1. Functional Test Generation using Constraint Logic Programming Zhihong Zeng, Maciej Ciesielski Dept. of Electrical & Comp. Engineering University of Massachusetts, Amherst Bruno Rouzeyre LIRMM, Université de Montpellier II

2. 3-5 Dec. 2001VLSI-SOC 20012 Outline Introduction Functional test generation for design validation Related work on satisfiability (SAT) Boolean and hybrid approaches to SAT New word-level SAT solver Unified Boolean and arithmetic domains Based on Constraint Logic Programming Experimental results, conclusions

3. 3-5 Dec. 2001VLSI-SOC 20013 Functional Simulation A viable approach to design validation Scales well with design size Predictable run time The way engineer thinks Downsides Incomplete, low “functional” coverage Confidence: when am I done? (coverage metrics are used to help)

4. 3-5 Dec. 2001VLSI-SOC 20014 Functional Validation - typical scenario Coverage Normalized verification test cycles 50 % 80 % 100 % 100.0 1.0 Deterministic tests Pseudo-random directed tests Manual directed tests Test development time ?

5. 3-5 Dec. 2001VLSI-SOC 20015 Deterministic Testing - Automation Functional test generation Given an RTL design and a coverage metric Goal: generate test vectors to reach predefined coverage goal Coverage metrics: code coverage, state coverage, transition coverage, etc Formulate as satisfiability (SAT) problem

6. 3-5 Dec. 2001VLSI-SOC 20016 Our Design Validation Framework Simulation Seed Environment Symbolic Simulation (bounded time model) Symbolic Simulation (bounded time model) Convert to a SAT Problem Word-level SAT BDDs Boolean SAT BDDs Boolean SAT Next Simulation Target Done Y N

7. 3-5 Dec. 2001VLSI-SOC 20017 Formulating a SAT Problem F s + D * A B C E 0 1 x Simulation target (value to be satisfied) state=s 3 F= b’0110 F> d’12 110011101110011101 Test vector Symbolic expression to be satisfied

8. 3-5 Dec. 2001VLSI-SOC 20018 Design Validation of RTL Designs Behavioral or RT-level designs contain datapath and control logic Word-level arithmetic operators Boolean logic Previous approaches are not efficient Gate-level ATPG, Boolean SAT, BDDs – bit-level Word-level information is not utilized Need more abstract, word-level models

9. 3-5 Dec. 2001VLSI-SOC 20019 Previous Work - Boolean SAT Based on Davis-Putnam procedure Tools: SATO, RelSat, GRASP, Chaff, … Conflict-based non-chronological back-jumping Conflict-based learning Variable ordering Restart Careful implementation BDD-based SAT WORD-LEVEL information is ignored!

10. 3-5 Dec. 2001VLSI-SOC 200110 Previous Work - Hybrid SAT HSAT: [Fallah et al. DAC’98] CNF-based SAT to solve Boolean constraints Linear Programming relaxation to check consistency of arithmetic constraints ATPG + Arithmetic solver [Huang et al. DAC’00] Gate-level ATPG to justify objectives in Boolean domain Modular arithmetic solver to solve arithmetic constraints Word-level implications between two domains are attempted: weak, not robust

11. 3-5 Dec. 2001VLSI-SOC 200111 Hybrid Approaches - Limitations Inefficient constraint propagation across domains: Boolean/arithmetic a=1 12>A A  3B+C B>C C>1 A<4C fghjfghj LKLK b  f=1, g=1 h=1, j=1 k 1  Arithmetic solver: infeasible  Conflict, backtrack 0 b

12. 3-5 Dec. 2001VLSI-SOC 200112 Constraint Propagation – a Problem A better constraint propagation: tighter interaction between the two domains a=1 12>A A  3B+C B>C C>1 A<4C fghjfghj LKLK b 0  f=1, g=1 h=1, j=1 b Boolean to Arithmetic Arithmetic to Boolean  A=11,C=2  k=0

13. 3-5 Dec. 2001VLSI-SOC 200113 Our approach: Word-level SAT in a unified domain Put arithmetic and Boolean constraints in same domain Constraint propagation is implicit to one engine More efficient than hybrid approach Use the state-of-the-art constraint solving techniques No need to develop your own Branch & Bound procedure Constraint solving is extensively researched by AI, CS Our previous work LPSAT: SAT based on Linear Programming A new word-level SAT Based on constraint logic programming (CLP-SAT)

