Application: Planning as Satisfiability (SATPlan) (Pre Lecture)

Application: Planning as Satisfiability (SATPlan) (Pre Lecture) Dr. Neil T. Dantam CSCI-561, Colorado School of Mines Fall 2018 Dantam (Mines CSCI-561) Application: Planning as Satisfiability (SATPlan) (Pre Lecture) Fall 2018 1 / 44

Outline Introduction to Planning Planning Domains First Order Logic Planning Domain Description Language Planning Algorithms Naïve Search Heuristic Search Constraint-Based Planning Dantam (Mines CSCI-561) Application: Planning as Satisfiability (SATPlan) (Pre Lecture) Fall 2018 2 / 44

Why? Foreshadows/applies the Cook-Levin Theorem Directly connects SAT with transition systems (languages and automata) Dantam (Mines CSCI-561) Application: Planning as Satisfiability (SATPlan) (Pre Lecture) Fall 2018 3 / 44

Introduction to Planning Outline Introduction to Planning Planning Domains First Order Logic Planning Domain Description Language Planning Algorithms Naïve Search Heuristic Search Constraint-Based Planning Dantam (Mines CSCI-561) Application: Planning as Satisfiability (SATPlan) (Pre Lecture) Fall 2018 4 / 44

Introduction to Planning A Planning Problem Illustration Start Goal Dantam (Mines CSCI-561) Application: Planning as Satisfiability (SATPlan) (Pre Lecture) Fall 2018 5 / 44

Introduction to Planning Planning Problem Definition State Space: Q = f 0 f 1... f m, for each fluent (state variable) f i Actions: U = {a 0,..., a n } Transitions: δ : Q U Q, where for δ(q 0, a) = q 1, Start: q 0 Q is the initial state Goal: G Q is the set of goal states Dantam (Mines CSCI-561) Application: Planning as Satisfiability (SATPlan) (Pre Lecture) Fall 2018 6 / 44

Introduction to Planning Planning vs. DFA Planning Domain DFA State Space: Q = f 0 f 1... f m Actions: U = {a 0,..., a n } Transitions: δ : Q U Q Start: q 0 Q is the initial state Goal: G Q is the set of goal states Q is a finite set call the states Σ is a finite set call the alphabet δ : Q Σ Q is the transition function q 0 Q is the start state F Q is the set of accept states What s different? Dantam (Mines CSCI-561) Application: Planning as Satisfiability (SATPlan) (Pre Lecture) Fall 2018 7 / 44

Planning Domains Outline Introduction to Planning Planning Domains First Order Logic Planning Domain Description Language Planning Algorithms Naïve Search Heuristic Search Constraint-Based Planning Dantam (Mines CSCI-561) Application: Planning as Satisfiability (SATPlan) (Pre Lecture) Fall 2018 8 / 44

Planning Domains First Order Logic First Order Logic aka Predicate Calculus Propositional Logic plus... Objects: instead of just Boolean values, Finite set of values Enumerated type Predicates: Function from objects to Booleans P : O... O B Functions: Function from objects to objects F : O... O O Quantifiers: express properties on collections Universal: all items have the property Existential: at least one item has the property Dantam (Mines CSCI-561) Application: Planning as Satisfiability (SATPlan) (Pre Lecture) Fall 2018 9 / 44

Planning Domains First Order Logic Logical Predicate Boolean-valued function N-ary arguments f : O... O B Examples: English Logic x is happy happy (x) The suitcase contains a bomb. contains (suitcase, bomb) x is less than y. less (x, y) Formal way of stating facts. Dantam (Mines CSCI-561) Application: Planning as Satisfiability (SATPlan) (Pre Lecture) Fall 2018 10 / 44

Planning Domains First Order Logic Exercise: Logical Predicates Material Properties Objects: {wood, glass, steel} Predicates: Statements: transparent (?x) flammable (?x) denser (?x,?y) transparent (wood) = transparent (glass) = transparent (steel) = flammable (wood) = flammable (glass) = flammable (steel) = denser (wood, glass) = denser (glass, wood) = denser (glass, steel) = Dantam (Mines CSCI-561) Application: Planning as Satisfiability (SATPlan) (Pre Lecture) Fall 2018 11 / 44

