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Taylor Expansion Diagrams: A Compact Canonical Representation for Symbolic Verification

Taylor Expansion Diagrams: A Compact Canonical Representation for Symbolic Verification. M. Ciesielski, P. Kalla, Z. Zeng B. Rouzeyre Electrical & Computer Engineering LIRMM University of Massachusetts, Amherst, USA Montpellier, France ciesiel@ecs.umass.edu rouzeyre@lirmm.fr.

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Taylor Expansion Diagrams: A Compact Canonical Representation for Symbolic Verification

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  1. Taylor Expansion Diagrams:A Compact Canonical Representationfor Symbolic Verification M. Ciesielski, P. Kalla, Z. Zeng B. Rouzeyre Electrical & Computer Engineering LIRMM University of Massachusetts, Amherst, USA Montpellier, France ciesiel@ecs.umass.edu rouzeyre@lirmm.fr TED

  2. Outline • Motivation – RTL verification • Background • BDD (binary), BMD (word-level) • New canonical representation: TED • Construction and manipulation • Properties • Applications • Conclusions • Limitations, future work TED

  3. + + B 1 0 * * - - A F 0 1 B > A F ak s D <= bk ak s D bk Motivation – RTL Verification • Complex RTL designs • Data flow and control • Arithmetic and Boolean • Equivalence verification • Need representation that can handle arithmetic/Boolean • Efficient, compact • Canonical TED

  4. Common Representations • Boolean functions ( f : B  B ) • Truth table, Karnaugh map • SoP, PoS, ESoP, Reed-Muller, etc. • Binary Decision diagrams (BDD, ZDD, KFDD, …) • Arithmetic functions ( f : B  Int ) • Binary Moment Diagrams (*BMD, K*BMD, …) • Multi-terminal, Algebraic Decision Diagrams (ADD) • Need more abstract representation for arithmetic functions (f : Int  Int ) TED

  5. f x fx’ fx Binary Decision Diagrams (BDD) • Based on recursive Shannon expansion: f = x fx + x’ fx’ where:fx = f(x=1), fx’=f(x=0) • Compact data structure for Boolean logic • Canonical representation • reduced ordered BDDs (ROBDD) • Essential for verification • equivalence checking, satisfiability (SAT) TED

  6. a a b b c c 0 0 1 1 Application to Verification F = a’bc + abc +ab’c • Canonicity: equivalence checking of logic circuits G = (a+b)c  • Limitations • Require bit-level expansion of word-level variables • Size explosion for some functions (arithmetic) TED

  7. Binary Moment Diagrams (BMD) • Devised for word-level, arithmetic operations • Based on modified Shannon expansion (pos. Davio) f = x fx + x’ fx’ = x fx + (1-x) fx’ = fx’ + x (fx - fx’ ) = fx’ + x fx wherefx’ = fx=0is zero moment fx = (fx - fx’ )is first moment (derivative) TED

  8. BMD *BMD x2 x3 x1 x0 8 4 2 0 1 8 0 4 2 1 x1 x2 x0 x3 1 BMD – Example • Unsigned integer: X = 8x3 + 4x2 + 2x1 + x0 TED

  9. X Y 4 X + Y 2 4 1 2 1 4 4 0 0 1 1 x1 x0 x2 x2 x0 x1 y1 y0 y2 y1 y0 y2 2 2 1 1 *BMD • Efficiently models word-level operators • Limitation: requires bit-level representation of a word TED

  10. X + Y X Y 4 2 4 1 2 1 4 4 0 0 0 0 1 1 1 1 x0 x1 x0 x1 x2 x2 y2 y1 y0 y0 y2 y1 Y X Y X 2 2 1 1 Symbolic Symbolic Word level Word level Symbolic Representation • Why expand words into bits? Can we do better? • Abstract words into symbolic variables TED

  11. F(x) x … F1(x) F0(x) F2(x) Taylor Expansion Diagram (TED) • F = arithmetic function (F : Int  Int ) • Treat F as a continuous function • Taylor Expansion (around x=0): F(x) = F(0) + x F’(0) + ½ x2 F’’(0) + … • Notation • F0(x) = F(x=0) 0-child - - - - - - • F1(x) = F’(x=0) 1-child---------- • F2(x) = ½ F’’(x=0) 2-child====== • etc. • F(x) = F0(x)+ x F1(x) + x2 F2(x) + … TED

  12. F0(A) =F|A=0 = 2C + 3 F1(A) =F’|A=0 = 2AB|A=0 = 0 C B A C A B F2(A) =½ F’’|A=0 = B B B0 = B(0) = 0 B1=B’= 1 0 1 2 3 G0(C) = (2C+3)|C=0 = 3 G1(C) = (2C+3)’= 2 Construction - Your First TED F = A2B + 2C + 3 G= 2C + 3 (without normalization) TED

  13. (A+B)(A+2C) (A+B)C +1 X2 = (8x3 + 4x2 + 2x1 + x0)2 64 A B B A 16 16 B 8 4 4 1 2 C C 4 0 0 1 1 1 1 0 1 1 1 1 x1 x1 x0 x0 x2 x3 x2 2 1 1 1 TED – a few Examples TED

