1. Taylor Expansion Diagrams (TED): Verification EC667: Synthesis and Verification of Digital Systems Spring 2011 Presented by: Sudhan

2. 2 In this presentation... 1. What is a TED? 2. What is the Motivation behind TED’s development? 3. How to create a TED? 4. How TED is used for Verification?

3. 3 1. Taylor Expansion Diagram Canonical representation for arithmetic expressions useful for equivalence checking of a design among others. Works at a higher level of abstraction. Built on Taylor series expansion. (Representation of a function as a sum of infinite terms) f(x) is a real, differentiable function corresponding to an algebraic expression F. F 1 (x=0) is the first derivative of the function w.r.t x and evaluated at x 0 =0. F(x,y,..) = F 0 (x=0) + x F 1 (x=0) + x 2 F 2 (x=0) + …

4. 4 2. Motivation for TED Development Size and complexity of digital designs steadily increasing. So does the need for verification tools capable of working at higher abstraction levels. Typically high level descriptions contain word-level arithmetic computations along with Boolean logic. This need for canonical representations working with symbolic, algebraic word-level computations was the motivation behind creating a new representation.

5. 2. Review of Previous work Logic of equality with uninterpreted functions. [J. Burch, ’94] Use of Propositional logic Formula. [R. Bryant, 01] Reduced, ordered Binary Decision Diagrams (ROBDD) and its variants. [R.E. Bryant, ‘86] Binary Moment Diagrams. [R.E. Bryant, 95] Symbolic Algebra Methods. [G. DeMicheli, 03] Drawbacks: Requires bit level functions for word level signals. Algebra tools require canonical representations for equivalence checking. 5 X + Y 1 0 X Y Symbolic 1 0 x0 x1 x2 y0 y1 y2 1 2 4 2 4 1 Bit level

6. 2.Reduced Ordered Binary Decision Diagrams It represents a set of binary valued decisions in a rooted directed acyclic graph(DAG). It is based on a recursive Shannon decomposition and has a set of reduction rules. Advantages : Widely available software Packages and used for combinational equivalence checking. BDD based verification systems successful in verifying control dominated circuits. Drawbacks: Not successful for data path design. With complex circuits, BDD representations face size limits. 6

7. Performs decomposition of a linear function based on its first two moments (linear moment) and (constant moment). Uses modified Shannon’s expansion, in which a binary variable is treated as a (0,1) integer. Drawbacks: For high degree polynomials defined over words with large bit- width, it becomes an expensive representation. It has a set of restrictions imposed to make it canonical and thus makes it difficult to construct. 7 2. Binary Moment Diagrams

8. f’(x) f’’(x) … 3. Creating a TED - Basics Notation:  f(x=0) 0-child  f’(x=0) 1-child  f’’(x=0) 2-child 8 f(x) x 1 x x2x2

9. 9 1.Start with the algebraic Expression and ordering of variables. 2. Recursively, from F find the values for all the constant, first and second derivative terms w.r.t to each variable. 3. Form the TED with this information based on the notation. 4. Apply reduction rules and Normalization to get Normalized TED. 3. Creating a TED - Steps

10. 10 3. Creating a TED - Example Arithmetic expression : F = (A+B) (A+2C) = A 2 + AB + 2AC + 2BC Consider ordering : A,B,C 1.Decomposition w.r.t A: F(A=0) = 2BC F’(A=0) = B + 2C 1/2F’’(A=0) = 1 2.Decomposition of G=F(A=0), H=F’(A=0), F’’(A=0) w.r.t B: G(B=0) = 0 H(B=0) = 2C G’(B=0) = 2C H’(B=0) = 1 3.Similarly, Decomposition w.r.t C: F(C=0) = 0 F’(C=0) = 2 H=B+2C G=2BC F A B B 1 0 C 2 2C From TED: F = A 2 +AB+2AC+2BC Every path from root node to non zero vertex is a non zero term in the expression 1 1 1 1 A B B C A2A2

