Dipartimento di Elettronica Informazione e Bioingegneria. Cognitive Robotics. SATplan. Act1. Pre1. Fact. G. Gini Act2

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1 Dipartimento di Elettronica Informazione e Bioingegneria Cognitive Robotics SATplan Pre Act1 Act2 Fact

2 why SAT (satisfability)? 2 Classical planning has been observed as a form of logical deduction. Planning as satisfiability offers a new solution to planning problems = formulates a planning problem as a propositional satisfiability problem. The Mathematical Analysis of Logic, Cambridge, 1847 The Calculus of Logic, Cambridge and Dublin Mathematical journal, vol. 3, 1848 George Boole

3 Propositional logic 3 Syntax The proposition symbols S 1, S 2 etc are sentences, as well as S 1,, S 1 S 2 S 1 S 2 S 1 S 2 Semantics Each model specifies true/false for each proposition symbol models can be enumerated automatically Rules for evaluating truth with respect to a model m are the truth tables

4 Valid, satisfiable, not satisfiable 4 Validity and satisfiability are connected to inference : 1. A sentence is valid if it is true in all models: KB αiff (KB α) is valid 2. A sentence is satisfiable if it is true in some model 3. A sentence is unsatisfiable if it is true in no models KB αiff (KB α) is unsatisfiable

5 Propositional satisfiability (SAT) 5 Given a propositional formula, does it have a model (satisfying assignment)? SAT is decidable Usually formulae in clausal normal form (CNF) CNF Formula is a conjunction of clauses C1 C2 Each clause is a disjunction of literals L1 v L2 v L3, Each literal is variable or its negation P, -P, Q, -Q,

6 Converting to CNF 6 Eliminate implies P implies Q => -P v Q Push negation down -(P Q) => -P v -Q -(P v Q) => -P -Q Clausify using De Morgan s laws E.g. P v (Q R) => (P v Q) (P v R)

7 How do we solve SAT? 7 Systematic methods Truth tables Davis Putnam procedure Local search methods GSAT WalkSAT Tabu search, SA, Exotic methods DNA, quantum computing,

8 Truth tables 8 Enumerate all possible truth assignments P Q R C1=P v Q C2=-Q v -R C1&C2 T T T T F F T T F T T T T F T T T T T F F

9 Davis Putnam procedure 9 Davis, Logemann and Loveland Consider (X or Y) & (-X or Z) & (-Y or Z) & try X=true remove clauses which are satisfied, obtain (-X or Z) & (-Y or Z) & Simplify clauses containing X, obtain ( Z) & (-Y or Z) & can now deduce that Z must be true at any point, may have to backtrack and try X=false instead

10 Procedure DPLL 10 DPLL/Davis-Putnam-Logemann-Loveland Backtraking depth-first search Unit propagation to reduce search function DPLL(C) if C is a consistent set of literals then return true; if C contains an empty clause then return false; for every unit clause l in C, C=unit-propagate(l, C); *l T for every literal l pure in C, C=pure-literal-assign(l, C); *-l F l := choose-literal(c); return DPLL(CΛl) OR DPLL(CΛnot(l)); Properties: Complete Space complexity: O(n) Time complexity: O(1.618^n)

11 Planning as SAT 11 Traditional view Planning = deducibility (theorem proving) New view Planning = propositional satisfiability of a sentence = finding a model of a set of axioms? Advantage: more efficient algorithms for satisfiability than for planning Problem: formalize planning problem in propositional logic

12 Planning as SAT 12 test the satisfiability of the sentence: initial state all possible action descriptions goal Sentence contains propositions for every action occurrence A model will assign T to the actions that are part of the correct plan and F to the others An assignment that corresponds to an incorrect plan will not be a model because of inconsistency with the assertion that the goal is T. If the planning is unsolvable the sentence will be unsatisfiable.

13 Planning as Propositional Formulas 13 Encoding a planning problem to a propositional formula is based on: Restrict the planning problem to a problem of finding a plan of known length n Open world assumption: unknown propoitions are left unspecified Transform the bounded planning problem into a satisfiability problem. Each state and action is mapped to propositions that describe states and actions at each step, from step 0 (initial state) to step n (goal state) Since planning as satisfiability can only deal with bounded planning problems, the algorithm is iteratively run for different tentative lengths until a plan is found

14 Architecture of SAT planner 14 Compiler take a planning problem as input, guess a plan length, and generate a propositional logic formula, which if satisfied, implies the existence of a solution plan Symbol table record the correspondence between propositional variables and the planning instance Simplifier use fast techniques such as unit clause propagation and pure literal elimination to shrink the CNF formula Solver use systematic or stochastic methods to find a satisfying assignment. If the formula is unsatisfiable, then the compiler generates a new encoding reflecting a longer plan length Decoder translate the result of solver into a solution plan.

