# Knowledge Compilation

Tree decision diagrams (TDDs) measure width by counting distinct subfunctions across each vtree cut, a count blind to linear dependence among those subfunctions. Replacing the count with the rank of the cut matrix yields algebraic tree decision diagrams (ATDDs), whose width is the factor rank and which have exactly one representation per function and vtree. ATDDs are exponentially more succinct than TDDs and quasi-polynomially more succinct than structured deterministic DNNFs, while keeping polynomial-time model counting, weighted model counting and equivalence checking.

Tree decision diagrams (TDDs) measure width by counting distinct subfunctions across each vtree cut, a count blind to linear dependence among those subfunctions. Replacing the count with the rank of the cut matrix yields algebraic tree decision diagrams (ATDDs), whose width is the factor rank and which have exactly one representation per function and vtree. ATDDs are exponentially more succinct than TDDs and quasi-polynomially more succinct than structured deterministic DNNFs, while keeping polynomial-time model counting, weighted model counting and equivalence checking.