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Logic Stuff & FV Basics

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

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Logic Stuff & FV Basics

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  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 0 1 b 1 0 0 c c 0 1 0 1 1 0 0 1

  18. BDD Reduction & Ordering • Always specify an order for the variables • Reduction: merge identical nodes a a 0 1 0 1 b b 1 1 0 0 c c 0 0 c 0 1 0 1 0 1 1 0 1 0 1 0

  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 0 1 b 1 0 10 c c 0 1 0 1 10 01 01 10

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

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

  23. Substitute Constant in BDD • Just eliminate irrelevant subtrees, connect correct nodes • Example: c = 1 a 0 1 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 1 0 APPLY(f1,f2,AND)| v1=0 APPLY(f1,f2,AND)| v1=1

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

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

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

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

  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 d 0 1

  31. BDDs: Exponential Blowup • (a&b) | (c&d) a c c b b d d 0 1

  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://ocw.mit.edu/OcwWeb/Electrical-Engineering-and-Computer-Science/6-042JFall-2005/LectureNotes/index.htm • http://en.wikipedia.org/wiki/Linear_temporal_logic • http://www.inf.unibz.it/~artale/FM/slide3.pdf • http://www.cerc.utexas.edu/~gnolkha/verif/BDD.ppt • http://www.comp.nus.edu.sg/~abhik/CS4271/lectures/Lec11-BDD.pdf • http://www.satlive.org/ • http://www.princeton.edu/~chaff/publication/cade_cav_2002.pdf

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