, f, naming_fn, env, width, fp)
.br
.B bdd_manager bddm;
.br
.B bdd f;
.br
.B char *(*naming_fn)();
.br
.B pointer env;
.br
.B int width;
.br
.B FILE *fp;
.in +4
Prints a function profile of a BDD in histogram form.  The arguments
are the same as those to \fBbdd_print_profile\fR.
.LP
.B int
.br
.B bdd_dump_bdd(bddm, f, vars, fp)
.br
.B bdd_manager bddm;
.br
.B bdd f;
.br
.B bdd *vars;
.br
.B FILE *fp;
.in +4
Writes an encoded description of the BDD or MTBDD
.B f
to the file given by \fBfp\fR.  The argument
.B vars
should be a null-terminated array of variables that include the
support of \fBf\fR.  These variables need not be in order of
increasing index.  The function returns a nonzero value if
.B f
was written to the file successfully.
.LP
.B bdd
.br
.B bdd_undump_bdd(bddm, vars, fp, error)
.br
.B bdd_manager bddm;
.br
.B bdd *vars;
.br
.B FILE *fp;
.br
.B int *error;
.in +4
Loads an encoded description of a BDD or MTBDD from the file given by
\fBfp\fR.  The argument
.B vars
should be a null-terminated array of variables that will become the
support of the BDD.  As in \fBbdd_dump_bdd\fR, these need not be in
order of increasing index.  If the same array of variables is used in
dumping and undumping, the BDD returned will be equal to the one that
was dumped.  More generally, if the array
.B v1
is used when dumping, and the array
.B v2
is used when undumping, the BDD returned will be equal to the original
BDD with the \fIi\fRth variable in
.B v2
substituted for the \fIi\fRth variable in
.B v1
for all \fIi\fR.  Null is returned if the operation fails for some
reason (node limit reached, I/O error, invalid file format, etc.).
In this case, an error code is stored in \fBerror\fR.  The code will
be one of the following.
.nf
.ta 3in
\fIValue\fR     \fIMeaning\fR
BDD_UNDUMP_FORMAT       Invalid file format
BDD_UNDUMP_OVERFLOW     Node limit exceeded
BDD_UNDUMP_IOERROR      File I/O error
BDD_UNDUMP_EOF  Unexpected EOF
.fi
.LP
.B int
.br
.B bdd_type(bddm, f)
.br
.B bdd_manager bddm;
.br
.B bdd f;
.in +4
Returns an integer classifying the BDD or MTBDD \fBf\fR.  The possible
return values and their meanings are as follows.
.nf
.ta 3in
\fIValue\fR     \fIMeaning\fR
BDD_TYPE_OVERFLOW       \fBf\fR is a null pointer
BDD_TYPE_ZERO   \fBf\fR is the constant FALSE
BDD_TYPE_ONE    \fBf\fR is the constant TRUE
BDD_TYPE_CONSTANT       \fBf\fR is an MTBDD constant
BDD_TYPE_POSVAR \fBf\fR is a variable
BDD_TYPE_NEGVAR \fBf\fR is the negation of a variable
BDD_TYPE_NONTERMINAL    \fBf\fR is not one of the above
.fi
.LP
.B void
.br
.B bdd_free(bddm, f)
.br
.B bdd_manager bddm;
.br
.B bdd f;
.in +4
Decreases the reference count of
.B f
by one.  When the reference count of a BDD or MTBDD node reaches 0,
the node and any of its children that are not otherwise referenced may
eventually be garbage collected and reused.  Intermediate results and
unused BDDs and MTBDDs should be freed whenever possible.  For
example:

