1. Logic Stuff & FV Basics Erik Seligman CS 510, Lecture 2, January 2009

2. Goals of This Session  Review basics of boolean logic, and some fundamental FV algorithms Logic should just be a review for people in this class!  Establish common symbols & terms Variety of ways to express common ops  Have basic foundation for discussing FV Getting a flavor for contents of tools NOT describing full internal algorithms NOT full mathematical rigor –If you want screenfuls of symbols, take Xie or Song class!

3. Basic Boolean Logic

4. Fundamental operations  For consistency, will use Verilog-like notation: AND: a & b OR: a | b NOT: ~a  Sometimes AND represented as multiplication, and OR as addition Like arithmetic, except 1+1 == 1  Implication: a -> b Same as: ~a | b Terms: a is the antecedent, b is the consequent

5. Basic Boolean Identities  Commutative, Associative  Distributive both ways a & (b|c) == (a&b) | (a&c) a | (b&c) == (a|b) & (a|c)  Idempotence: a&a == a, a|a == a  DeMorgan ~(a&b) = ~a | ~b ~(a|b) = ~a & ~b

6. Implication relationships a -> b  Converse: b -> a  Inverse: ~a -> ~b  Contrapositive: ~b -> ~a Which pairs are identical in truth value?  Can be useful when restating for FV  Use |= (“logically entails”) symbol as distinct from implication when appropriate (a -> b) |= (~b -> ~a)

7. Inference Rules  Rules to derive new statements  Some basic rules (a -> b), (a) (modus ponens) b (a | b), (~a) b a -> F (contradiction) ~a

8. What is a Proof?  Apply sequence of inference rules  Example: Known: S1: a, S2: (a -> b), S3: (d -> ~b)) Prove: ~d –C1: S1, S2 |= b –C2: S3 |= (~d | ~b) –C3: C1, D3 |= ~d

9. Predicate Logic  Add predicates, or functions, and quantifiers: For All (A), Exists (E)  Examples: A(x) Cat(x) -> Mammal(x) E(x) Cat(x) & ~Black(x)

10. Linear Temporal Logic (LTL)

11. What Is Linear Temporal Logic?  Add notion of time to predicate logic X = Next time G = Globally / always F = Future / eventually U = Until  Statements evaluated at points in time Discrete, “clocked” machine model  Lots of power for stating properties Useful in real-life designs In upcoming 2009 SVA standard

12. Equivalent operations in LTL  Ga == ~(F(~a))  Fa == ~(G(~a))  Fa == T U a  Distributive laws G(a &b) = Ga & Gb F(a | b) = Fa | Fb But be careful… –can G(a|b) be distributed? –How about F(a&b)?

13. LTL examples  Eventually bus grant will occur F(grant)  Requests will be held until there is a grant or a power down req -> (req U (grant | power_down))  Deadlock free Ai. req[i] -> F(grant[i])  At some point after reset, the reset signal will stay low forever reset -> F(G(~reset))

14. LTL: Strong and Weak Statements  If the machine may exit/terminate, and an “until” is waiting, did it pass? Example: (a -> b U c)  Strong property: must finish  Weak property: considered true if evaluation may never complete Usually the default

15. Types of Properties  Safety: “Something bad won’t happen.” G(~ (grant & busy))  Liveness: “Something good will happen.” F(grant) Be careful: weak or strong?  Fairness: “Something happens infinitely often.” G(F(!busy)) Usually considered subset of liveness Often required as assumption on design inputs

16. Binary Decision Diagrams (BDDs)

17. BDD Example  (a & c) | (~a & b & ~c) a b cc 1 1 0 0 0 1 101 0 001

18. BDD Reduction & Ordering  Always specify an order for the variables  Reduction: merge identical nodes a b cc 0 1 0 0 0 1 101 0 011 a b c 1 0 0 0 1 1 0 10

19. Why are BDDs useful?  Canonical: unique for given var ordering Assuming they are reduced Two formulas equivalent iff same BDD!  Easy to define operations Complement Substitute constant (“Restrict”) Apply any boolean operator (&, |, etc)  Many cases proven efficient in practice But danger of exponential blowup

