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Functional Vector Generation for HDL Models Using Linear Programming and 3-Satisfiability

Functional Vector Generation for HDL Models Using Linear Programming and 3-Satisfiability. Farzan Fallah Srinivas Devadas Laboratory for Computer Science MIT. Kurt Keutzer Department of EECS UC Berkeley. Outline. Introduction Functional test vector generation strategy

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Functional Vector Generation for HDL Models Using Linear Programming and 3-Satisfiability

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  1. Functional Vector Generation for HDL Models Using Linear Programming and 3-Satisfiability Farzan FallahSrinivas Devadas Laboratory for Computer Science MIT Kurt Keutzer Department of EECS UC Berkeley

  2. Outline • Introduction • Functional test vector generation strategy • New hybrid SAT method • Results • Ongoing work

  3. Introduction Design Verification if (C) A = 1; else A = 2; $display(“%d”, A);

  4. Introduction Design Verification • How do we solve the design verification problem? • Formal verification • Who specifies correctness property? • Simulation • How do you get the test vectors?

  5. Introduction Coverage Directed Simulation • Choose a coverage metric, e.g., line/branch/path coverage. • Automatically generate vectors that achieve targeted coverage under chosen metric.

  6. W > Z Functional Vector Generation Strategy • We view HDL descriptions as interconnections of combinational modules which include Boolean and word-level operators. f X + Y c G x2 0 d 1 x0

  7. Functional Vector Generation Strategy • Boolean operators: AND, OR, and INVERTER gates. • Word-level operators: Comparison: Addition/Subtraction: Increment/Decrement: Scalar multiplication: Left/Right shift:

  8. Functional Vector Generation Strategy • Any other operator must be converted into a collection of the previous operators. Example can be modeled as,

  9. Functional Vector Generation Strategy • can be modeled as, and 4 constraints over Y and individual bits of X for each pi , where piis a partial product. Z = 2n-1pn-1+ 2n-2pn-2+ … + p0

  10. Functional Vector Generation Strategy • We choose path coverage metric. • A path is a set of alternating modules and signals. S1 S2 S3 m1 m2 P={S1, m1, S2, m2, S3}

  11. Functional Vector Generation Strategy • Sensitizing a sub-path through a module means the value of the input to the path should affect the value at the output. A 0 0 • Sensitizing a path means sensitizing each module .

  12. AND 1 Functional Vector Generation Strategy • Sensitization of a sub-path through a module will require values at some of the module inputs. • Value of side-inputs for logic gates, OR 0 Side-inputs

  13. Value of side-input for word-level operators (assuming unsigned numbers), A > C C A A B k B Functional Vector Generation Strategy C

  14. Functional Vector Generation Strategy b X > c Y a Constraints: b = 1 c = a AND b b = X > Y • Find a solution that satisfies the set of constraints. • Write module input-output relationship for every module in the circuit. • Write sensitization requirement on intermediate signals. • Select a path.

  15. Functional Vector Generation Strategy • How do we find a solution to the constraints? • Use Boolean clauses to modelall modules and solve resulting 3-SAT problem. • Use Boolean clauses andlinear constraints to model modules and solve the problem using 3-SAT solver andinteger programsolver. • Both methods use Branch and Bound technique.

  16. X Z Y X Y 3-SAT Approach • Use Boolean clauses to model input-output relationship of each gate.

  17. 3-SAT Approach • Solve the resulting 3-SAT problem to find a satisfying assignment for all input variables. • 3-SAT approach is not efficient because word-level operators have to be modeled using Boolean operators.

  18. x2 x2 0 0 d d 1 1 x0 x0 c c X W + > Y Z e e a a b b New Hybrid Satisfiability Approach 1 1 0 0 f f g g G h h • We want to exercise a path in the circuit. • Use linear word-level operators (Linear arithmetic constraints, LACs) to model data-path.

  19. x2 1 0 d 1 x0 0 c f g h e a b New Hybrid Satisfiability Approach • Choose values for side-inputs.

  20. x2 0 d 1 x0 c e a b New Hybrid Satisfiability Approach 1 0 f g h • Control part is modeled using Boolean operations. ?? • Choose values for side-inputs. • Gates are modeled using Boolean clauses.

  21. New Hybrid Satisfiability Approach Correlation: x0 is first bit of X x2 is third bit of X g is equal to G • Gates are modeled using Boolean clauses. • There is correlation between Boolean variables and integer variables.

