A binary decision diagram (BDD), like a negation normal form (NNF) or a propositional directed acyclic graph (PDAG), is a data structure that is used to represent a Boolean function. A Boolean function can be represented as a rooted, directed, acyclic graph, which consists of decision nodes and two terminal nodes called 0-terminal and 1-terminal. Each decision node is labeled by a Boolean variable and has two child nodes called low child and high child. The edge from a node to a low (high) child represents an assignment of the variable to 0 (1). Such a BDD is called 'ordered' if different variables appear in the same order on all paths from the root. It is called 'reduced' if the graph is reduced according to two rules: A logical formula is in negation normal form if negation occurs only immediately above elementary propositions. ... A propositional directed acyclic graph (PDAG) is a data structure that is used to represent a Boolean function. ... A binary tree, a simple type of branching linked data structure. ... A Boolean function describes how to determine a Boolean value output based on some logical calculation from Boolean inputs. ... A child node or descendant node is a node in a tree data structure that is linked to by a parent node. ...

• Merge any isomorphic subgraphs.
• Eliminate any node whose two children are isomorphic.

In popular usage, the term BDD almost always refers to Reduced Ordered Binary Decision Diagram (ROBDD in the literature, used when the ordering and reduction aspects need to be emphasized). The advantage of an ROBDD is that it is canonical(unique) for a particular functionality. This property makes it useful in functional equivalence checking and other operations like functional technology mapping. In mathematics, an isomorphism (in Greek isos = equal and morphe = shape) is a kind of mapping between objects, devised by Eilhard Mitscherlich, which shows a relation between two properties or operations. ... In mathematics, an isomorphism (in Greek isos = equal and morphe = shape) is a kind of mapping between objects, devised by Eilhard Mitscherlich, which shows a relation between two properties or operations. ...

A path from the root node to the 1-terminal represents a (possibly partial) variable assignment for which the represented Boolean function is true. As the path descends to a low child (high child) from a node, then that node's variable is assigned to 0 (1).

BDDs are extensively used in CAD (Computer Aided Design) software to synthesize circuits (logic synthesis) and in formal verification.

Example

The left figure below shows a binary decision tree (the reduction rules are not applied), and a truth table, each representing the function f (x1, x2, x3). In the tree on the left, the value of the function can be determined for a given variable assignment by following a path down the graph to a terminal. In the figures below, a dotted (solid) line represents an edge to a low (high) child. Therefore, to find (x1=0, x2=1, x3=1), begin at x1, traverse down the dotted line to x2 (since x1 has an assignment to 0), then down two solid lines (since x2 and x3 each have an assignment to one). This leads to the terminal 1, which is the value of f (x1=0, x2=1, x3=1). In operations research, specifically in decision analysis, a decision tree is a decision support tool that uses a graph or model of decisions and their possible consequences, including chance event outcomes, resource costs, and utility. ... Truth tables are a type of mathematical table used in logic to determine whether an expression is true or whether an argument is valid. ...

The binary decision tree of the left figure can be transformed into a binary decision diagram by maximally reducing it according to the two reduction rules. The resulting BDD is shown in the right figure.

Binary decision tree and truth table for the function f(x1, x2, x3) = -x1 * -x2 * -x3 + x1 * x2 + x2 * x3

BDD for the function f

Image File history File links Drawn by me January 31, 2005, released under GNU Free Documentation License. ... Image File history File links Drawn by me January 31, 2005, released under GNU Free Documentation License. ... Image File history File links No higher resolution available. ... Image File history File links No higher resolution available. ...

History

The basic idea from which the data structure was created is the Shannon expansion. A switching function is split into two sub-functions (cofactors) by assigning one variable (cf. if-then-else normal form). If such a sub-function is considered as sub-tree, it can be represented by a binary decision tree. Binary decision diagrams (BDD) were introduced by Lee (Lee 1959), and further studied and made known by Akers (Akers 1978) and Boute (Boute 1976). The Shannon expansion develops the idea that Boolean functions can be reduced by means of the identity: , where F is any function and and are positive and negative Shannon cofactors of F, respectively. ...