Planning Domains First Order Logic Grammar for First Order Logic (excluding quantifiers) Sentence AtomicSentence Sentence Connective Sentence [(] Sentence [)] [ ] Sentence AtomicSentence Predicate [(] TermList [)] Connective [ ] [ ] [ = ] [ ] TermList Term Term [,] TermList Term Constant Variable Predicate [A] [B]... Constant [a] [b]... Variable [?x] [?y]... Dantam (Mines CSCI-561) Application: Planning as Satisfiability (SATPlan) (Pre Lecture) Fall 2018 12 / 44

Planning Domains First Order Logic First-Order Logic Description Constants: A, B, C C Predicates: on (?x,?y) clear (?x) ontable (?x) B A State: clear (B) clear (C) ontable (B) ontable (A) on (C, A) Dantam (Mines CSCI-561) Application: Planning as Satisfiability (SATPlan) (Pre Lecture) Fall 2018 13 / 44

Planning Domains First Order Logic First Order vs. Propositional Logic Every sentence in first-order logic can be converted into an equivalent sentence in propositional logic (modulo functions) Dantam (Mines CSCI-561) Application: Planning as Satisfiability (SATPlan) (Pre Lecture) Fall 2018 14 / 44

Planning Domains First Order Logic Example: Reduction to Propositional Logic (Grounding) Objects: {glass, steel} Predicates: Propositions: transparent (?x) denser (?x,?y) FOL predicate proposition {}}{{}}{ transparent (glass) transparent-glass transparent (steel) transparent-steel denser (glass, steel) denser-glass-steel denser (glass, glass) denser-glass-glass denser (steel, glass) denser-steel-glass denser (steel, steel) denser-steel-steel Dantam (Mines CSCI-561) Application: Planning as Satisfiability (SATPlan) (Pre Lecture) Fall 2018 15 / 44

Planning Domains First Order Logic Exercise: Grounding Constants: A, B, C Predicate: clear (?x), ontable (?x), and on (?x,?y) Propositions: Dantam (Mines CSCI-561) Application: Planning as Satisfiability (SATPlan) (Pre Lecture) Fall 2018 16 / 44

Planning Domains Planning Domain Description Language A Planning Problem Start B Start C A Goal A B C Goal ontable (b) on (c, a) ontable (a) handempty () ontable (c) on (b, c) on (a, b) Dantam (Mines CSCI-561) Application: Planning as Satisfiability (SATPlan) (Pre Lecture) Fall 2018 17 / 44

Planning Domains Planning Domain Description Language Example: Actions / Operators Name: pickup-up Parameters:?x Precondition: clear (?x) ontable (?x) handempty () Effect: ontable (?x) clear (?x) handempty () holding (?x) B B C A pick-up (B) C A ontable (B) ontable (A) on (C, A) clear (B) clear (C) handempty () ontable (B) handempty () clear (B) holding (B) Dantam (Mines CSCI-561) Application: Planning as Satisfiability (SATPlan) (Pre Lecture) Fall 2018 18 / 44

Planning Domains Planning Domain Description Language Exercise: Actions / Operators Name: unstack Parameters:?x,?y Precondition: on (?x,?y) clear (?x) handempty () Effect: holding (?x) clear (?y) clear (?x) handempty () on (?x,?y) C B C A unstack (C, A) B A ontable (B) ontable (A) on (C, A) clear (B) clear (C) handempty () Dantam (Mines CSCI-561) Application: Planning as Satisfiability (SATPlan) (Pre Lecture) Fall 2018 19 / 44

Planning Domains Planning Domain Description Language Task Language pick-up (B) B C A...... B B C A put-down (B) unstack (C, A) pick-up (B) A C stack (C, A) B C A put-down (C) pick-up (C) B A C put-down (B)... Dantam (Mines CSCI-561) Application: Planning as Satisfiability (SATPlan) (Pre Lecture) Fall 2018 20 / 44

