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  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!

Basic Boolean Logic

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

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

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)

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

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

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)

Linear Temporal Logic (LTL)

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

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)?

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))

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

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

Binary Decision Diagrams (BDDs)

BDD Example  (a & c) | (~a & b & ~c) a b cc

BDD Reduction & Ordering  Always specify an order for the variables  Reduction: merge identical nodes a b cc a b c

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

Complement A BDD  Replace f with ~f: just reverse terminals a b cc

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

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

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

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

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

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

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

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

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

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

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

SAT Algorithms

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

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

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

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?

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!

References   and-Computer-Science/6-042JFall- 2005/LectureNotes/index.htm and-Computer-Science/6-042JFall- 2005/LectureNotes/index.htm     ec11-BDD.pdf ec11-BDD.pdf   pdf pdf