CS137: Today Electronic Design Automation SAT Davis-Putnam - PDF document

CS137: Today Electronic Design Automation • SAT • Davis-Putnam • Data Structures Day 10: October 19, 2005 • Optimizations Modern SAT Solvers – Watch2 – VSIDS ({z}Chaff, GRASP,miniSAT) – ?restarts • Learning 1 2 CALTECH CS137 Fall2005 -- DeHon CALTECH CS137 Fall2005 -- DeHon Problem CNF • SAT: Boolean Satisfiability • Conjunctive Normal Form • Given: logical formula f in CNF • Logical AND of a set of clauses • Find a set of variable assignments that • Clauses: logical OR of a set of literals makes f true • Literal: a variable or its complement • Or conclude no such assignment exists • E.g. (A+B+/C)*(/B+D)*(C+/A+/E) 3 4 CALTECH CS137 Fall2005 -- DeHon CALTECH CS137 Fall2005 -- DeHon CNF Search • Conjunctive Normal Form • Can be solved with pruning search • Logical AND of a set of clauses – Pick an unassigned variable – Branch on true/false • To be satisfied: – Compute implications – Every clause must be made true • (A+B+/C)*(/B+D)*(C+/A+/E) – If know D= false � B must be false 5 6 CALTECH CS137 Fall2005 -- DeHon CALTECH CS137 Fall2005 -- DeHon 1

Davis-Putnam Previously while (true) { • Also looked at PODEM if (!decide()) // no unassigned vars – Backtracking search on variable return(satisfiable); assignment while ( !bcp()) { // constraint propagation if (!resolveConflict()) // backtrack return(not satisfiable); } } 7 8 CALTECH CS137 Fall2005 -- DeHon CALTECH CS137 Fall2005 -- DeHon decide() Data Structures • Picks an unassigned variable • Variable “array” • Gives it a value • Clause “DB” • Push on decision stack • Each clause is a set of variables – Efficient structure for depth- • Decision “stack” first search tree 9 10 CALTECH CS137 Fall2005 -- DeHon CALTECH CS137 Fall2005 -- DeHon bcp() bcp • Q=new queue(); • Q.insert(top of decision stack); • What do we need to do on each variable assignment? • while (!Q.empty()) – V=Q.pop(); – Find implications – For each clause C in DB with V • Implication when all other literals in a clause are false • If C has one unassigned literal, rest false • Look through all clauses this assignment effects – Vnew=unassigned literal in C • See if any now have all false and one unassigned – val=value Vnew must take – Assign implied values – If (Vnew assigned to value other than val) » return ( false ); // conflict – Propagate that assignment – Q.add(Vnew=val); – Conflict if get implications for true and false • return( true ) 11 12 CALTECH CS137 Fall2005 -- DeHon CALTECH CS137 Fall2005 -- DeHon 2

Tracking Implications Variable array • Each implication made at • Each variable has a list pointing to all some tree level clauses in which it appears? – Associated with some entry on decision stack – Avoid need to look at every clause – Has associated decision stack height • On backtrack – Unassign implications above changed decision level 13 14 CALTECH CS137 Fall2005 -- DeHon CALTECH CS137 Fall2005 -- DeHon Track Variable Assignment Track Variable Assignment • Each clause has • Each clause has counter counter – Count number of unassigned literals – Count number of – Decrement when assign false literal unassigned literals – Decrement when – Mark clause as satisfied when assign true assign false literal literal (remove from clause database?) – Mark clause as satisfied when assign true literal (remove from clause database?) 15 16 CALTECH CS137 Fall2005 -- DeHon CALTECH CS137 Fall2005 -- DeHon Track Variable Assignment resolveConflict() • Each clause has counter • What does resolveConflict need to do? – Count number of unassigned literals – Decrement when assign false literal – Look at most recent decision – Mark clause as satisfied when assign true – If can go other way, switch value literal • (clear implications to this depth) – Watch for counter decrement 2 � 1 – Else pop and recurse on previous decision • That’s when a literal is implied. – If pop top decision, • unsatisfiable 17 18 CALTECH CS137 Fall2005 -- DeHon CALTECH CS137 Fall2005 -- DeHon 3