.nf
bdd
f_or_g_and_h(bddm, f, g, h)
     bdd_manager bddm;
     bdd f, g, h;
{
  bdd temp, result;
  temp=bdd_and(bddm, g, h);
  result=bdd_or(bddm, f, temp);
  bdd_free(bddm, temp);    /* Free intermediate */
  return (result);
}
.fi
.LP
.B void
.br
.B bdd_unfree(bddm, f)
.br
.B bdd_manager bddm;
.br
.B bdd f;
.in +4
Increases the reference count of
.B f
by one.  This is usually used in conjunction with
\fBbdd_clear_refs\fR.  Works with MTBDDs.
.LP
.B void
.br
.B bdd_clear_refs(bddm)
.br
.B bdd_manager bddm;
.in +4
Sets the reference counts of all BDD and MTBDD nodes (except for the
node for TRUE/FALSE) to 0.  Calling this routine and then immediately
calling
.B bdd_unfree
on a set of BDDs has the effect of disposing of all BDDs except those
in the set.
.LP
.B void
.br
.B bdd_gc(bddm)
.br
.B bdd_manager bddm;
.in +4
Forces a BDD garbage collection; all nodes not reachable from a node
with a nonzero reference count are disposed of.  (Garbage collections
also occur internally at various times.)
.LP
.B long
.br
.B bdd_total_size(bddm)
.br
.B bdd_manager bddm;
.in +4
Returns the number of BDD and MTBDD nodes in existence (including
those which are eligible for garbage collection).
.LP
.B long
.br
.B bdd_vars(bddm)
.br
.B bdd_manager bddm;
.in +4
Returns the number of variables in existence.
.LP
.B int
.br
.B bdd_cache_ratio(bddm, ratio)
.br
.B bdd_manager bddm;
.br
.B int ratio;
.in +4
Sets the BDD operation cache size ratio to
.B ratio
and returns the old cache size ratio.  The number of cache entries is
constrained to be (roughly) less than the cache size ratio divided by
16 times the number of BDD nodes in existence.  The default size ratio
is 4, which gives about 1 cache entry per 4 BDD nodes.  The amount of
memory required per node will be about 17+(\fBratio\fR/16)*20 bytes on
a machine with 32-bit words.
.LP
.B void
.br
.B bdd_node_limit(bddm, limit)
.br
.B bdd_manager bddm;
.br
.B long limit;
.in +4
Sets the number of allowed BDD nodes to
.B limit
and returns the old limit.  A value of 0 specifies no limit.  If in
the course of an operation, the number of nodes reaches the limit, an
internal garbage collection takes place.  If this does not free enough
nodes to continue, the operation is aborted and a null value is
returned.  When dynamic reordering is used to shift around large
variable block, this limit may be exceeded during reordering.
.LP
.B int
.br
.B bdd_overflow(bddm)
.br
.B bdd_manager bddm;
.in +4
Returns 1 if any operation has caused an overflow in the number of
nodes, and 0 otherwise.  Calling this routine clears the internal
overflow flag, so subsequent calls will return 0 until the next
overflow occurs.
.LP
.B void
.br
.B bdd_overflow_closure(bddm, overflow_fn, overflow_env)
.br
.B bdd_manager bddm;
.br
.B void (*overflow_fn)();
.br
.B pointer overflow_env;
.in +4
Sets the closure to invoke when an overflow occurs.  The function
given by
.B overflow_fn
will be invoked as the last stage in the cleanup after the overflow.
The function is passed the BDD manager and the pointer given by
\fBoverflow_env\fR.  Typically, the function will jump to a
user-provided error recovery routine.
.LP
.B void
.br
.B bdd_abort_closure(bddm, abort_fn, abort_env)
.br
.B bdd_manager bddm;
.br
.B void (*abort_fn)();
.br
.B pointer abort_env;
.in +4
Sets a closure to invoke when the next node creation is attempted.
All temporary results will be cleaned up just before the function
given by
.B abort_fn
is called.  The function is passed the BDD manager and the pointer
given by \fBabort_env\fR.  Typically, the function will jump to a
user-provided error recovery routine.  This functionality is intended
to be used to cleanly interrupt BDD operations.  Typically,
.B bdd_abort_closure
will be called within a signal handler.
.LP
.B void
.br
.B bdd_stats(bddm, fp)
.br
.B bdd_manager bddm;
.br
.B FILE *fp;
.in +4
Prints some statistics to the file given by \fBfp\fR.
.LP
.B void
.br
.B bdd_dynamic_reordering(bddm, reorder_fn)
.br
.B bdd_manager bddm;
.br
.B void (*reorder_fn)();
.in +4
Selects the method for dynamic reordering.  When dynamic reordering is
being used, the library may attempt to rearrange the BDD variable
ordering in the midst of an operation so as to reduce the number of
nodes in use.  There are currently two available reordering methods.