14. 3-5 Dec. 2001VLSI-SOC 200114 Why Another Unified Approach? LPSAT [DATE’01]: SAT based on ILP Non-linear constraints must be linearized Numerical convergence problem (integers) Optimization engine CPLEX ($$, not a SAT solver) Alternative: SAT based on Constraint Logic Programming (CLP-SAT)

15. 3-5 Dec. 2001VLSI-SOC 200115 CLP-SAT: SAT using Constraint Logic Programming Use Gnu-prolog as solving engine Publicly available Constraint solving over finite domain A native Prolog compiler: fast execution time Modeling is straightforward No need to linearize complex operators (MPY) Built-in predicates for Boolean gates

16. 3-5 Dec. 2001VLSI-SOC 200116 Design/Constraint Modeling Identify Arithmetic and Boolean domains Word-level signals: B, D, E, X, Z (Partial) bit expansion: A, C, Y Y = Y[0] + 2*Y[1] + 4*Y[2] + … Single-bit Boolean signals: s A Z X Y < s + B D * C E 1 0

17. 3-5 Dec. 2001VLSI-SOC 200117 CLP-SAT – Modeling the Constraints Arithmetic/BooleanGnu-Prolog Predicates Z = A * BZ #= A * B Z = and (A, B) A #  B #  Z Z = or (A, B) A #  B #  Z Z = not (A)A #\ Z Z = ite(s, A, B) (s #= 1) #  (Z #= A), (s #= 0) #  (Z #= B)

18. 3-5 Dec. 2001VLSI-SOC 200118 Handling Modular Arithmetic A 16-bit adder: Z = A + B Z <= 2 8 + 16 8 A BZ ? D 9 Adding a free variable, D A+B = Z + D*2 8 D <= 2 9 Z <= 2 8 Z = (A+B) mod 256

19. 3-5 Dec. 2001VLSI-SOC 200119 Handling Wide Arithmetic Operators > 56-bit s A B > 28-bit AHAH BHBH == 28-bit > 28-bit ALAL BLBL s Internal representation (computer): 32-bit words Largest domain in CLP is 28-bit Decompose using Boolean logic

20. 3-5 Dec. 2001VLSI-SOC 200120 Experiments - Benchmarks M13 – 13-bit multipliers (sat, non) M16 – 16-bit multipliers, decomposed (sat, non) Square - compute Z 2 = X 2 + Y 2, 16-bit variables Quadratic - solve X 2 = a X + b for 16-bit X Linear-k- k-wide chain of comparators (k=40, 90), simple structure, large # inputs (over 1200) GCD-k- sequential circuit, 24-bit gcd unrolled k=20, 40 time frames Mdpe- multiplier feeding a dynamic priority encoder, taken from realistic design

21. 3-5 Dec. 2001VLSI-SOC 200121 Benchmarks - Statistics BenchmarksCLP-SAT # lines LPSAT #constraints CNF-SAT #clauses M16x1611614924720 square52971077361 quadratic41346972015 Linear(2)3527274977887 gcd20876542117785 gcd4015151062248449 Mpde(2)685367330851

22. 3-5 Dec. 2001VLSI-SOC 200122 Experimental Results Run-time speed-up TestcaseCLP-SATLPSATSATOGRASPCHAFF M16*16(s)3124.70.280.071 M16*16(n)214848.60--1 Square(1)520-- 0.231 Square(0)4135-- 1 Quadratic11020.480.351 Linear(1)0.83.20.240.401 Gcd201.1sec*0.03sec*-- Gcd401.9sec*0.08sec*-- Mdpe(1)12 0.180.021 *: real CPU time

23. 3-5 Dec. 2001VLSI-SOC 200123 Conclusions Functional test generation for design validation Symbolic simulation + word-level SAT A new word-level SAT in a unified domain Based on constraint logic programming A good alternative to LPSAT Use the state-of-the-art constraint solving techniques Preliminary results are promising: Good for mixed arithmetic and Boolean logic Limitation: not competitive for designs with large portions of random logic