15 Architecture of SAT planner 15 Increment time bound if Unsatisfiable Init State Goal Actions Compiler CNF Simplifier CNF Solver Satisfying Assignment Decoder Plan Symbol Table CNF = conjunctive normal form

16 Encoding the problem 16 Reduction of first order logic to propositions KB in first order logic : x King(x) Greedy(x) Evil(x) King(John), King(Richard), Greedy(John), Greeedy (Richard) Instantiating in all possible ways, we have a propositionalized KB: King(John), King(Richard), Greedy(John), Greeedy (Richard), Evil(John), Evil(Richard) Efficiency of SAT compilers depends on the number of clauses (time complexity can be exponential in the size of the formula)

17 Encoding actions Regular = each ground action is represented by a different logical variable Drive(truck3,Seattle,Renton,t) p1 2. Simple action splitting Drive(truck3,Seattle,Renton,t) Drive1(truck3,t)&Drive2(Seattle,t)&Drive3(Renton,t) p1 p2 p3 Each action described by a conjunction of variables

18 the dinner example 18 Initial Conditions: (and (garbage) (cleanhands) (quiet)) Goal: (and (dinner) (present) (not (garbage)) Actions: Cook :p (cleanhands) :e (dinner) Wrap :p (quiet) :e (present) Carry :p :e (and (not (garbage)) (not (cleanhands)) Dolly :p :e (and (not (garbage)) (not (quiet)))

19 Encoding dinner planning 19 INIT: the initial state is completely specified at time zero, including all properties false garb0 ^ cleanh0 ^ quiet0 ^ dinner0 ^ present0 GOAL: all desired goal properties are asserted to be true at time 2n (n is the length of plan) garb2 ^ dinner2 ^ present2 A P,E: Actions imply their preconditions and effects (preconditions at t-1, effects at t) ( carry1 v dinner2) ^ ( carry1 v cleanh0)

20 axioms 20 Precondition axioms: action occurrence requires the preconditions to be satisfied Action execution axioms: prevent simultaneous actions use only if they interfere with each other Successor-state axioms: all the situations in which a proposition is true see situation calculus

21 Frame Axioms 21 Frame axioms constrain unaffected fluents 1. Classical frame State which fluents are left unchanged by an action At(truck1,Kent,t1)&Drive(truck3,Seattle,Renton,t)=> At(truck1,Kent,t+1) combine with at-least-one axiom to ensure that some actions occurs 2. Explanatory frame Enumerate the set of actions that could have occurred in order to account for a state change combine with exclusion axiom to enforce the constraints in the resulting plan

22 SATPLAN algorithm 22 a planning problem upper limit to the plan length function SATPLAN(problem, T max )- return solution or failure for T= 0 to T max do cnf,mapping TRANSLATE-TO_SAT(problem,T) assignment SAT-SOLVER(cnf) if assignment is not null then return EXTRACT-SOLUTION(assignment, mapping) return failure Example for Tmax = set-level heuristics from graphplan (all literals of the goal are in the graph and are not mutex

23 Complexity of SAT encoding 23 Number of propositions from the original planning problem? Given act = number of operators (actions) o = number of objects in the domain P = maximum number of arguments in any action T = steps The number of propositional symbols generated is T X act X o P often many millions. To reduce, symbol splitting

24 Compare Graphplan with SAT 24 Both convert parameterized action schemata into a finite propositional structure representing the space of possible plans up to a given length planning graph a CNF formula Both use local consistency methods before resorting to exhaustive search mutex propagation propositional simplification Both iteratively expand their propositional structure until they find a solution planning graph is extends when no solution is found propositional logic formula is recreated for a longer plan length Planning graph can be automatically converted into CNF notation for solution with SAT solvers

25 Translation of Plan Graph to SAT 25 Pre1 Act1 Fact Pre2 Act2 Fact Act1 Act2 Act1 Pre1 Pre2 Act1 Act2

26 Graphplan-based Encodings 26 Example: rocket domain load and move operators Translation begins at goal-layer of the graph, and works backwards. 1. Initial state holds at layer 1 and goals hold at the highest layer. 2. Precondition axiom: operators imply their preconditions Load(A,R,L,2) (At(A,L,1) At(R,L,1) 3. Each fact at level i implies the disjunction of all the operators at level i-1 that have it as an add-effect, e.g. In(A,R,3) (Load(A,R,L,2) v Load(A,R,P,2) v Maintain(In(A,R),2)) 4. Conflicting actions are mutually exclusive, e.g. Load(A,R,L,2) v Move(R,L,P,2)

27 SAT plan - conclusion 27 Expanded with extensions to planning language good results in competitions Real application: Interleaving Planning and execution monitoring see Partially SAT based reactive control system for the NASA Deep SpaceOne autonomous aircraft