Planning Domains Planning Domain Description Language Planning Domain Definition Language (PDDL) PDDL: S-Expression based format for planning domains Operators File: Defines the predicates and actions Facts File: Defines the objects (constants), start state, and goal Dantam (Mines CSCI-561) Application: Planning as Satisfiability (SATPlan) (Pre Lecture) Fall 2018 21 / 44

Planning Domains Planning Domain Description Language Planning Domain Definition Language Operators ( d e f i n e ( domain b l o c k s ) ( : p r e d i c a t e s ( on? x? y ) ( o n t a b l e? x ) ( c l e a r? x ) ( handempty ) ( h o l d i n g? x ) ) ( : a c t i o n pick up : parameters (? x ) : p r e c o n d i t i o n ( and ( c l e a r? x ) ( o n t a b l e? x ) ( handempty ) ) : e f f e c t ( and ( not ( o n t a b l e? x ) ) ( not ( c l e a r? x ) ) ( not ( handempty ) ) ( h o l d i n g? x ) ) ) ( : a c t i o n put down : parameters (? x ) : p r e c o n d i t i o n ( h o l d i n g? x ) : e f f e c t ( and ( not ( h o l d i n g? x ) ) ( c l e a r? x ) ( handempty ) ( o n t a b l e? x ) ) ) ( : a c t i o n s t a c k : parameters (? x? y ) : p r e c o n d i t i o n ( and ( h o l d i n g? x ) ( c l e a r? y ) ) : e f f e c t ( and ( not ( h o l d i n g? x ) ) ( not ( c l e a r? y ) ) ( c l e a r? x ) ( handempty ) ( on? x? y ) ) ) ( : a c t i o n u n s t a c k : parameters (? x? y ) : p r e c o n d i t i o n ( and ( on? x? y ) ( c l e a r? x ) ( handempty ) ) : e f f e c t ( and ( h o l d i n g? x ) ( c l e a r? y ) ( not ( c l e a r? x ) ) ( not ( handempty ) ) ( not ( on? x? y ) ) ) ) ) Dantam (Mines CSCI-561) Application: Planning as Satisfiability (SATPlan) (Pre Lecture) Fall 2018 22 / 44

Planning Domains Planning Domain Description Language Planning Domain Definition Language Facts Start ( d e f i n e ( problem sussman anomaly ) ( : domain b l o c k s ) ( : o b j e c t s a b c ) ( : i n i t ( on c a ) ( o n t a b l e a ) ( o n t a b l e b ) ( c l e a r c ) ( c l e a r b ) ( handempty ) ) ( : g o a l ( and ( on b c ) ( on a b ) ) ) ) C B A Goal A B C Dantam (Mines CSCI-561) Application: Planning as Satisfiability (SATPlan) (Pre Lecture) Fall 2018 23 / 44

Planning Algorithms Outline Introduction to Planning Planning Domains First Order Logic Planning Domain Description Language Planning Algorithms Naïve Search Heuristic Search Constraint-Based Planning Dantam (Mines CSCI-561) Application: Planning as Satisfiability (SATPlan) (Pre Lecture) Fall 2018 24 / 44

Planning Algorithms Naïve Search Explicit Automaton Construction 54A (MOVE B C G) (MOVE B G C) 64E start (MOVE C A B) (MOVE C B A) (MOVE C G B) (MOVE C A G) (MOVE C B G) (MOVE C G A) 124F (MOVE A G C) A4D 122B (MOVE A C G) (MOVE A G C) (MOVE B G A) (MOVE B A G) 10AA (MOVE A B C) A29 (MOVE A C G) (MOVE A C B) (MOVE A B G) 121D (MOVE C G A) (MOVE A G B) (MOVE C A G) 61C C B A Start A B C Goal (MOVE B C G) (MOVE B G C) (MOVE B A G) (MOVE A G B) (MOVE B C A) 114B (MOVE A B G) 1119 (MOVE B G A) 10CE (MOVE B A C) (MOVE C B G) (MOVE C G B) 8CC Dantam (Mines CSCI-561) Application: Planning as Satisfiability (SATPlan) (Pre Lecture) Fall 2018 25 / 44