The first, \fBbdd_reorder_stable_window3\fR, permutes the variables
within windows of three adjacent variables so as to minimize the
overall BDD size.  This process is repeated until no more reduction in
size occurs.  The second method, \fBbdd_reorder_sift\fR, moves each
variable throughout the order to find an optimal position for that
variable (assuming all other variables are fixed).  This generally
achieves greater size reductions than the window-based method, but is
slower.  The
.B reorder_fn
may also be
.B bdd_reorder_none
(an appropriately cast null pointer), in which case dynamic reordering
is turned off.  Also see the discussion on variable blocks in
\fBbdd_new_var_block\fR.
.LP
.B void
.br
.B bdd_reorder(bddm)
.br
.B bdd_manager bddm;
.in +4
Invoke the current dynamic reordering method.
.LP
.B block
.br
.B bdd_new_var_block(bddm, v, n)
.br
.B bdd_manager bddm;
.br
.B bdd v;
.br
.B long n;
.in +4
Groups the variable
.B v
and the \fBn\fR-1 variables after it in the ordering into a single
block for purposes of dynamic reordering.  The purpose of blocks is to
provide control over the possible orders that dynamic reordering will
consider.  In general, the variable blocks form a hierarchy.  For
example, a block consisting of the variables with indexes 0 through 3
might be made up of two sub-blocks, one for the variables with index 0
and 1, and one for the variables with index 2 and 3.  When dynamic
reordering is invoked, it is actually applied to each block within the
hierarchy.  Reordering a block involves shuffling around the
sub-blocks within it.  Thus, dynamic reordering actually moves groups
of variables rather than single variables.  If you know that a group
of variables should be together in the ordering, you should collect
them together into a block.  As an example, in BDD-based sequential
verification algorithms, the variables representing the current state
and next state of a state-holding element should generally be adjacent
in a good ordering.  By grouping these variables into a block, we can
ensure that only orderings with this property are considered.  After a
block has been reordered, each sub-block within it is recursively
reordered as well.  You can also specify that certain blocks should
not be reordered (see
.B bdd_var_block_reorderable
below).
.LP
.B void
.br
.B bdd_var_block_reorderable(bddm, b, reorderable)
.br
.B bdd_manager bddm;
.br
.B block b;
.br
.B int reorderable;
.in +4
If
.B reorderable
is non-zero, turns on reordering for the given block, otherwise turns
it off.  By default, blocks are not reorderable.  As an example,
suppose we are building the BDDs representing a circuit with distinct
control and data path.  In such a case, we typically want to have the
control variables at the top of the ordering.  For the data path, we
probably want to have the variables for each bit slice grouped
together, and we want the bit slices to be ordered from
most-significant to least-significant.  However, we want to allow
reordering within the control part and within each slice.  To do this,
we create the variables in the following order: control variables
first, down to LSB slice variables.  Then we create separate variable
blocks for the control part and for each slice.  We then turn on
reordering for these blocks.  Next, we create a block containing all
of the variables, and we leave reordering off for this block.  When
dynamic reordering is invoked, it will rearrange the control variables
and the variables within each slice, but will not move the control
variables or the slices in relation to each other.
.LP
.B void
.br
.B bdd_free_terminal_closure(bddm, free_terminal_fn, free_terminal_env)
.br
.B bdd_manager bddm;
.br
.B void (*free_terminal_fn)();
.br
.B pointer free_terminal_env;
.in +4
Sets a closure to invoke when freeing an MTBDD terminal node.  The
function receives the BDD manager, two longs representing the value of
the terminal, and the pointer given by \fBfree_terminal_env\fR.  If
you using the terminal value to hold pointers to other data
structures, you can set up this routine to free those structures.
.LP
.B bdd
.br
.B mtbdd_get_terminal(bddm, value1, value2)
.br
.B bdd_manager bddm;
.br
.B long value1;
.br
.B long value2;
.in +4
