# Common Lisp binding to CUDD

This is a fork of original CL-CUDD using the modern common lisp convension.

**Supported implementations**: SBCL, CCL and ECL.**Requirements**: make, curl**Developmental State**: After some refurbishment, now it loads reliably and all tests pass.**TODOs**:~~Automatic variable reordering~~~~Variable grouping API~~~~GC hook and control API~~- Higher-order layer for set manipulation
- benchmarking

**Related work**: trivialib.bdd is another common lisp library for BDDs, which is entirely written in lisp. CUDD is more on the state-of-the-art side.

## What is BDDs and CUDD?

BDDs (Binary Decision Diagrams) are awesome datastructures that can compactly represent exponentially large number of datasets, as well as allowing the direct computation over the compressed representation, i.e., you can take the sum/product/union/intersection of the datasets without decompressing the data!

CUDD is a famous C implementation of BDDs and its relatives: Multi-Terminal Binary Decision Diagrams (MTBDDs, also known as Algebraic DD / ADDs) and Zero-suppressed Decision Diagrams.

References:

- Wikipedia contains lots of information.
- Binary Decision Diagrams by Akers et al
- Symbolic Boolean manipulation with ordered binary-decision diagrams by RE Bryant (this sorta made BDDs truely practical I guess)
- ADDs : Multi-Terminal Binary Decision Diagrams: An Efficient Data Structure for Matrix Representation by Clarke et al
- ZDDs (survey) : An Introduction to Zero-Suppressed Binary Decision Diagrams by Alan Mishchenko

trivialib.bdd is another common lisp library for BDDs, which is entirely written in lisp. CUDD is more on the state-of-the-art side.

## Building/Loading the system

The system is asdf-loadable. This version of CL-CUDD automatically fetches CUDD v3.0.0 from http://vlsi.colorado.edu/~fabio/CUDD/ via curl. The archive is expanded in the ASDF system directory and builds its dynamic library, which is then loaded by CL-CUDD.

To test the system, evaluate `(asdf:test-system :cl-cudd.test)`

. It also writes the visualizations of the decision diagrams to the system directory in DOT format. If you have Graphviz installed, the test script also tries to convert the results into pdfs.

## The binding(s)

The binding consists of two layers: The lower layer has `cl-cudd.baseapi`

package. This layer is a very thin wrapper around the C library, passes raw pointers around and requires that you take care of reference counting.

Above this layer there is a package named `cl-cudd`

(with a nickname `cudd`

). It wraps the pointers from the lower layer, takes care of reference counting for you, and also adds documentation from the CUDD manual.

## DD Construction Examples

### Representing a binary function using BDD

Suppose you have a binary function denoted as `b = f(x0,x1,...xn)`

which can be expressed in a DNF formula such as `(or (and x0 (not x1) x2) ...)`

. To begin with, we construct the first disjunction, which is in this case `(and x0 (not x1) x2)`

. It can be implemented as follows:

```
;; functional
(reduce #'node-and
(list (make-var 'bdd-node :index 0)
(node-complement (make-var 'bdd-node :index 1))
(make-var 'bdd-node :index 2))
:from-end t ;; more efficient when BDDs are build bottom-up
:initial-value (one-node 'bdd-node))
;; imperative
(let ((f (one-node 'bdd-node)))
(setf f (node-and f (make-var 'bdd-node :index 2)))
(setf f (node-and f (node-complement (make-var 'bdd-node :index 1))))
(setf f (node-and f (make-var 'bdd-node :index 0)))
f)
;; with a threading macro
(ql:quickload :arrow-macros) (use-package :arrow-macros)
(-> (one-node 'bdd-node)
(node-and (make-var 'bdd-node :index 2))
(node-and (node-complement (make-var 'bdd-node :index 1)))
(node-and (make-var 'bdd-node :index 0)))
```

To take the disjunction of conjunctions, there are similarly named `node-or`

function and `zero-node`

function.

```
(-> (zero-node 'bdd-node)
(node-or bdd1)
(node-or bdd2)
...)
```

`(plot pathname dd)`

function (accepts bdd, add and zdd) generates a dot file and a pdf, but requires graphviz.

### Representing real functions using ADD and taking their symbolic sum

In ADD, the terminal node can have a value instead of boolean (in BDD), and you are able to perform various arithmetic operations in the closed form, i.e. for two ADDs (functions) `f`

and `g`

, you can obtain `f+g`

efficiently.

