12,705 research outputs found
Chain Reduction for Binary and Zero-Suppressed Decision Diagrams
Chain reduction enables reduced ordered binary decision diagrams (BDDs) and
zero-suppressed binary decision diagrams (ZDDs) to each take advantage of the
others' ability to symbolically represent Boolean functions in compact form.
For any Boolean function, its chain-reduced ZDD (CZDD) representation will be
no larger than its ZDD representation, and at most twice the size of its BDD
representation. The chain-reduced BDD (CBDD) of a function will be no larger
than its BDD representation, and at most three times the size of its CZDD
representation. Extensions to the standard algorithms for operating on BDDs and
ZDDs enable them to operate on the chain-reduced versions. Experimental
evaluations on representative benchmarks for encoding word lists, solving
combinatorial problems, and operating on digital circuits indicate that chain
reduction can provide significant benefits in terms of both memory and
execution time
The Complexity of Reasoning with FODD and GFODD
Recent work introduced Generalized First Order Decision Diagrams (GFODD) as a
knowledge representation that is useful in mechanizing decision theoretic
planning in relational domains. GFODDs generalize function-free first order
logic and include numerical values and numerical generalizations of existential
and universal quantification. Previous work presented heuristic inference
algorithms for GFODDs and implemented these heuristics in systems for decision
theoretic planning. In this paper, we study the complexity of the computational
problems addressed by such implementations. In particular, we study the
evaluation problem, the satisfiability problem, and the equivalence problem for
GFODDs under the assumption that the size of the intended model is given with
the problem, a restriction that guarantees decidability. Our results provide a
complete characterization placing these problems within the polynomial
hierarchy. The same characterization applies to the corresponding restriction
of problems in first order logic, giving an interesting new avenue for
efficient inference when the number of objects is bounded. Our results show
that for formulas, and for corresponding GFODDs, evaluation and
satisfiability are complete, and equivalence is
complete. For formulas evaluation is complete, satisfiability
is one level higher and is complete, and equivalence is
complete.Comment: A short version of this paper appears in AAAI 2014. Version 2
includes a reorganization and some expanded proof
SDDs are Exponentially More Succinct than OBDDs
Introduced by Darwiche (2011), sentential decision diagrams (SDDs) are
essentially as tractable as ordered binary decision diagrams (OBDDs), but tend
to be more succinct in practice. This makes SDDs a prominent representation
language, with many applications in artificial intelligence and knowledge
compilation. We prove that SDDs are more succinct than OBDDs also in theory, by
constructing a family of boolean functions where each member has polynomial SDD
size but exponential OBDD size. This exponential separation improves a
quasipolynomial separation recently established by Razgon (2013), and settles
an open problem in knowledge compilation
On the Error Resilience of Ordered Binary Decision Diagrams
Ordered Binary Decision Diagrams (OBDDs) are a data structure that is used in
an increasing number of fields of Computer Science (e.g., logic synthesis,
program verification, data mining, bioinformatics, and data protection) for
representing and manipulating discrete structures and Boolean functions. The
purpose of this paper is to study the error resilience of OBDDs and to design a
resilient version of this data structure, i.e., a self-repairing OBDD. In
particular, we describe some strategies that make reduced ordered OBDDs
resilient to errors in the indexes, that are associated to the input variables,
or in the pointers (i.e., OBDD edges) of the nodes. These strategies exploit
the inherent redundancy of the data structure, as well as the redundancy
introduced by its efficient implementations. The solutions we propose allow the
exact restoring of the original OBDD and are suitable to be applied to
classical software packages for the manipulation of OBDDs currently in use.
Another result of the paper is the definition of a new canonical OBDD model,
called {\em Index-resilient Reduced OBDD}, which guarantees that a node with a
faulty index has a reconstruction cost , where is the number of nodes
with corrupted index
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