Creates an MTBDD terminal node corresponding to the value given by
.B value1
and \fBvalue2\fR.  If a terminal node with the value already exists,
its reference count is increased.  See also
\fBbdd_free_terminal_closure\fR.
.LP
.B void
.br
.B mtbdd_terminal_value(bddm, f, value1, value2)
.br
.B bdd_manager bddm;
.br
.B bdd f;
.br
.B long *value1;
.br
.B long *value2;
.in +4
.B f
should be an MTBDD terminal node.  The value of the node is stored in
.B value1
and \fBvalue2\fR.
.LP
.B bdd
.br
.B mtbdd_ite(bddm, f, g, h)
.br
.B bdd_manager bddm;
.br
.B bdd f;
.br
.B bdd g;
.br
.B bdd h;
.in +4
.B f
should be a BDD and
.B g
and
.B h
should be MTBDDs.  Returns the MTBDD for the operation IF
.B f
THEN
.B g
ELSE \fBh\fR.
.LP
.B bdd
.br
.B mtbdd_substitute(bddm, f)
.br
.B bdd_manager bddm;
.br
.B bdd f;
.in +4
Does the analog of
.B bdd_substitute
for the MTBDD \fBf\fR.  The elements in the variable association must
be BDDs.
.LP
.B bdd
.br
.B mtbdd_equal(bddm, f, g)
.br
.B bdd_manager bddm;
.br
.B bdd f;
.br
.B bdd g;
.in +4
Returns the BDD which is true for those valuations on which the MTBDDs
.B f
and
.B g
are equal.  That is, this is the analog of a logical XNOR for MTBDDs.
.LP
.B bdd
.br
.B mtbdd_transform(bddm, f)
.br
.B bdd_manager bddm;
.br
.B bdd f;
.in +4
Conceptually applies the user-defined transform to all terminals of
the specified MTBDD.  (This is actually done by just flipping the
pointer flag, so this routine is really a macro for \fBbdd_not\fR.)
See \fBmtbdd_transform_closure\fR.
.LP
.B void
.br
.B mtbdd_transform_closure(bddm, canonical_fn, transform_fn, env)
.br
.B bdd_manager bddm;
.br
.B int (*canonical_fn)();
.br
.B void (*transform_fn)();
.br
.B pointer env;
.in +4
Sets the MTBDD terminal transformation closure.  Currently in the
library, the pointer representing a boolean function and the pointer
representing the negation of that function are identical except for
the low-order bit.  Complementing a function is done by simply
toggling that bit.  The MTBDD terminal transformation allows this
mechanism to be extended to MTBDDs.  Whenever a terminal is created,
.B canonical_fn
will be called.  It is passed the BDD manager, two longs representing
the terminal being created, and the pointer given by \fBenv\fR.  The
function should return zero if the value is already canonical, and a
non-zero result if it needs to be transformed.  If the value needs to
be transformed, then
.B transform_fn
will be called, with the BDD manager, two longs representing the value
to be transformed, pointers to two longs to hold the result, and the
pointer given by \fBenv\fR.  The actual terminal node that is created
will contain the transformed value.  The original terminal requested
will be represented by a pointer to this node, with the low-order bit
of the pointer set.  Also see \fBmtbdd_one_data\fR.  If you are going
to call this function, you should do it before creating any MTBDD
terminals.
.LP
.B void
.br
.B mtbdd_one_data(bddm, value1, value2)
.br
.B bdd_manager bddm;
.br
.B long value1;
.br
.B long value2;
.in +4
If you are planning to use MTBDDs that contain TRUE and FALSE as well
as other values, you may need to use this function to set the MTBDD
value for the node representing TRUE.  In this case, also keep in mind
that the when the transformation function is applied to this value, it
should yield the value that you want for FALSE.  Also, the value for
TRUE should be regarding as canonical, i.e., TRUE must be represented
by a pointer with the low-order bit cleared.  As an example, suppose
that we are planning to use MTBDDs to represent spectral transforms of
boolean functions [4].  In this case, the MTBDD terminal values will
conceptually be integers.  Further, it is convenient for TRUE to be
represented by the value -1, and FALSE to be represented by +1.  We
will represent terminal values using two longs, with the first long
representing the most-significant part of the integer.  We will also
assume a 2's complement representation, so TRUE should be represented
by the data values -1 and -1.  Since the value for FALSE is the
negation of that for TRUE, we will have our transform function
represent integer negation.  Also, since we want the value for TRUE to
be canonical, we will regard nonnegative values as canonical.  Thus,
we define

