Daniel Kroening and Ofer Strichman 1 Decision Procedures An Algorithmic Point of View BDDs

Decision Procedures An algorithmic point of view2 Part I Reminders -  What is Logic  Proofs by deduction  Proofs by enumeration  Decidability, Soundness and Completeness  some notes on Propositional Logic Deciding Propositional Logic  SAT tools  BDDs      

Decision Procedures An algorithmic point of view3 Next: Binary Decision Diagrams SAT looked for a satisfying solution to CNF We will now examine a graph-based data structure called Binary Decision Diagrams. It has several advantages and disadvantages comparing to SAT Developed by Bryant [1986]. Next few slides are from the source …

Decision Procedures An algorithmic point of view4 Alternate Approach Generate complete representation of function Canonicity: functions are equal iff representations are identical (A Æ C) Ç (C Æ B) (A Ç B) Æ (C) B 01C AB 01C A

Decision Procedures An algorithmic point of view5 Decision Structures Truth TableDecision Tree  Vertex represents decision  Follow green (dashed) line for value 0  Follow red (solid) line for value 1  Function value determined by leaf value.

Decision Procedures An algorithmic point of view6 Variable Ordering  Assign arbitrary total ordering to variables e.g., x 1 < x 2 < x 3  Variables must appear in ascending order along all paths OKNot OK x 3 x 2 x 1 x 1 x3x3

Decision Procedures An algorithmic point of view7 Reduction Rule #1 Merge equivalent leaves aa a

Decision Procedures An algorithmic point of view8 Reduction Rule #2 y x z x Merge isomorphic nodes y x z x y x z x

Decision Procedures An algorithmic point of view9 Reduction Rule #3 Eliminate Redundant Tests y x y

Decision Procedures An algorithmic point of view10 Example OBDD Initial GraphReduced Graph Canonical representation of Boolean functions o For a given variable ordering  Two functions are equivalent iff graphs are isomorphic Can be tested in linear time  Desirable property: simplest form is canonical. (x 1 Ç x 2 ) Æ x 3

Decision Procedures An algorithmic point of view11 Satisfiability etc. Constants Unique unsatisfiable function Unique tautology Conclusion: given a BDD it takes constant time to check:  Validity  Contradiction  Satisfiability Is this a free lunch ? …

Decision Procedures An algorithmic point of view12 Effect of Variable Ordering Good OrderingBad Ordering Linear GrowthExponential Growth

Decision Procedures An algorithmic point of view13 Selecting Good Variable Ordering Intractable Problem  Even when problem represented as OBDD i.e., to find optimum improvement to current ordering Application-Based Heuristics  Exploit characteristics of application  e.g., Ordering for functions of combinational circuit Traverse circuit graph depth-first from outputs to inputs

Decision Procedures An algorithmic point of view14 Building BDDs ‘from below’ Starting from a binary decision tree is too hard for formulas with many variables. Goal: construct the BDD ‘from below’.

Decision Procedures An algorithmic point of view15 Building BDDs ‘from below’ For this we will need a function called APPLY:  Given the BDDs for f 1 and f 2,  and a binary connective F 2 { Æ, Ç, !, $...} (any one of the 16 binary connectives),  Construct the BDD for f 1 F f 2,

Decision Procedures An algorithmic point of view16 Building BDDs ‘from below’ Def: a restriction of a function f to x = d, denoted f | x = d where x 2 vars( f ), d 2 {0,1}, is equal to f after assigning x = d. Given the BDD of f, deriving the BDD of f | x =0 is simple: f : (x 1 Ç x 2 ) Æ x 3 01 x 3 x 1 01 x 3 f | x2=1

Decision Procedures An algorithmic point of view17 Now, APPLY (1/3) Let v 1, v 2 denote that root nodes of f 1, f 2, respectively, with var( v 1 ) = x 1 and var( v 2 )= x If v 1 and v 2 are leafs, f 1 F f 2 is a leaf node with value val(v 1 ) F val(v 2 ) Ç = Æ =

Decision Procedures An algorithmic point of view18 Now, APPLY (2/3) 2. If x 1 = x 2 = x, apply Shanon expansion: f 1 F f 2 = ( : x Æ f 1 | x=0 F f 2 | x=0 Ç x Æ f 1 | x=1 F f 2 | x=1 ) x BDD for f 1 | x=0 BDD for f 1 | x=1 x BDD for f 2 | x=0 BDD for f 2 | x=1 Æ = x BDD for f 1 | x=1 Æ f 2 | x=1 BDD for f 1 | x=0 Æ f 2 | x=0

Decision Procedures An algorithmic point of view19 Now, APPLY (3/3) 3. else, suppose x 1 < x 2 in the variable order. f 1 F f 2 = ( : x 1 Æ f 1 | x=0 F f 2 Ç x 1 Æ f 1 | x=1 F f 2 ) x1x1 BDD for f 1 | x1=0 BDD for f 1 | x1=1 x2x2 BDD for f 2 | x2=0 BDD for f 2 | x2=1 Æ = x1x1 BDD for f 1 | x=1 Æ f 2 BDD for f 1 | x=0 Æ f 2

Decision Procedures An algorithmic point of view20 BDDs from below: example. x 2 01 x 1 x 2 f 1 : x 1 $ x 2 x 2 10 Ç f 2 : : x 2 = x 1 BDD for f 1 | x1=0 Ç f 2 BDD for f 1 | x1=1 Ç f 2 BDD for f 1 | x1=0 Ç f 2 = x2x2 0 1 Ç x 2 10 = x Ç 0 = 0 1 Ç 1 = 1

Decision Procedures An algorithmic point of view21 BDDs from below: example. x 2 01 x 1 x 2 f 1 : x 1 $ x 2 x 2 10 Ç f 2 : : x 2 = x 1 BDD for f 1 | x=0 Ç f 2 BDD for f 1 | x=1 Ç f 2 = x 1 x 2 10 x 2 1 = x 1 x 2 10 x 2 f 1 Ç f 2 = x 1 Ç ( : x 1 Æ : x 2 )

Decision Procedures An algorithmic point of view22 Comparing SAT to BDDs BDD is a canonical data structure that represents the semantic of a function, i.e. all its solutions  Some applications (e.g. symbolic model checking) need canonicity to detect when two sets are equivalent.  Can require exponential space & time (highly sensitive to variable ordering) SAT searches through CNF for a single solution  CNF is not canonical.  Poly-space algorithms exists. Time can be exponential.