A unified theory of terminal-valued and edge-valued decision diagrams

Transcription

1 A unified theory of terminal-valued and edge-valued decision diagrams Gianfranco Ciardo Andrew S. Miner Department of Computer Science Iowa State University September 2, 214 Work supported in part by the National Science Foundation under grant CCF Ciardo, Miner (ISU) Unifying decision diagrams September 2, / 14

2 Many types of decision diagrams Extending the domain from B L to X = X L X 1 is straightforward type domain range BDD B L B MDD X B MTBDD B L Z, R MTMDD X Z, R ADD B L any type edge values combinator domain range EVBDD Z sum B L Z EV + MDD N { } sum X Z { } PDG probabilities multiply B L [, 1] EV MDD [, 1] multiply X R AADD R R (a,b) (c,d)=(a+bc,bd) B L R Ω Ciardo, Miner (ISU) Unifying decision diagrams September 2, / 14

3 Motivating questions Can we unify terminal and edge valued decision diagrams? Can we achieve more elegance, simplicity, and generality? Why can MTBDDs encode any partial function B L Z { }? Why can t EVBDDs (in their original definition) do that? Why can EV + MDDs (our canonical definition) do that? This is why we introduced EV + MDDs, but what is the key issue? Why can EV + MDDs have range R but EV MDDs must have range R? Can EV MDDs encode CTMC generators, not just rate matrices? What are the advantages/disadvantages of terminal vs. edge valued DDs? Which decision diagram encoding should I use for a particular application? Ciardo, Miner (ISU) Unifying decision diagrams September 2, / 14

4 ET-monoid Semigroup (S, ): set S is closed w.r.t. the associative binary operator Monoid (S, ): semigroup where S contains identity element e Group: monoid (S, ) where every element of S has an inverse in S Total order : transitive, antisymmetric, and total binary relation Definition ET-monoid M = (S, S E, S T,, ): (S, ) is a monoid, {e} S E S, S T S, S E S T =, is a total order on S, and Axiom 1: S E is closed over S E S E S E Axiom 2: S T terminates S E from the right S E S T S T Axiom 3: Each element in S E has an inverse in S a S E, a 1 S Axiom 4: For any sequence Σ C + M, there exists some σ Σ and mthe total D(σ) order such that defines σ is an a desirability optimal sequence on (sequences for m in Σof) edge values Axiom 5: For any sequence Σ C + M and any a S E, if σ is an optimal sequence this is quite forcomplex, m in Σ, then but necessary a σ is an foroptimal canonicity sequence in ourfor general a msetting a Σ Ciardo, Miner (ISU) Unifying decision diagrams September 2, / 14

5 A few ET-monoids (S, S E, S T,, ) (B, {}, {1},, any) (B, {1}, {},, any) (B, B,,, any) is exclusive or, is the identity (Z, Z,, +, ) a b if a < b or a = b and a b (Z, {}, Z \ {}, +, ) is the identity and the most desirable (Z { + }, Z, { + }, +, ) (Z { }, Z, { }, +, ) (Z { +,, µ}, Z, { +, }, +, ) µ means undefined substitute Z with Q or R in the above four (R, Z, {n + 2 : n Z}, +, ) (Z, N,, +, ) is the identity and the most desirable (Z { + }, N, { + }, +, ) substitute Z with Q or R and N with Q or R in the above two Ciardo, Miner (ISU) Unifying decision diagrams September 2, / 14

6 A few more ET-monoids (S, S E, S T,, ) (Q, Q >,,, ) a b if ln a < ln b or ln a = ln b and a 1 b (Q, Q >, {},, ) 1 is the identity and the most desirable (Q { + }, Q >, { + },, ) (Q { +, µ}, Q >, {, + },, ) substitute Q with R and Q > with R > in the above four (Q, N \ {},,, ) 1 is the identity and the most desirable (Q, N \ {}, {},, ) (Q { +, µ + }, N \ {}, { + },, ) (Q { +, µ + }, N \ {}, {, + },, ) (R R, R R >,,, ) (a, b) (c, d) = (a + bc, bd) is such that the identity (, 1) is the most desirable Ciardo, Miner (ISU) Unifying decision diagrams September 2, / 14