.nf
int
canonical_fn(bddm, value1, value2, env)
     bdd_manager bddm;
     long value1;
     long value2;
     pointer env;
{
  return (value1 > 0 || (!value1 && value2 > 0));
}

void
transform_fn(bddm, value1, value2, result1, result2, env)
     bdd_manager bddm;
     long value1;
     long value2;
     long *result1;
     long *result2;
     pointer env;
{
  if (!value2)
    /* Will be a carry when taking 2's complement of value2.  Thus, */
    /* take 2's complement of high part. */
    value1= -value1;
  else
    {
      value2= -value2;
      value1= ~value1;
    }
  *result1=value1;
  *result2=value2;
}
.fi

We then call
.B mtbdd_transform_closure
to register these functions, and use

.nf
bdd_one_data(bddm, -1l, -1l);
.fi

to set the value for TRUE to -1.  (The default canonical checking and
transformation functions and the default MTBDD values for TRUE and
FALSE are actually as given in this example.)  If you are going to
call \fBbdd_one_data\fR, you should do it before creating any MTBDD
terminals.
.SH "ADDING NEW ROUTINES"
If you want to add new routines to the library, you would be
well-advised to look at some of the existing ones to get a feel for
how they operate.  Good ones include \fBbdd_ite\fR (the basic logical
operation) and \fBbdd_exists\fR (a routine using variable
associations).  Some basic points are explained below.  To get the
declarations of the internal library data structures and routines, you
should
.B #include <bddint.h>
instead of using \fBbdduser.h\fR.  You will probably want to study
this file to become familiar with the data structures.

Pointers to BDD nodes and cache entries are tagged using the low three
bits of the pointer.  Because of this, all structures must be aligned
on eight byte boundaries.  The storage allocation routines guarantee
this alignment.  The tag field of a tagged pointer is extracted with
the
.B TAG
macro.  The
.B POINTER
macro masks off the tag to get the actual pointer.  If the pointer is
a pointer to a BDD node, you can use
.B BDD_POINTER
instead; this just casts the result to a
.B bdd
after masking off the tag.  The tag can be set using \fBSET_TAG\fR, and
individual tag bits can be manipulated with \fBTAG0\fR,
.B FLIP_TAG0
and \fBSET_TAG0\fR for tag bit 0, and the analogous macros for tag
bits 1 and 2.  More commonly, slightly higher level macros are used
for manipulating tags.  For BDD nodes, there is only one tag bit that
is actually used.  When it is set, it indicates the pointer should be
interpreted as representing the complement of the node that it points
to.  (Or for MTBDDs, that it should be interpreted as transformed
using the user-definable transformation function).  There are macros
for testing, clearing, and flipping the negation flag.

