By Tariq Bashir Ahmad

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Taylor Expansion Diagrams (TED)

• By Tariq Bashir Ahmad

Adapted from the paper M. Ciesielski, P. Kalla,
Z. Zeng, B. Rouzeyre,Taylor Expansion
DiagramsA Compact Canonical Representation for
Symbolic Verification, in DATE, 2002.
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Presentation Structure

• Motivation RTL verification
• Background
• BDD (binary), BMD (word-level)
• New canonical representation TED
• Construction and manipulation
• Properties
• Applications

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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

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Levels of Abstraction
We can design at a higher level of abstraction,
Can we verify at a higher level of abstraction ?
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Common Representations

• Boolean functions ( f B ? B )
• Truth table, Karnaugh map
• SoP, PoS etc.
• Binary Decision diagrams (BDD)
• Arithmetic functions ( f B ? Int )
• Binary Moment Diagrams (BMD)
• Need more abstract representation for arithmetic
  functions (f Int ? Int )

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Binary Decision Diagrams (BDD)

• Based on recursive Shannon expansion f
  x fx x fx
• where fx f(x1), fxf(x0)
• Compact data structure for Boolean logic
• Canonical representation
• reduced ordered BDDs (ROBDD)
• Essential for verification
• equivalence checking, satisfiability (SAT)

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Application to Verification

• Canonicity equivalence checking of logic circuits

F abc abc abc
G (ab)c
?

• Limitations
• Require bit-level expansion of word-level
  variables
• Size explosion for some functions (arithmetic)

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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
• where fx fx0 is zero moment
• f? x (fx - fx ) is first moment
  (derivative)

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BMD Example

• Efficiently models word-level operators

X Y X2x1x0 y2y1y0 0 1 1 1 0 1 8
X Y X2x1x0 . y2y1y0 0 1 1 . 1 0 1 15

• Limitation requires bit-level representation of
  a word

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Symbolic Representation

• Why expand words into bits? Can we do better?
• Abstract words into symbolic variables

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Taylor Expansion Diagram(TED)

• F arithmetic function (F Int ? Int )
• Treat F as a continuous function
• Taylor Expansion (around x0)
• F(x) F(0) x F(0) ½ x2 F(0)
• Notation
• F0(x) F(x0) 0-child - - - - - -
• F1(x) F(x0) 1-child ----------
• F2(x) ½ F(x0) 2-child
• So

F(x) F0(x) x F1(x) x2 F2(x)
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Construction of TED example
F A2B 2C 3 with order AltBltC
G 2C 3
(without normalization)
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Few more examples
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TED Reduction Rules

• 1. Eliminate redundant nodes
• Nodes with all edges ? 0
• Nodes with only constant term

f 0 a2 0 a g(b) g(b)
2. Merge isomorphic subgraphs
(A2 5A 6)(B C)
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TED Normalization

• TED can be normalized
• weights of edges of a given node must be
  relatively prime (to allow sharing isomorphic
  graphs)

2A 2B 6
2(A B 3)
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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 gt y, then z x, and
• h(x) f0(x) OP g(y) x f1(x) OP g(y)
  x2 f2(x) OP g,
• else .

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COMPOSE Operator ADD/SUB

• Nodes indexed by same variable

• Nodes indexed by different variable (x gt y)

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Compose Operation Example
B


C
2AB2C
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Comparison of BDD and TED
Property BDD TED
Decomposition Shannon Taylor
Composition And/OR Multiply/Add
Canonicity Yes Yes
Canonicity Reduction rules Reduction rules normalization
Satisfiability Yes No
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TED for Arithmetic Circuits

• Arithmetic circuits contain related word-level
  (A, B) and Boolean (ak, bk) variables
• A an-1, , ak , ,a0 2(k1)Ahi 2k
  ak Alo

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Application 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(AB)(A-B) (1-s1)D s1 (ak gt bk) ak
(1-bk)
A Ahi,ak,Alo, B Bhi,bk,Blo
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Result RTL Verification

• Related word-level and Boolean variables
• F1 s1(AB)(A-B) (1-s1)D
• s1 (ak gt bk) ak (1-bk)

F1 (ak-akbk) (2k1Ahi 2kak Alo)2 -
(2k1Bhi 2k bk Blo)2 (1ak akbk)
D
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Questions?