4 th Nov, Oct 23 rd Happy Deepavali!

10/23 SAT & CSP

Average: 33 St Dev: 13.2 Max: 55.5 Min: Midterm Histogram Marks

Entailment by model checking We said entailment can be checked by truth table enumeration. Can we do the check without full enumeration? –SAT problem Given a set of propositions And a set of (CNF) clauses Find a model (an assignment of t/f values to propositions) that satisfies all clauses –k-SAT is a SAT problem where all clauses are length less than or equal to k »SAT is NP-complete; »1-SAT and 2-SAT are polynomial »k-SAT for k> 2 is NP-complete –If we have a procedure for solving SAT problems, we can use it to compute entailment The senstence S is entailed, if negation of S, when added to the KB, gives a SAT theory that is unsatisfiable (NO MODEL) –CO-NP-Complete –SAT is useful for modeling many other “assignment” problems We will see use of SAT for planning; it can also be used for Graph coloring, n- queens, Scheduling and Circuit verification etc (the last thing makes SAT VERY interesting for Electrical Engineering folks) Review

Compiling Planning into SAT Init: At-R-E-0 & At-A-E-0 & At-B-E-0 Goal: In-A-1 & In-B-1 Graph: “cond at k => one of the supporting actions at k” In-A-1 => Load-A-1 In-B-1 => Load-B-1 At-R-M-1 => Fly-R-1 At-R-E-1 => P-At-R-E-1 Load-A-1 => At-R-E-0 & At-A-E-0 “Actions at k => preconds at k-1” Load-B-1 => At-R-E-0 & At-B-E-0 P-At-R-E-1 => At-R-E-0h ~In-A-1 V ~ At-R-M-1 ~In-B-1 V ~At-R-M-1 “Mutexes” Goals: In(A),In(B) One way of finding a k-length plan is to grow a k-length planning graph (with mutexes) and looking for a valid subgraph of this graph. If it is not found, extend the graph and try again Review

Planning as SAT (another example) Init: J-0 Goal: P-2 & Q-2 Do the following for every cond; action and mutex at every level Graph: “cond at k => one of the supporting actions at k” P-2=> O1-2 V p-P-2 Q-2 => O2-2 V p-Q-2…. “Actions at k => preconds at k-1” O2-2 => J-1 p-P-2 => P-1 Mutexes ~P-2 V ~J-2 ~P-1 V ~Q-1 …. A solution assignment will be J-0; O2-1; Q-1; p-Q-2; O1-2; P-2; Q-2;

SAT as Search Search in the space of “partial assignments” (P=T,Q=F..) –Extend a partial assignment by picking a variable and generating the t/f children of the assignment Nodes: partial assignments Operators: assigning values to variables –We can assign variables in any order (commutative). So no need to consider more than one order –If a partial assignment is already inconsistent, you can kill the search –Depth first search is good enough depth of the search tree is finite (= number of propositions) all satisfying models are equally good. All assignments are at depth d (d = # propositions)

Improvements? Need hints on –Which prop to pick for assignment next? –Which value to try for it next? –Can we detect inevitable failures early on? Involves inference –Notice the “circularity” »Inference can be done by SAT which involves inference (which can be done by SAT which involves inference… –Can we avoid previously encountered failures? –Can we improve the mobility of the search?

DPLL—does lookahead by unit propagation DPLL is a procedure that basically implements this with 2 enhancements: –Unit propagation [Trying to detect inevitable failure early] Before expanding a partial assignment, run unit resolution to completion –Apply inference of type [P ; ~P V Q VJ ] => Q V J repeatedly –Whenever a unit clause is found, set the corresponding variable to True, and remove all clauses that have that variable –If an empty clause is detected, the node is “dead”--Backtrack –Pure-literal elimination If a proposition appears with the same polarity in all clauses it appears, set its value to that polarity, and remove all the clauses that it is part of. –Makes sense only if all satisfying assignments are equally desirable – E.g. be-god V study; be-god V practice

Davis-Putnam-Logeman-Loveland Procedure  detect failure

Improvements to DPLL Order in which propositions are selected –Select the most constraining prop first (the one that takes part in most clauses) –Select the prop which, when set, leads to most unit-propagation cascade [Costly, but works wonders—e.g SATZ) Learning from mistakes (“Nogood” learning) –If we are forced to backtrack over particular partial assignment, remember the reason so if that reason holds in another branch, we can cut search. –Problem: Need to carefully control the amount of nogoods you remember (storage and matching costs). Works great in RelSAT. Using random-restart policies to increase the “mobility” of the search Many many many more.. (still continuing) Read the overview paper on the quest for efficient boolean solvers