Planning Algorithms Naïve Search Breadth-First Search q 1 q 0 q 1,1 q 1,2... q 1,n q 2... q n Is it complete? (always solves/terminates) Is it optimal? (finds best solution) Issues with BFS? Search Queue: q 2,1, q 2,2,..., q 2,m2, q 3,1, q 3,2,..., q 3,m3,..., q n,1, q n,2,..., q n,mn q 1,1,1, q 1,1,2,... Dantam (Mines CSCI-561) Application: Planning as Satisfiability (SATPlan) (Pre Lecture) Fall 2018 26 / 44

Planning Algorithms Heuristic Search A* Search q 0 q 1 q 2... q n Low Cost High Cost Priority Queue q i q j... q l Need: heuristic cost-to-go Is it optimal? Admissible heuristic: Never over-estimate cost A* vs. Djikstra s Algorithm: Djikstra: Exact cost-to-state A*: Exact cost-to-state plus heuristic cost-to-go Dantam (Mines CSCI-561) Application: Planning as Satisfiability (SATPlan) (Pre Lecture) Fall 2018 27 / 44

Planning Algorithms Heuristic Search Greedy Heuristic Search Give up on optimality Greedy Search: Expand best node according to (non-admissible) heuristic Pro: easier / faster non-admissable heuristic Con: No longer optimal Fast Forward (FF): Hoffmann, Jörg, and Bernhard Nebel. The FF planning system: Fast plan generation through heuristic search. Journal of Artificial Intelligence Research (2001). Fast Downward (FD): M. Helmert, The fast downward planning system. Journal of Artificial Intelligence Research (2006). Dantam (Mines CSCI-561) Application: Planning as Satisfiability (SATPlan) (Pre Lecture) Fall 2018 28 / 44

Planning Algorithms Constraint-Based Planning Satisfiability Given: A Boolean formula: Variables P = p 1... p n Formula φ : B n B Find: Is φ(p) satisfiable? P, (φ(p) = 1) What is P? Solution: Davis-Putnam-Logeman-Loveland (DPLL) Backtracking Search Modern DPLL solvers are effective. Dantam (Mines CSCI-561) Application: Planning as Satisfiability (SATPlan) (Pre Lecture) Fall 2018 29 / 44

Planning Algorithms Constraint-Based Planning Planning as Boolean Satisfiability 1. Ground first-order logic domain (PDDL) to propositional logic 2. Encode planning problem as Boolean formula: Unroll for fixed steps, n. One Boolean variable per state/action per step 3. If SAT: return action variable assignments 4. Else: increment n and repeat Dantam (Mines CSCI-561) Application: Planning as Satisfiability (SATPlan) (Pre Lecture) Fall 2018 30 / 44

Planning Algorithms Constraint-Based Planning Example: State Variable Unrolling First-Order Logic Objects: A, B, C Predicate: ontable (?x) Propositional Logic ontable-a ontable-b ontable-c Unrolled For n = 3: ontable-a-0 ontable-b-0 ontable-c-0 ontable-a-1 ontable-b-1 ontable-c-1 ontable-a-2 ontable-b-2 ontable-c-2 Dantam (Mines CSCI-561) Application: Planning as Satisfiability (SATPlan) (Pre Lecture) Fall 2018 31 / 44

Planning Algorithms Constraint-Based Planning Example: Action Variable Unrolling First-Order Logic Objects: A, B, C Predicate: pick-up (?x) Propositional Logic pick-up-a pick-up-b pick-up-c Unrolled For n = 3: pick-up-a-0 pick-up-b-0 pick-up-c-0 pick-up-a-1 pick-up-b-1 pick-up-c-1 pick-up-a-2 pick-up-b-2 pick-up-c-2 Dantam (Mines CSCI-561) Application: Planning as Satisfiability (SATPlan) (Pre Lecture) Fall 2018 32 / 44