Imagine `f(x1,x2)`

and `g(x1,x2)`

takes the following values:

x1 | x2 | f(x1,x2) | g(x1,x2) |
---|---|---|---|

0 | 0 | 2 | 4 |

0 | 1 | 2 | 4 |

1 | 0 | 3 | 3 |

1 | 1 | 5 | 7 |

```
;; f
(-> (-> (ADD-constant 2)
(node-and (node-complement (make-var 'ADD-node :index 0)))
(node-and (node-complement (make-var 'ADD-node :index 1))))
(node-or
(-> (ADD-constant 2)
(node-and (node-complement (make-var 'ADD-node :index 0)))
(node-and (make-var 'ADD-node :index 1))))
(node-or
(-> (ADD-constant 3)
(node-and (make-var 'ADD-node :index 0))
(node-and (node-complement (make-var 'ADD-node :index 1)))))
(node-or
(-> (ADD-constant 5)
(node-and (make-var 'ADD-node :index 0))
(node-and (make-var 'ADD-node :index 1)))))
;; g
(-> ...) ;; omitted
(ADD-apply +plus+ f g) ;; -> an ADD representing (f+g)(x1,x2)
```

Other possible arguments to `ADD-apply`

are

`+XNOR+`

(originally Cudd_addXnor) -- XNOR of two 0-1 ADDs.`+XOR+`

(originally Cudd_addXor) -- XOR of two 0-1 ADDs.`+NOR+`

(originally Cudd_addNor) -- NOR of two 0-1 ADDs.`+NAND+`

(originally Cudd_addNand) -- NAND of two 0-1 ADDs.`+OR+`

(originally Cudd_addOr) -- Disjunction of two 0-1 ADDs.`+AGREEMENT+`

(originally Cudd_addAgreement) -- f op g, where f op g is f if f==g; background if f!=g.`+DIFF+`

(originally Cudd_addDiff) -- f op g , where f op g is plusinfinity if f=g; min(f,g) if f!=g.`+ONE-ZERO-MAXIMUM+`

(originally Cudd_addOneZeroMaximum) -- 1 if f > g and 0 otherwise.`+MAXIMUM+`

(originally Cudd_addMaximum) -- Integer and floating point maximum.`+MINIMUM+`

(originally Cudd_addMinimum) -- Integer and floating point minimum.`+MINUS+`

(originally Cudd_addMinus) -- Integer and floating point substraction.`+DIVIDE+`

(originally Cudd_addDivide) -- Integer and floating point division.`+SET-NZ+`

(originally Cudd_addSetNZ) -- This operator sets f to the value of g wherever g != 0.`+THRESHOLD+`

(originally Cudd_addThreshold) -- Threshold operator for Apply (f if f >=g; 0 if f<g)`+TIMES+`

(originally Cudd_addTimes) -- Integer and floating point multiplication.`+PLUS+`

(originally Cudd_addPlus) -- Integer and floating point addition

Another way to construct an ADD is to convert the corresponding BDD by `bdd->add`

function. (This case applies only to the case where 0-1 ADDs suffice, i.e. ADDs with only 0 or 1 terminal nodes).

### Representing a family of set using ZDD

Recently, ZDD is famous especially among Japanese CS researchers due to this hyped youtube video https://www.youtube.com/watch?v=Q4gTV4r0zRs made by people funded by JST ERATO (Exploratory Research for Advanced Technology). No, it is actually quite powerful and is a better option than BDD when *most of the paths leads to the zero-node*, and is particularly useful for representing a family of sets.

One way to construct a ZDD is to call either of conversion functions `bdd->zdd-simple`

and `bdd->zdd-cover`

to an existing BDD.

`bdd->zdd-simple`

directly maps one BDD variable to one ZDD variable. This is called a*unate*algebra representation, which is suitable for representing a binary function and a set of subsets.`bdd->zdd-cover`

maps one BDD variable to two ZDD variables representing true and false. This is called*binate*algebra representation, which is suitable for representing cube covers because it can efficiently encode the missing variables. See section 3.3 of [Mishchenko 14] section 3.11 of CUDD manual.