Before using the macros below on a pointer \fBf\fR, you need to use
\fBBDD_SETUP(f)\fR.  This actually declares a new variable to hold the
masked pointer \fBBDD_POINTER(f)\fR.  Hence, it needs to be placed at
some point where a variable declaration could legally go.  If you
change \fBf\fR, you can reset this internal variable using
\fBBDD_RESET\fR.

BDD pointers are generally manipulated using the following macros.
Below, ``node'' refers to the node referenced by the pointer.
.LP
.B BDD_IS_CONST
.in +4
Tests if the node represents the constant TRUE or FALSE or an MTBDD
terminal node.
.LP
.B BDD_INDEX
.in +4
Returns the index of the node, or
.B BDD_MAX_INDEX
if given a constant node.
.LP
.B BDD_INDEXINDEX
.in +4
Returns the index index of a node.  This field is the value returned
by \fBbdd_if_id\fR and is invariant; when you create a new variable,
the index of old nodes may change, but the index index stays the same.
When you call \fBbdd_find\fR, you pass the desired index index of the
new node, not the index.
.LP
.B BDD_NOT
.in +4
Flips the negation flag on a pointer.
.LP
.B BDD_THEN, BDD_ELSE
.in +4
Return the THEN and ELSE pointers of a node, taking proper account of
pointer flags.  These are used for doing Shannon expansions on a node.
.LP
.B BDD_TOP_VAR2
Takes a \fBbdd_manager\fR, a variable that can hold an index index,
and two \fBbdd\fRs.  Sets the index index variable to the index index
of the variable with the lowest index among the variables at the roots
of the BDDs.  This index index can then be used with...
.LP
.B BDD_COFACTOR
Takes an index index, a BDD, and two variables of type \fBbdd\fR, and
sets the two variables either to the original BDD or to the cofactors
of the original BDD with respect to its top variable, depending on
whether the index index of the first BDD matches that specified.  You
can do a Shannon expansion on the top variable of two BDDs by using
.B BDD_TOP_VAR2
to get the index index of the highest variable and then using
.B BDD_COFACTOR
to take the appropriate cofactors.
.LP
.B BDD_MARK
.in +4
Accesses the mark field of a node.  This expands to a l-value, so you
can set the mark with this as well.  (But see BDD_TEMP_REFS below.)
.LP
.B BDD_ONE, BDD_ZERO
.in +4
Take a BDD manager and give back the BDDs for TRUE and FALSE.
.LP
.B BDD_REFS
.in +4
Accesses the reference count field of a node.
.LP
.B BDD_INCREFS, BDD_DECREFS
.in +4
Increment and decrement the reference count.
.LP
.B BDD_TEMP_REFS
.in +4
Accesses the temporary reference count field of a node.  The temporary
reference count and the mark actually share storage, so you can't use
both at once!  That is, unless you are very clever, you can't write a
routine that builds temporary nodes and uses the marks.
.LP
.B BDD_TEMP_INCREFS, BDD_TEMP_DECREFS
.in +4
Increment and decrement the temporary reference count.
.LP