  14. 2. Merge isomorphic subgraphs Eliminate redundant nodes: Nodes with all edges  0 Nodes with only constant term f A a A 1 1 6 6 5 g 5 B B B B b C C C C g 1 1 0 1 0 0 1 0 b TED - Reduction Rules • f = 0 a2 + 0 a + g(b) = g(b) (A2 + 5A + 6)(B + C) TED

  15. A B A B 2 6 1 2 2 A 1 2 1 3 2 B 3 1 0 1 TED Normalization • TED can be normalized • weights of edges of a given node must be relativelyprime (to allow sharing isomorphic graphs) 2A + 2B + 6 2(A + B + 3) TED

  16. h = f OP g z OP OP= (+, - , •) f g u x y q v TED: Composition • Recursive composition of nodes, starting at the top • Operation depends on relative order of variables x, y • if x = y, then z = x, and h(x) = f(x) OP g(x) = f0(x) OP g0(y) + x [f1(x) OP g1(y)] + x2[f2(x) OP g2], … • if x > y, then z = x, and h(x) = f0(x) OP g(y) + x [f1(x) OP g(y)] + x2[f2(x) OP g], … • else …. = TED

  17. x x x u v u + v = + u0+v0 u1+v1 u0 u1 v0 v1 x y x u u + v v = + u0+ v u0 u1 v0 v1 u1 COMPOSE Operator – ADD/SUB • Nodes indexed by same variable • Nodes indexed by different variable (x > y) TED

  18. x x x u • v u v u1•v1 u0•v0 u0 u1 v0 v1 u0•v1+u1•v0 x y u x u • v v u0 u1 v0 v1 u0 • v u1 • v COMPOSE Operator – MULT • Nodes indexed by same variable = • • Nodes indexed by different variable (x > y) = • TED

  19. A+B A+2C A A A B B C 4 6 C 1•5 1•5 0+7 4•6 3•5 3•1 0•5 3 8 7 5 A 1 1 1 2 2 1 0 0 0 2 0 0 0•2 1•2 1•2 1•1 0•1 0•0 1•0 1•1 1•0 1 B B B B C C 8+7 C 2 0+2 0+0 COMPOSE Operation - Example B = + * C (A+B)(A+2C) + = TED

  20. Canonical Compact Linear for polynomials of arbitrary degree TED for Xk, k = const, with n bits, hask(n-1)+1 nodes. Can contain symbolic, word-level, and Boolean variables It is not a Decision Diagram X2=(8x3+4x2+2x1+x0)2 64 16 16 8 4 4 4 1 1 1 1 0 1 x1 x0 x1 x0 x3 x2 x2 2 1 1 1 Properties of TED n = 4, k = 2 TED

  21. x x x x x y = x y x y = (x + y – x y) x y = (x + y – 2 x y) x’ = (1-x) 1 -1 y y y y y 0 0 0 0 1 1 1 1 -1 -2 1 1 NOT XOR AND OR TED for Boolean logic • Needed to model arithmetic-Boolean interface TED

  22. B + A F1 Ahi Alo 1 0 * - Ahi ak s1 D > bk 2(k+1) ak 2k Alo s1 = ak (1-bk) 1 0 TED for Arithmetic Circuits • Arithmetic circuits contain related word-level (A, B) and Boolean (ak, bk) variables A = [ an-1, …, ak , …,a0 ] = 2(k+1)Ahi + 2k ak + Alo TED

  23. B A + * F2 A F1 - 1 0 * 0 1 B - * D s2 ak s1 ak D > bk bk Applications to RTL Verification • Equivalence checking with TEDs • interacting word-level and Boolean variables F2 = (1-s2) (A2-B2) + s2 D s2 = ak’ bk = 1 - ak + ak bk F1 = s1(A+B)(A-B) + (1-s1)D s1 = (ak > bk) = ak (1-bk) A = [Ahi,ak,Alo], B = [Bhi,bk,Blo] TED

  24. F1 = F2 -1 -1 22k+2 2k+2 Blo Alo Ahi Alo ak ak bk bk 1 1 1 0 2k 1 D -22k+2 -2k+2 Bhi 2k+1 2k RTL Verification – Result • Related word-level and Boolean variables F1 = s1(A+B)(A-B) + (1-s1)D s1 = (ak > bk) = ak (1-bk) F1 = (ak-akbk){ (2k+1Ahi + 2kak +Alo)2 - (2k+1Bhi + 2k bk + Blo)2 } + (1–ak + akbk) D TED

  25. Conclusions • TED representation, features • Canonical • Compact (linear for polynomials) • Represents arithmetic (word-level) operators + Boolean logic • Applications • Equivalence checking, symbolic simulation • Limitations • TED still needs to be implemented: efficiency, robustness? • How to represent relations (X < Y) without bit expansion? • Normalization: more complicated than in *BMD • Infinite series for some functions (ax) • Open problems • Application to satisfiability (SAT), functional test generation TED

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