Processing Succinct Matrices and Vectors

Abstract

Abstract. We study the complexity of algorithmic problems for matrices that are represented by multi-terminal decision diagrams (MTDD). These are a variant of ordered decision diagrams, where the terminal nodes are labeled with arbitrary elements of a semiring (instead of 0 and 1). A simple example shows that the product of two MTDD-represented matrices cannot be represented by an MTDD of polynomial size. To overcome this deficiency, we extended MTDDs to MTDD+ by allowing componentwise symbolic addition of variables (of the same dimension) in rules. It is shown that accessing an entry, equality checking, matrix multiplication, and other basic matrix operations can be solved in polynomial time for MTDD+-represented matrices. On the other hand, testing whether the determinant of a MTDD-represented matrix vanishes is PSPACE-complete, and the same problem is NP-complete for MTDD+-represented diagonal matrices. Computing a specific entry in a product of MTDD-represented matrices is #P-complete.

The complexity of counting models of linear-time temporal logic

“…Some of these classes appeared in the literature, e.g., #Pspace was shown to be equal to FPspace [11] (if the output is encoded in binary). Also, computing a specific entry of a matrix power A n is in #Pspace, if A is represented succinctly and n in binary [14], and counting self-avoiding walks in succinctly represented hypercubes is complete for #Exptime [12] under rightbit-shift reductions.…”