  22. Simplifying SAT New Hybrid Satisfiability Approach Correlation: x0 is first bit of X x2 is third bit of X g is equal to G • Essential variable: If there is a single variable in TRUE (complemented) form in a clause it must be set to 1(0). • c must be set to 0.

  23. Simplifying SAT New Hybrid Satisfiability Approach Correlation:x0-X, x2-X, g-G Current assignment: c=0, e=1 • New essential variables might be created.

  24. Simplifying SAT New Hybrid Satisfiability Approach Correlation:x0-X, x2-X, g-G Current assignment: c=0, e=1 , b=1 • Unate variable: If a variable only appears in true (complemented) form it can be set to 1(0). • Note that there was no integer variable correlated to b. • Use unate variable rule recursively. • We can set b to 1.

  25. Linear infeasibility New Hybrid Satisfiability Approach Correlation:x0-X, x2-X, g-G Current assignment: c=0, e=1, b=1, f=0 • If SAT is infeasible, there is no solution for the constraints, otherwise we continue search for the solution. • Use polynomial time sufficiency check for SAT infeasibility. • If linear relaxation of LACs is infeasible, there is no solution to the constraints. This can be checked in polynomial time.

  26. Branching New Hybrid Satisfiability Approach Correlation:x0-X, x2-X, g-G Current assignment: c=0, e=1, b=1, f=0 • Choose an input variable which appears in clauses most (x2). • Select a Boolean variable heuristically and set it to 1(0), if it fails to find a feasible solution, set it to 0(1). • Set x2 to 1.

  27. Simplifying SAT New Hybrid Satisfiability Approach Correlation:x0-X, x2-X, g-G Current assignment: c=0, e=1, b=1, f=0, x2=1 • Set d to 1.

  28. Branching New Hybrid Satisfiability Approach Correlation:x0-X, x2-X, g-G Current assignment: c=0, e=1, b=1, f=0, x2=1, d=1 , h=1 • Set h to 1. • There is a correlation between x2 and X, use new integer variables and modify LACs.

  29. Branching New Hybrid Satisfiability Approach Correlation: x0-X, x2-X, g-G Current assignment: c=0, e=1, b=1, f=0, x2=1, d=1, h=1 • Find a solution to LACs using Integer Programming.

  30. Branching New Hybrid Satisfiability Approach Solution Boolean Variables: c=0, e=1, b=1, f=0, x2=1, d=1, h=1, x0=X, g=X Integer Variables: X7-3=0, X1-0=0, Y=0, W=4, Z=0, G=1 • Free Boolean variables correlated to integer variables, are set to the appropriate values.

  31. Branching New Hybrid Satisfiability Approach Solution Boolean Variables: c=0, e=1, b=1, f=0, x2=1, d=1, h=1, x0=0, g=X Integer Variables: X7-3=0, X1-0=0, Y=0, W=4, Z=0, G=1 • Free Boolean variables correlated to integer variables, are set to the appropriate values.

  32. Branching New Hybrid Satisfiability Approach Solution Boolean Variables: c=0, e=1, b=1, f=0, x2=1, d=1, h=1, x0=0, g=1 Integer Variables: X7-3=0, X1-0=0, Y=0, W=4, Z=0, G=1 • Free Boolean variables correlated to integer variables, are set to the appropriate values.

  33. Results HSAT 3-SAT Example Exp. Time/ Sat. Time #Clauses/ #LACs #Clauses Sat. Time pport 17/18 685 <1 s 1.2/4.7 s schsm <1 s 841 141/24 3.1/261 s ctla 1355 <1 s 1/24 >1000 s ctlbc <1 s 1341 1/60 4.9/142 s 0/104 mult16 8466 16 s >1000 s

  34. Ongoing Work • Improvement on satisfiability checking using seamless integration of linear programming and satisfiability. • Currently we use conventional time frame expansion strategy for sequential HDL models. • We are exploring more efficient algorithms. • We are working on generating functional vectors for a variety of code coverage metrics.

  35. OCCOM talk Ongoing Work Future system overview Simulation driver (vectors) Simulation model (HDL) Simulation monitor (yes/no) Simulation engine LP-3SAT Vector Generation No OCCOM Coverage Analysis Is it enough? Diagnosis of Unverified Portions Yes Stop

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