Planning Algorithms Constraint-Based Planning Exercise: Unrolling First-Order Logic Objects: A, B Predicate: on (?x,?y) Action: stack (?x,?y) Grounding Unrolled Dantam (Mines CSCI-561) Application: Planning as Satisfiability (SATPlan) (Pre Lecture) Fall 2018 33 / 44

Planning Algorithms Constraint-Based Planning SATPlan Start and Goal Start: State state holds at step 0 Goal: Goal expression holds at step n Dantam (Mines CSCI-561) Application: Planning as Satisfiability (SATPlan) (Pre Lecture) Fall 2018 34 / 44

Planning Algorithms Constraint-Based Planning Example: Start and Goal PDDL Facts ( d e f i n e ( problem sussman anomaly ) ( : domain b l o c k s ) ( : o b j e c t s a b c ) ( : i n i t ( on c a ) ( o n t a b l e a ) ( o n t a b l e b ) ( c l e a r c ) ( c l e a r b ) ( handempty ) ) ( : g o a l ( and ( on b c ) ( on a b ) ) ) ) Start State on-c-a-0 ontable-a-0 ontable-b-0 clear-c-0 clear-b-0 handempty-0 on-a-a-0 on-a-b-0 on-a-c-0... Goal Unroll for n = 4 on-a-b-4 on-b-c-4 Dantam (Mines CSCI-561) Application: Planning as Satisfiability (SATPlan) (Pre Lecture) Fall 2018 35 / 44

Planning Algorithms Constraint-Based Planning Exercise: Start and Goal PDDL Facts ( d e f i n e ( problem sussman anomaly alt ) ( : domain b l o c k s ) ( : o b j e c t s a b c ) ( : i n i t ( o n t a b l e c ) ( o n t a b l e a ) ( on b a ) ( c l e a r c ) ( c l e a r b ) ( handempty ) ) ( : g o a l ( and ( on c b ) ( on b a ) ) ) ) Start State Goal Unroll for n = 4: Dantam (Mines CSCI-561) Application: Planning as Satisfiability (SATPlan) (Pre Lecture) Fall 2018 36 / 44

Planning Algorithms Constraint-Based Planning SATPlan Transition Function Operator Encoding: Selected operator s preconditions and effects must hold: precondition at step k effect at step k+1 {}}{{}}{ o [k] i = pre(o i ) [k] eff(o i ) [k+1] Operator Exclusion: One operator per step: o i [k] = ( o 0 [k] o (i 1) [k] o (i+1) [k] o m [k] ) Frame Axioms: Each proposition p is unchanged unless set by an effect: operators changing p {}}{ ( p [k] = p [k+1]) (o ) [k] [k] j... o l Dantam (Mines CSCI-561) Application: Planning as Satisfiability (SATPlan) (Pre Lecture) Fall 2018 37 / 44

Planning Algorithms Constraint-Based Planning Example: Operator Encoding pick-up-a-0 pick-up-a-0 ( : a c t i o n pick up : parameters (? x ) : p r e c o n d i t i o n ( and ( c l e a r? x ) ( o n t a b l e? x ) ( handempty ) ) : e f f e c t ( and ( not ( o n t a b l e? x ) ) ( not ( c l e a r? x ) ) ( not ( handempty ) ) ( h o l d i n g? x ) ) ) pick-up-a-0 = ( clear-a-0 ontable-a-0 handempty-0 ontable-a-1 clear-a-1 handempty-1 ) holding-a-1 Dantam (Mines CSCI-561) Application: Planning as Satisfiability (SATPlan) (Pre Lecture) Fall 2018 38 / 44