How will this perform? • 10,000’s of variables Chaff Optimizations • 100,000’s of clauses (millions) • Every assignment walks to the clause database • Cache performance? 19 20 CALTECH CS137 Fall2005 -- DeHon CALTECH CS137 Fall2005 -- DeHon Challenge 1 Avoiding Clause Visits • Idea: watch only 2 variables in each • Currently, visit every clause on each clause assignment • Only care about final set of next to last – Clause with K variables variable – Visited K-1 times • If set other k-2, won’t force an implication – K-2 of which just to discover it’s not the last • When set one of these (and everything • Can we avoid visiting every clause on else set) every assignment? – Every clause in which a variable appears? – Then we have an implication 21 22 CALTECH CS137 Fall2005 -- DeHon CALTECH CS137 Fall2005 -- DeHon Watch 2 Data Structure Avoiding Clause Visits • Idea: watch only 2 variables in each clause • Only care about final set of next to last variable • What if we set of these two “watched” variables? – If not last, change the watch to one of the unset variables 23 24 CALTECH CS137 Fall2005 -- DeHon CALTECH CS137 Fall2005 -- DeHon 4

Watch 2 Note • If watched literal becomes false • Watch pair is arbitrary – Check if all non-watched are set • Unassigning a variable (during • if so, set implication on other watched backtrack) • else, update watch literal – Does not require reset of watch set – Constant time to “unset” a variable 25 26 CALTECH CS137 Fall2005 -- DeHon CALTECH CS137 Fall2005 -- DeHon Challenge 2: Variable Ordering Variable Ordering • How do we decide() which variable to • Old Ideas: use next? – Random – Want to pick one that facilitates lots of – (DLIS) Dynamic largest individual sum pruning • Used most frequently in unresolved clauses • BAD? – Must re-sort with every variable assignment? – …none clearly superior • DLIS competitive • Rand good on CAD benchmarks? 27 28 CALTECH CS137 Fall2005 -- DeHon CALTECH CS137 Fall2005 -- DeHon New: VSIDS New: VSIDS • Variable State Independent Decaying • Variable State Independent Decaying Sum Sum – Each literal has a counter – Each literal has a counter – When clause added to DB, increment – When clause added to DB, increment counter for each literal counter for each literal • Remove clauses when satisfied? – Select unassigned literal with highest count • Reinsert on backtrack – Select unassigned literal with highest count – Periodically, all counters are divided by a constant – Periodically, all counters are divided by a constant 29 30 CALTECH CS137 Fall2005 -- DeHon CALTECH CS137 Fall2005 -- DeHon 5

New: VSIDS VSIDS • Variable State Independent Decaying • Goal: satisfy recent conflict clauses Sum • Decaying sum weights things being – Each literal has a counter added – When clause added to DB, increment counter for – Clauses not conflicting for a while, have each literal values reduced – Select unassigned literal with highest count • (? Avoid walking through them by increasing • Don’t need to re-sort each selection weight on new stuff rather than decreasing all • Only re-sort on backtrack old?) • Maybe priority queue insert? • Impact: order of magnitude speedup – Periodically, all counters are divided by a constant 31 32 CALTECH CS137 Fall2005 -- DeHon CALTECH CS137 Fall2005 -- DeHon Restarts Overall • Periodically restart • Two orders of magnitude benefit on – Clearing the state of all variables unsatisfiable instances • i.e. clear decision stack • One order of magnitude on satisfiable – Leave clauses in clause database instances • ? Keep ordering based on recent costs • ? Re-insert clauses must reinsert on restart? – State of clause database drives variable ordering • Benefit: new variable ordering based on lessons of previous search 33 34 CALTECH CS137 Fall2005 -- DeHon CALTECH CS137 Fall2005 -- DeHon Learning • When encounter a conflict Learning – Determine variable assignment contributing to conflict – Add new clause to database • New clause allows pruning 35 36 CALTECH CS137 Fall2005 -- DeHon CALTECH CS137 Fall2005 -- DeHon 6

Davis-Putnam w/ Learning Implication Graph while (true) { • As perform bcp propagation if (!decide()) // no unassigned vars – When set variable, insert back link to return(satisfiable); previous variable set forcing this variable while ( !bcp()) { // constraint propagation set analyzeConflicts(); // learning – Graph captures what this implication if (!resolveConflict()) // backtrack depends upon return(not satisfiable); • When encounter a conflict } – Identify what variable values caused } 37 38 CALTECH CS137 Fall2005 -- DeHon CALTECH CS137 Fall2005 -- DeHon Example Conflict Resolution • x1 & /x9 & /x10 & /x11 lead to conflict • /(x1 & /x9 & /x10 & /x11) • /x1+x9+x10+x11 � new clause for DB Marques-Silva/Sakallah TRCOMP v48n5p506 1999 39 40 CALTECH CS137 Fall2005 -- DeHon CALTECH CS137 Fall2005 -- DeHon New Clause More Implications •New clause does not include x12, x13 •May encounter this case again • x4 & /x10 & /x11 lead to conflict • /x4+x10+x11 � new clause for DB • Also (/x1+x9+x4) 41 42 CALTECH CS137 Fall2005 -- DeHon CALTECH CS137 Fall2005 -- DeHon 7