New BDD nodes are created using \fBbdd_find\fR.  This routine takes a
BDD manager, an index index, and two subBDDs as arguments.  New MTBDD
terminals can be created with \fBbdd_find_terminal\fR.  The result
cache is manipulated using the
.B bdd_lookup_in_cache
and
.B bdd_insert_in_cache
routines.  There are different versions of these routines depending on
exactly what is being cached.  The basic ones are
\fBbdd_lookup_in_cache31\fR and \fBbdd_insert_in_cache31\fR.
The first of these takes a cache entry type (CACHE_TYPE_ITE,
CACHE_TYPE_TWO, etc.), three arguments of unspecified type (passed as
longs), and a pointer to an unspecified type of result (a pointer to a
long).  It returns a nonzero result if the lookup succeeds.  The
corresponding insert routine is similar except that the result is
passed in as a long, and nothing is returned.  There are similar
functions that are for routines that take two arguments and return two
results (or a single double-word result), or for routines that take
one argument and return three results.  There are also macros such as
\fBbdd_lookup_in_cache2\fR that are wrappers for things like
two-argument functions, etc.  In general, some action must be taken
when results are returned from the cache, when entries are purged from
the cache, when entries are garbage collected, and when a variable
association ID is reclaimed.  For the built-in cache entry types,
these actions are done automatically.  For example, when a BDD is
returned from an entry with CACHE_TYPE_TWO, the temporary reference
count of the BDD is incremented.  Some of the entry types are
available for customization.  The actions to take for these entry
types are specified by calling \fBbdd_cache_functions\fR.  This
function takes a BDD manager, an integer between 1 and 3 specifying
the number of arguments you want to cache on, and four function
pointers.  When returning a result, purging an entry, garbage
collecting, or reclaiming an association ID, these functions are
called.  The first three functions are passed the BDD manager and the
entry.  (The tag bits will have already been masked off the entry
pointer.)  The last receives these plus the association ID being freed
(cast to a pointer).  The garbage collection function should return a
nonzero result if the entry should be garbage collected.  If the entry
contains some BDD nodes, they should be tested with \fBBDD_IS_USED\fR.
The function called when an association ID is reclaimed should return
a nonzero result if the entry should be flushed from the cache.  This
function and the purge function and return functions may be null,
specifying that no action need be taken.  \fBbdd_cache_functions\fR
returns an integer that represents a tag to use with the cache
insertion and lookup routines, or -1 if there are no more free tags
available.  The routine \fBbdd_free_cache_tag\fR makes a tag available
again.

Routines that build new BDD nodes must take into account the
possibility of running into the node limit.  The package is set up to
make this easy if you use the following strategy.  Organize your
routine as a top-level (user-callable) procedure and an internal
procedure for performing the actual computation.  The top-level
procedure should check its arguments before calling the internal
routine.  The
.B bdd_check_arguments
function can be used to test for null arguments (indicating a prior
overflow) or arguments with a zero reference count (indicating a bug).
It should also use the
.B FIREWALL
macro to set up an overflow trap.  The internal routine should use
temporary reference counts to keep track of the nodes it is using.
When a node is returned from the internal routine, increment its
temporary reference count once.  (You don't have to do this for the
constants or for variables, since they can't be garbage collected.)
When you pass a node to \fBbdd_find\fR, its temporary reference count
is decremented once automatically, and its reference count is
incremented.  Also, the result of \fBbdd_find\fR has its temporary
reference count incremented once automatically.  Hence, if you your
routine has the standard organization (Shannon's expansion followed by
\fBbdd_find\fR on the subresults), you usually don't have to worry
about incrementing or decrementing the reference counts yourself.  If
you don't use a subresult, or if you want a subresult to stick around
after calling \fBbdd_find\fR, you'll have to do the appropriate
twiddling.  When the internal routine finally returns, you should have
a BDD with a single temporary reference count.  Use
.B RETURN_BDD
to convert this temporary reference count to an external one and
return the result to the user.  If you follow this strategy, you won't
have to deal with overflow; when the node limit is reached,
\fBbdd_find\fR will try garbage collecting, and if that doesn't work,
will call the overflow trap set up by \fBFIREWALL\fR.  The overflow
trap handler will automatically zero all temporary reference counts
and return a null pointer to the user.  Note: if you want to call
other routines, such as the IF-THEN-ELSE routine, within your internal
procedure, you should call the internal procedure for the routine.
That way, the overflow handler will give control back to the user
if the routine you are calling causes an overflow.