20. Complement A BDD  Replace f with ~f: just reverse terminals a b cc 1010 1 0 0 0 1 101 1010 0101 0101 1010

21. Substitute Constant in BDD  Just eliminate irrelevant subtrees, connect correct nodes  Example: c = 1 a b cc 1 1 0 0 0 1 101 0 001

22. Substitute Constant in BDD  Just eliminate irrelevant subtrees, connect correct nodes  Example: c = 1 a b 1 0 0 1 001

23. Substitute Constant in BDD  Just eliminate irrelevant subtrees, connect correct nodes  Example: c = 1 a 10 0 1 Don’t forget to reduce

24. Apply Operation to BDDs (AND, OR, etc)  Basic idea: recursively examine, with one var restricted to constant Each recursive call reduced #vars by 1 At terminal apply obvious function  APPLY(f1,f2,AND) = v1 0 1 APPLY(f1,f2,AND)| v1=0 APPLY(f1,f2,AND)| v1=1

25. APPLY example  Goal: BDD1(a,b) AND BDD2(a,b) a 10 0 1 b 10 0 a 10 10 BDD1 = a&b BDD2 = !a

26. APPLY example: Step 1 a 10 0 1 b 10 0 a 10 10 Use restrictions for a=0, a=1 a 10 APPLY| a=0 APPLY| a=1

27. APPLY example: Step 2 a 10 0 1 b 10 0 a 10 10 Use restrictions for a=0, a=1 a 10 0 AND 1 BDD1.b AND 0

28. APPLY example: Step 3 a 10 0 1 b 10 0 a 10 10 Compute results using constants if available a 10 00

29. APPLY example: Result 0 2 recursive calls per variable But always reduces size of problem So eventual constants guaranteed

30. BDDs: Exponential Blowup  (a&b) | (c&d) a b c 10 d

31. BDDs: Exponential Blowup  (a&b) | (c&d) a c 10 c bb d d

32. SAT Algorithms

33. What is SAT?  SAT= general problem: can boolean statement be satisfied? Known NP-complete But good heuristics known  FV Focus was on BDDs in 1990s Now seen as too restrictive Modern tools have BDD + SAT engines

34. SAT Example: DPLL Algorithms  Algorithms first proposed in 1960’s But renewed interest due to FV application  Start by converting formula to CNF form: product-of-sums (clauses) (a+b+c)(a+~d+e)(~b+~c)… Reminder: multiplication=AND, addition=OR Target: assignment satisfying every term If some clause is 0, assignment fails

35. Outline of DPLL algs (from Zhang/Malik paper, see ref slide)

36. Sub-functions  Deduction: find what must be true Example: (a+b)(~c+d) If c was assigned 1, then d must be 1 Can spend compute cycles to be more aggressive  Choose_free_variable: tricky part! Look for var that affects most clauses? Weight clauses strategically? Learn from conflicts/backtracks?

37. Other DPLL SAT Aspects  Capacity: How to store set of clauses? Direct: sparse matrix representation BDDs, tries, other options  Preprocessing First pass: gather high-level data hints  Randomization Random restart if seem to be dying?  Other approaches: SAT is still an active research area! www.satlive.org

38. References  http://www.jimloy.com/logic/logic.htm http://www.jimloy.com/logic/logic.htm  http://ocw.mit.edu/OcwWeb/Electrical-Engineering- and-Computer-Science/6-042JFall- 2005/LectureNotes/index.htm http://ocw.mit.edu/OcwWeb/Electrical-Engineering- and-Computer-Science/6-042JFall- 2005/LectureNotes/index.htm  http://en.wikipedia.org/wiki/Linear_temporal_logic http://en.wikipedia.org/wiki/Linear_temporal_logic  http://www.inf.unibz.it/~artale/FM/slide3.pdf http://www.inf.unibz.it/~artale/FM/slide3.pdf  http://www.cerc.utexas.edu/~gnolkha/verif/BDD.ppt http://www.cerc.utexas.edu/~gnolkha/verif/BDD.ppt  http://www.comp.nus.edu.sg/~abhik/CS4271/lectures/L ec11-BDD.pdf http://www.comp.nus.edu.sg/~abhik/CS4271/lectures/L ec11-BDD.pdf  http://www.satlive.org/ http://www.satlive.org/  http://www.princeton.edu/~chaff/publication/cade_cav_ 2002.pdf http://www.princeton.edu/~chaff/publication/cade_cav_ 2002.pdf