Propositional directed acyclic graph

From Wikipedia, the free encyclopedia

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 of the following form:

• Leaves are labeled with (true), (false), or a Boolean variable.
• Non-leaves are (logical and), (logical or) and (logical not).
• - and -nodes have at least one child.
• -nodes have exactly one child.

Leaves labeled with () represent the constant Boolean function which always evaluates to 1 (0). A leaf labeled with a Boolean variable x is interpreted as the assignment x = 1, i.e. it represents the Boolean function which evaluates to 1 if and only if x = 1. The Boolean function represented by a -node is the one that evaluates to 1, if and only if the Boolean function of all its children evaluate to 1. Similarly, a -node represents the Boolean function that evaluates to 1, if and only if the Boolean function of at least one child evaluates to 1. Finally, a -node represents the complemenatary Boolean function its child, i.e. the one that evaluates to 1, if and only if the Boolean function of its child evaluates to 0.

[edit] PDAG, BDD, and NNF

Every binary decision diagram (BDD) and every negation normal form (NNF) is also a PDAG with some particular properties. The following pictures represent the Boolean function f(x1,x2,x3) = − x1 * − x2 * − x3 + x1 * x2 + x2 * x3:

BDD for the function f

PDAG for the function f obtained from the BDD

PDAG for the function f

[edit] See also

• Data structure
• Boolean satisfiability problem
• Proposition

[edit] References

• M. Wachter & R. Haenni, "Propositional DAGs: a New Graph-Based Language for Representing Boolean Functions", KR'06, 10th International Conference on Principles of Knowledge Representation and Reasoning, Lake District, UK, 2006.
• M. Wachter & R. Haenni, "Probabilistic Equivalence Checking with Propositional DAGs", Technical Report iam-2006-001, Institute of Computer Science and Applied Mathematics, University of Bern, Switzerland, 2006.
• M. Wachter, R. Haenni & J. Jonczy, "Reliability and Diagnostics of Modular Systems: a New Probabilistic Approach", DX'06, 18th International Workshop on Principles of Diagnosis, Peñaranda de Duero, Burgos, Spain, 2006.