11. f = 0 v 2 + 0 v + y(u) = y(u) 11 3. Creating a TED – Reduction Rules 1. Eliminate redundant nodes: Nodes with all non zero edges connected to terminal 0 y v u 0 u y x 2. Merge Isomorphic Graph: Isomorphic if structure and attributes of the two TED’s match. All 3 subgraphs are isomorphic i.e. one to one mapping between vertex sets and edge sets. ‘Taylor Expansion Diagrams: A Canonical Representation for Verification of Data Flow Designs’ – Maciej Ciesielski, Priyank kalla, Serkan Askar

12. 3. Creating a TED - Normalization Applied to reduce the graph. Move numeric values from nonzero terminal nodes to terminal edges. A reduced ordered TED representation is normalized when: The weights assigned to edges spanning Out of a node are prime. Numeric value 0 appears only in the terminal nodes. Graph contains no more than two terminal nodes. Applying Normalization to example 12 ‘Taylor Expansion Diagrams: A Canonical Representation for Verification of Data Flow Designs’ – Maciej Ciesielski, Priyank kalla, Serkan Askar F A B B 1 0 C 2 2

13. 13 3. Creating a TED - Composition Like the apply operator in BDD, TED’s can be similarly composed, following specific rules. Starting from root node of the two TED’s, the final TED is constructed by recursively: constructing all the non zero terms from two functions. combining the terms according to a given operation. Rules: Case 1 : Nodes w and y with same index: 0-child is obtained by pairing 0-child of w,y. 1-child is obtained by the sum of two cross products of 0 and 1 children. 2-child is obtained by pairing the 1-children of w,y. Case 2 : Nodes w and y with different indices: The node with the lower index is added to the 0-child of the node corresponding to variable with higher index. Case 3 : Both the nodes are terminal nodes: val = val(w) + val(y) if add operation, val = val(w)*val(y) for multiplication.

14. 14 Apply composition and verify if the final TED is the same as the one constructed by expansion of the expression. F = (A+B) (A+2C) = A 2 + AB + 2AC + 2BC Ordering : A,B,C 3. Creating a TED – Composition Example(1/2) A+B 01 w x A B * C A+2C 01 y z A 2 w.y x.z 1.z x.1 01 0 1 2 + 1 0.z 1.z 00 0 1 2 A B C B C C 0-child is obtained by pairing 0-child of w,y. 1-child is obtained by the sum of two cross products of 0 and 1 children. For terminal nodes, val = val(u)*val(v) For terminal nodes, val = val(u)*val(v)

15. 3. Creating a TED – Composition Example(2/2) 15 1.z x.1 01 0 1 2 + C B B 1 0.z 2 0 0 C On replacing w.y x.z 1.z 0 2 B C B x.1 1 A w.y x.z 1.z x.1 01 0 1 2 + 1 0.z 1.z 00 0 1 + 2 A B C C The two TED’s are equal The node with the lower index is added to the 0-child of the node corresponding to variable with higher index.

16. 4. TED as a tool for Verification Given RTL design, Build TED’s for Primary Inputs Expansion of word level signals needed When one or more bits from i/p used in other parts of design. Then, TED’s constructed for all components of the design. TED’s for the primary outputs are generated by composing the TED’s in topological order from PI to PO. After constructing normalized TED for each design, the test for functional equivalence is performed by checking for isomorphism of the resulting graphs. 16

17. Partitioning of word level signals into subvectors creates an issue for polynomial representation. Only finite Taylor expansions permitted. They cannot represent relational operators in symbolic form. TEDs cannot represent modular arithmetic. 17 Limitations of TED

18. 18 Summary Salient features of TED: Represents arithmetic (word-level) operators and Boolean logic. Canonical (Based on Taylor’s theorem which states the uniqueness of the Taylor’s series representation of a function, evaluated at a particular point) Compact (linear for polynomials) In the worst case, TED for f requires O(k n-1 ) nodes and O(k n )edges. Applications Useful for High-level synthesis, equivalence checking, symbolic simulation.

19. References Priyank Kalla, Maciej Ciesielski, Emmanuel Boutillon, Erric Martin, ‘High-Level Design Verification using Taylor Expansion Diagrams: first Results’ Maciej Ciesielski, Priyank Kalla, Serkan Askar, ‘ Taylor Expansion Diagrams: A Canonical Representation for Verification of Data Flow Designs’ 19

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