A typical routine looks like:

.nf
bdd
foo_step(bddm, f, g)
     bdd_manager bddm;
     bdd f, g;
{
  bdd_indexindex_type top_indexindex;
  bdd f1, f2;
  bdd g1, g2;
  bdd temp1, temp2;
  bdd result;
  
  BDD_SETUP(f);
  BDD_SETUP(g);
  if (<terminal case>)
    {
      BDD_TEMP_INCREFS(f);
      return (f);
    }
  if (bdd_lookup_in_cache2(bddm, <op>, f, g, &result))
    return (result);
  BDD_TOP_VAR2(top_indexindex, bddm, f, g);
  BDD_COFACTOR(top_indexindex, f, f1, f2);
  BDD_COFACTOR(top_indexindex, g, g1, g2);
  temp1=foo_step(bddm, f1, g1);
  temp2=foo_step(bddm, f2, g2);
  result=bdd_find(bddm, top_indexindex, temp1, temp2);
  bdd_insert_in_cache2(bddm, <op>, f, g, result);
  return (result);
}

bdd
foo(bddm, f, g)
     bdd_manager bddm;
     bdd f, g;
{
  if (bdd_check_arguments(2, f, g))
    {
      FIREWALL(bddm);
      RETURN_BDD(foo_step(bddm, f, g));
    }
  return ((bdd)0);
}
.fi

In the case of dynamic variable reordering, the same abort mechanism
is used.  After reordering, all reference counts are reset to their
original values and the operation is retried.  This is handled
automatically by the FIREWALL macro.  (The operation is aborted since
after reordering, the implicit ordering represented in the C
subroutine call stack may be different from the new variable order.
Reordering occurs before freeing the temporaries, since we want to
minimize the aggregate size of the operands plus the result that is
being constructed.)

Storage can be allocated through a number of mechanisms.  The routines
\fBmem_get_block\fR, \fBmem_free_block\fR, and \fBmem_resize_block\fR
are generally used for large single items.  For smaller uniformly
sized items, you probably should use a record manager.
.B mem_new_rec_mgr
will return a record manager that handles blocks of a given size.
Use
.B mem_new_rec
and
.B mem_free_rec
to obtain and free individual records.  Finally,
.B mem_free_rec_mgr
will dispose of the record manager and all of its associated records.
These routines are documented in more detail in the storage management
library man page.  If your structures are at most 64 bytes in size,
you can use the macros
.B BDD_NEW_REC
and \fBBDD_FREE_REC\fR.  These obtain records from the internal BDD
record managers.
.SH "PORTABILITY NOTES"
Since pointer tagging is heavily used, you'll have major problems if
you can't cast back and forth between pointers and longs without
losing something.  The low-level storage management routines are
fairly UNIX specific; they call
.B sbrk
directly.  If you don't have something similar, you may have to
rewrite them.  The storage management routines also need to be able to
move and clear blocks of memory whose size is given by a long.  You
may have to fiddle with these, especially if you have a machine where
int and long are different.  If you encounter portability problems,
let me know; maybe the next release will be able to accommodate your
machine.
.SH "SEE ALSO"
mem(3)
.SH BUGS
Surely you're joking.
.SH REFERENCES
[1] R. E. Bryant.  Graph Based Algorithms for Boolean Function
Manipulation.  \fIIEEE Transactions on Computers\fR, C-35(8):677-691,
August 1986.
.LP
[2] H. J. Touati, H. Savoj, B. Lin, R. K. Brayton, and A.
Sangiovanni-Vincentelli.  Implicit State Enumeration of Finite State
Machines using BDD's.  In \fIProceedings of the 1990 IEEE
International Conference on Computer-Aided Design\fR, November, 1990.
.LP
[3] K. S. Brace, R. L. Rudell, and R. E. Bryant.  Efficient
Implementation of a BDD Package.  In \fIProceedings of the 27th
ACM/IEEE Design Automation Conference\fR, June, 1990.
.LP
[4] E. M. Clarke, K. L. McMillan, X. Zhao, M. Fujita, and J. C.-Y.
Yang.  Spectral Transforms for Large Boolean Functions with
Applications to Technology Mapping.  In \fIProceedings of the 30th
ACM/IEEE Design Automation Conference\fR, June, 1993.
.SH AUTHOR
David E. Long
.br
long@research.att.com