- Propositional directed acyclic graph
A

**propositional directed acyclic graph (PDAG)**is adata structure that is used to represent aBoolean function . A Boolean function can be represented as a rooted,directed acyclic graph of the following form:

* Leaves are labeled with $op$ (true), $ot$ (false), or a Boolean variable.

* Non-leaves are $mathcal\{4\}$ (logical and), $mathcal\{5\}$ (logical or) and $Diamond$ (logical not).

* $mathcal\{4\}$- and $mathcal\{5\}$-nodes have at least one child.

* $Diamond$-nodes have exactly one child.Leaves labeled with $op$ ($ot$) 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 $mathcal\{4\}$-node is the one that evaluates to 1, if and only if the Boolean function of all its children evaluate to 1. Similarly, a $mathcal\{5\}$-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 $Diamond$-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.

**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 $f(x1,\; x2,\; x3)\; =\; -x1\; *\; -x2\; *\; -x3\; +\; x1\; *\; x2\; +\; x2\; *\; x3$:**See also***

Data structure

*Boolean satisfiability problem

*Proposition **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.

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