The full potential for efficient algorithms based on the data structure was investigated by Bryant at Carnegie Mellon University: his key extensions were to use a fixed variable ordering (for canonical representation) and shared sub-graphs (for compression). Applying these two concepts results in an efficient data structure and algorithms for the representation of sets and relations (Bryant 1986, Bryant 1992). By extending the sharing to several BDDs, i.e. one sub-graph is used by several BDDs, the data structure Shared Reduced Ordered Binary Decision Diagram is defined (Brace, Rudell, Bryant 1990). The notion of a BDD is now generally used to refer to that particular data structure. Randal E. Bryant (born October 27, 1952) is an American computer scientist and academic noted for his research on formally verifying digital hardware, and more recently some forms of software. ... Carnegie Mellon University is a private research university in Pittsburgh, Pennsylvania, United States. ...

On a more abstract level, BDDs can be considered as a compressed representation of sets or relations. Unlike other compressed representations, the actual operations are performed directly on that compressed representation, i.e. without decompression.

Variable ordering

The size of the BDD is determined both by the function being represented and the chosen ordering of the variables. For some functions, the size of a BDD may vary between a linear to an exponential range depending upon the ordering of the variables. Simply put, if we have a boolean function then depending upon the ordering of the variables we would end up getting a graph whose number of nodes would be linear at the best and exponential at the worst case. Let us consider the Boolean function . Using the variable ordering , the BDD needs nodes to represent the function. Using the ordering , the BDD consists of 2n nodes.

BDD for the function f(x1, ..., x8) = x1x2 + x3x4 + x5x6 + x7x8 using bad variable ordering

Good variable ordering

It is of crucial importance to care about variable ordering when applying this data structure in practice. The problem of finding the best variable ordering is NP-hard [Bollig and Wegener 1996]. For any constant c>1 it is even NP-hard to compute a variable ordering resulting in an OBDD with a size that is at most c times larger than an optimal one [Sieling 2002]. However there exist efficient heuristics to tackle the problem. Image File history File links No higher resolution available. ... Image File history File links No higher resolution available. ... Image File history File links No higher resolution available. ... Image File history File links No higher resolution available. ...

There are functions for which the graph size is always exponential — independent of variable ordering. This holds e. g. for the multiplication function (an indication^{[citation needed]} as to the apparent complexity of factoring ). Researchers have of late suggested refinements on the BDD data structure giving way to a number of related graphs: BMD (Binary Moment Diagrams), ZDD (Zero Suppressed Decision Diagram), FDD (Free Binary Decision Diagrams), PDD (Parity decision Diagrams), etc. In math, see Factorization. ... A binary moment diagram (BMD) is a generalization of the binary decision diagram (BDD) to linear functions over domains such as booleans (like BDDs), but also to integers or to real numbers. ... A zero suppressed decision diagram (ZSDD) is a version of binary decision diagram (BDD) where instead of nodes being introduced when the positive and the negative part are different, they are introduced when negative part is different from constant 0. ...

Logical operations on BDDs

Many logical operations on BDDs can be implemented by polynomial-time graph manipulation algorithms.

• conjunction
• disjunction
• negation
• existential abstraction
• universal abstraction

See also

• Boolean satisfiability problem
• Data structure
• Model checking
• Negational normal form (NNF)
• Propositional directed acyclic graph (PDAG)
• Radix tree
• Binary key - a method of species identification in biology using binary trees

To meet Wikipedias quality standards, this article or section may require cleanup. ... A binary tree, a simple type of branching linked data structure. ... Model checking is the process of checking whether a given model satisfies a given logical formula. ... Negative Normal form is way to represent the formula by keeping the negation symbol only to the literals. ... A propositional directed acyclic graph (PDAG) is a data structure that is used to represent a Boolean function. ... A radix tree, Patricia trie/tree, or crit bit tree is a specialized set data structure based on the trie that is used to store a set of strings. ...