7 ETDDs: edge-and-terminal valued decision diagrams First, a non-canonical version of an ETDD (forest): Definition Given domain X = X L X 1 and ET-monoid M = (S, S E, S T,, ), an ordered, ET-valued decision diagram (ETDD) over (X, M) is an acyclic, node-labeled, and edge-labeled multi-graph where: Each node p is at a level p.lvl = k, with L k Only terminal node Ω is at level Node p at level k > has n k = X k edges; for i k X k, p[i k ] = a, q means edge i k has value a S E S T and points to node q at level h <k; let p[i k ].val = a and p[i k ].node = q There is a non-empty set of root edges R; for any root edge a, p R, a S E S T and p is a node at level k, L k For any edge a, q, including a root edge, if a S T, then q = Ω Every node in the graph is reachable from some root edge Ciardo, Miner (ISU) Unifying decision diagrams September 2, / 14

8 Toward canonical ETDDs As usual, no duplicate or redundant nodes Our general setting also requires constraints on edge values and nodes: If (S E, ) is a group and S T =, force p[].val = e (e.g., EVBDDs) If not, canonicity is surprisingly elusive, we need to use desirability e.g., ET-monoid (Q { +,, µ +, µ }, Z\{},{, +, },, ) p p p p p q + q - q + q + q q - + Edge normalization Node normalization use the representative for the equivalence class divide the node by its most desirable divisor Ciardo, Miner (ISU) Unifying decision diagrams September 2, / 14

9 Canonicity Definition An ETDD is (fully) reduced if its edges and non-terminal nodes are normalized and contains no duplicate nodes Theorem Given a nonempty, finite set of vectors V X S E S T, there exists a reduced ETDD with root edges R such that V(L, R) = V Definition An ETDD is scalar independent if, for any nodes p, q with p.lvl =q.lvl =k, if v(p) = a x and v(q) = b x for vector x : X k X 1 S E S T and scalars a, b S E then p = q Theorem Every reduced ETDD is scalar independent Ciardo, Miner (ISU) Unifying decision diagrams September 2, / 14

10 Space comparisons: ETDDs over the same ET-monoid Theorem Given an ET-monoid M = (S, S E, S T,, ), an ETDD G over (X, M) with root edges R G, and a reduced ETDD H over (X, M) with root edges R H, if V(L, R G ) = V(L, R H ), then G is homomorphic to H... The reduced ETDD encoding is minimal (for a given X and M) Theorem...if G is scalar independent with no redundant nodes, it is isomorphic to H Any normalization that guarantees scalar independence works equally well Ciardo, Miner (ISU) Unifying decision diagrams September 2, / 14

11 Implications for ETDDs over the same ET-monoid Applying desirability to sequences left-to-right vs. right-to-left works equally well b a c d 1 9 k -6-2 m -2 n o -1 Using EVBDDs with p[].val = vs. min{p[i k ].val :i k X k }= works equally well Ω Ω Ciardo, Miner (ISU) Unifying decision diagrams September 2, / 14

12 Space comparisons: ETDDs over different ET-monoids Definition Given M G = (G, G E, G T,, G ) and M H = (H, H E, H T,, H ), M G is homomorphic to M H via function f : G H if a G E and b G E G T f (a b) = f (a) f (b) a G E f (a) H E t G T f (t) H E H T If f is one-to-one, M G is lossless homomorphic to M H via f Otherwise, M H is lossy homomorphic to M H via f M G is isomorphic to M H via bijection f if M G is homomorphic to M H via f and M H is homomorphic to M G via f 1 Ciardo, Miner (ISU) Unifying decision diagrams September 2, / 14

13 Implications for ETDDs over different ET-monoids Theorem If M G is homomorphic to M H via f, the reduced ETDD over (X, M G ) with root edges R G s.t. V(L, R G ) = V, is homomorphic to the reduced ETDD over (X, M H ) with root edges R H s.t. V(L, R H ) = f (V) Compl.-edge BDDs (B,B,,, ) never worse than BDDs (B,{},{1},, ) Lemma If G is isomorphic to H via some f, the reduced ETDD over (X, G) is isomorphic to the reduced ETDD over (X, H) encoding the same vector Any isomorphic ET-monoid works equally well Lemma ET-monoid M = (S, {e}, S E S T \ {e},, ) is lossless homomorphic to any ET-monoid M = (S, S E, S T,, ) via the identity function A reduced ETDD is never worse than the equivalent reduced MTMDD Ciardo, Miner (ISU) Unifying decision diagrams September 2, / 14

14 Conclusion and ongoing work Our ETDD framework unifies many types of decision diagrams We found the key canonicity requirements in a general setting Our general theorems provide results for popular decision diagram classes We are still completing work on the time complexity of ETDD algorithms (complexity improves if ET-monoid has more structure, e.g., S E is a group) The current paper is already 53 pages so far :-( Ciardo, Miner (ISU) Unifying decision diagrams September 2, / 14