Another option is to build a ZDD directly, starting from {{}} and "flipping" the true/false by `(zdd-change zdd variable)`

operator. Consider we are encoding a family of sets F={{1,2}, {1,3}, {3}}. First, A={{1,2}} is built as follows:

```
(defvar *a* (zdd-set-of-emptyset))
(setf *a* (zdd-change *a* 1))
(setf *a* (zdd-change *a* 2))
;; alternatively, using threading macro:
(defvar *a*
(-> (zdd-set-of-emptyset)
(zdd-change 1)
(zdd-change 2)))
```

Then take their union.

```
(defvar *b* ..) ; construct *b* and *c* similarly.
(defvar *c* ..)
(defvar *f* (reduce #'zdd-union (list *a* *b* *c*) :initial-value (zdd-emptyset)))
;; alternatively,
(defvar *f*
(-> (zdd-emptyset)
(zdd-union *a*)
(zdd-union *b*)
(zdd-union *c*)))
;; or,
(defvar *f* (zdd-emptyset))
(setf *f* (zdd-union *f* *a*))
(setf *f* (zdd-union *f* *b*))
(setf *f* (zdd-union *f* *c*))
;; of course, this is equivalent and is more efficient:
(defvar *f* (zdd-union *a* *b*))
(setf *f* (zdd-union *f* *c*))
```

## System structure

### Low-level

This is loosely based on the SWIG-extracted information and is using CFFI-Grovel to actually map C symbols to lisp symbols. If you want to use this layer, then it would be best to have a look at the CUDD manual.

You can use the low-level system just as you would use the C API of CUDD. This also means that you have to do all the reference counting yourself, with one exception: The reference count of the return value each CUDD function that returns a node is increased if it is not `null`

. If it is `null`

, a signal of type `cudd-null-pointer-error`

is raised.

### High-level

The high level API automatically wraps the CUDD nodes in an instance of class `node`

. ADD nodes are wrapped in an instance of `add-node`

and BDD nodes are wrapped in an instance of type `bdd-node`

.

This enables runtime type checking (so that you don't stick ADD nodes into BDD functions or vice-versa) and also automatic reference counting.

Almost all CUDD functions need to refer to a CUDD manager. In the high-level API this manager is contained in special variable `*manager*`

. You can bind a manager using the macro `with-manager`

. You can also create a manager by `(make-instance 'manager :pointer (cudd-init 0 0 256 262144 0))`

.

All functions of package `CL-CUDD`

are documented using the original or slightly modified documentation of CUDD.

## History

The initial version was automatically generated using SWIG by Utz-Uwe Haus. The second version was adapted to the needs by Christian von Essen . Later, @Neronus made a git repository on Github and @rpgoldman made a few bugfixes. Finally @guicho271828 (Masataro Asai) has modernized the repository according to the recent practice in common lisp: unit tests, Travis-CI support, better documentation and additional support for ZDDs.

## Known problems

Using the GC to do reference counting automatically has its own share of problems:

- References may be freed very late.

Nodes will be dereferenced only if your CL implementation thinks that it's time for it. This is usually when itself is running out of memory. Because you are usually only holding on to the top of a diagram, you are not using as much memory in CL as you are using in CUDD. Hence the GC might come pretty late while CUDD is happily accumulating memory.

The solution to that is to try to call the garbage collector manually every so often using for example TRIVIAL-GARBAGE:GC

## Solved problems

~~References may be freed too early~~

The old text below is wrong. CUDD's reference counting GC does not work this way. According to CUDD's manual, its GC happens when:

- A call to cuddUniqueInter , to cuddUniqueInterZdd , to cuddUnique- Const, or to a function that may eventually cause a call to them.
- A call to Cudd RecursiveDeref , to Cudd RecursiveDerefZdd , or to a function that may eventually cause a call to them.

Thus the GC does not occur at arbitrary code path, as assumed below.

```
The following two examples demonstrate the problem.
(defun foo (dd)
(let ((ptr (node-pointer dd)))
;; please don't GC me here
(cudd-do-something ptr)))
In this example the GC might decide to run where there is the
comment.
In that case, provided that nothing outside of the function call
holds on to `dd`, the reference count of `ptr` might be decreased,
go down to zero and the node vanishes before `cudd-do-something` is
called.
```

- Author
- Masataro Asai
- License
- LLGPL
- Categories
- ai, electronics, theorem provers