Implementation

This is a crude way to build a BDD in C. Declare the data structure as follows and then proceed accordingly. Wikibooks has a book on the topic of C Programming The C programming language (often, just C) is a general-purpose, procedural, imperative computer programming language developed in the early 1970s by Dennis Ritchie for use on the Unix operating system. ...

 /* The basic data structure */ struct vertex { char *φ; struct vertex *hi, *lo; .. } /* The interface to the Unique Table */ struct vertex *old_or_new(char *φ, struct vertex *hi, *lo) { if(“a vertex v = (φ, hi, lo) exists”) return v; else { v <- “new vertex pointing at (φ, hi, lo); return v; } } 

Data Structure for Building the ROBDD

 struct vertex *robdd_build(struct expr f, int i) { struct vertex *hi, *lo; struct char *φ; if(equal(f, '0')) return v0; else if (equal(f, '1')) return v1; else{ φ ← п(i); hi ← robdd_build( f(xi = 1), i+1); lo ← robdd_build( f(xi = 0), i+1); if(lo == hi) return lo; else return old_or_new(φ, hi, lo); } } 

Available Packages

• ABCD: The ABCD package by Armin Biere.
• BuDDy: A BDD Package by Jørn Lind-Nielsen.
• CMU BDD, BDD package, Carnegie Mellon University, Pittsburgh
• CUDD: BDD package, University of Colorado, Boulder
• JavaBDD, a Java port of BuDDy that also interfaces to CUDD, CAL, and JDD.
• The Berkeley CAL package which does breadth-first manipulation.
• TUD BDD: A BDD Package and a World-Level package by Stefan Höreth.
• Vahidi's JDD, a java library that supports common BDD and ZBDD operations.
• Vahidi's JBDD, a Java interface to BuDDy and CUDD packages.
• Maiki & Boaz BDD-PROJECT, a Web Application for BDD reduction and visualization.

References

• C. Y. Lee. "Representation of Switching Circuits by Binary-Decision Programs". Bell Systems Technical Journal, 38:985–999, 1959.
• Sheldon B. Akers. Binary Decision Diagrams, IEEE Transactions on Computers, C-27(6):509–516, June 1978.
• Raymond T. Boute, "The Binary Decision Machine as a programmable controller". EUROMICRO Newsletter, Vol. 1(2):16–22, January 1976.
• Beate Bollig, Ingo Wegener. Improving the Variable Ordering of OBDDs Is NP-Complete , IEEE Transactions on Computers, 45(9):993––1002, September 1996.
• Detlef Sieling. "The nonapproximability of OBDD minimization." Information and Computation 172, 103–138. 2002.
• Randal E. Bryant. "Graph-Based Algorithms for Boolean Function Manipulation". IEEE Transactions on Computers, C-35(8):677–691, 1986.
• R. E. Bryant, "Symbolic Boolean Manipulation with Ordered Binary Decision Diagrams", ACM Computing Surveys, Vol. 24, No. 3 (September, 1992), pp. 293–318.
• Karl S. Brace, Richard L. Rudell and Randal E. Bryant. "Efficient Implementation of a BDD Package". In Proceedings of the 27th ACM/IEEE Design Automation Conference (DAC 1990), pages 40–45. IEEE Computer Society Press, 1990.
• Ch. Meinel, T. Theobald, "Algorithms and Data Structures in VLSI-Design: OBDD - Foundations and Applications", Springer-Verlag, Berlin, Heidelberg, New York, 1998.
• R. Ubar, "Test Generation for Digital Circuits Using Alternative Graphs (in Russian)", in Proc. Tallinn Technical University, 1976, No.409, Tallinn Technical University, Tallinn, Estonia, pp.75–81.

EUROMICRO is an international scientific, engineering and educational organization dedicated to advancing the arts, sciences and applications of information technology and microelectronics. ...

External links

• H. Andersen "An Introduction to Binary Decision Diagrams," Lecture Notes, http://www.itu.dk/people/hra/bdd97-abstract.html, October 1997.
• Ch. Meinel, T. Theobald, "Algorithms and Data Structures in VLSI-Design: OBDD - Foundations and Applications" (complete text), Springer-Verlag, Berlin, Heidelberg, New York, 1998.