Planning Algorithms Constraint-Based Planning Example: Operator Exclusion pick-up-a-0 ( d e f i n e ( domain b l o c k s ) ( : p r e d i c a t e s ( o n t a b l e? x ) ( c l e a r? x ) ( handempty ) ( h o l d i n g? x ) ) ( : a c t i o n pick up : parameters (? x ) : p r e c o n d i t i o n ( and ( c l e a r? x ) ( o n t a b l e? x ) ( handempty ) ) : e f f e c t ( and ( not ( o n t a b l e? x ) ) ( not ( c l e a r? x ) ) ( not ( handempty ) ) ( h o l d i n g? x ) ) ) ( : a c t i o n put down : parameters (? x ) : p r e c o n d i t i o n ( h o l d i n g? x ) : e f f e c t ( and ( not ( h o l d i n g? x ) ) ( c l e a r? x ) ( handempty ) ( o n t a b l e? x ) ) ) ) o i [k] = ( o 0 [k] o (i 1) [k] o (i+1) [k] o m [k] ) pick-up-a-0 = ( pick-up-b-0 pick-up-c-0 put-down-a-0 put-down-b-0 put-down-c-0) Dantam (Mines CSCI-561) Application: Planning as Satisfiability (SATPlan) (Pre Lecture) Fall 2018 39 / 44

Planning Algorithms Constraint-Based Planning Example: Frame Axioms ontable-a-0 ( d e f i n e ( domain b l o c k s ) ( : p r e d i c a t e s ( o n t a b l e? x ) ( c l e a r? x ) ( handempty ) ( h o l d i n g? x ) ) ( : a c t i o n pick up : parameters (? x ) : p r e c o n d i t i o n ( and ( c l e a r? x ) ( o n t a b l e? x ) ( handempty ) ) : e f f e c t ( and ( not ( o n t a b l e? x ) ) ( not ( c l e a r? x ) ) ( not ( handempty ) ) ( h o l d i n g? x ) ) ) ( : a c t i o n put down : parameters (? x ) : p r e c o n d i t i o n ( h o l d i n g? x ) : e f f e c t ( and ( not ( h o l d i n g? x ) ) ( c l e a r? x ) ( handempty ) ( o n t a b l e? x ) ) ) ) ( p [k] = p [k+1]) (o j [k]... o l [k] ) (ontable-a-0 = ontable-a-1) pick-up-a-0 put-down-a-0 Dantam (Mines CSCI-561) Application: Planning as Satisfiability (SATPlan) (Pre Lecture) Fall 2018 40 / 44

Planning Algorithms Constraint-Based Planning Exercise: Transition Function (1/3) Operator Encoding for put-down-b-1 put-down-b-1 ( : a c t i o n put down : parameters (? x ) : p r e c o n d i t i o n ( h o l d i n g? x ) : e f f e c t ( and ( not ( h o l d i n g? x ) ) ( c l e a r? x ) ( handempty ) ( o n t a b l e? x ) ) ) Dantam (Mines CSCI-561) Application: Planning as Satisfiability (SATPlan) (Pre Lecture) Fall 2018 41 / 44

Planning Algorithms Constraint-Based Planning Exercise: Transition Function (2/3) Operator Exclusion for put-down-b-1 ( d e f i n e ( domain b l o c k s ) ( : p r e d i c a t e s ( o n t a b l e? x ) ( c l e a r? x ) ( handempty ) ( h o l d i n g? x ) ) ( : a c t i o n pick up : parameters (? x ) : p r e c o n d i t i o n ( and ( c l e a r? x ) ( o n t a b l e? x ) ( handempty ) ) : e f f e c t ( and ( not ( o n t a b l e? x ) ) ( not ( c l e a r? x ) ) ( not ( handempty ) ) ( h o l d i n g? x ) ) ) ( : a c t i o n put down : parameters (? x ) : p r e c o n d i t i o n ( h o l d i n g? x ) : e f f e c t ( and ( not ( h o l d i n g? x ) ) ( c l e a r? x ) ( handempty ) ( o n t a b l e? x ) ) ) ) o i [k] = ( o 0 [k] o (i 1) [k] o (i+1) [k] o m [k] ) Dantam (Mines CSCI-561) Application: Planning as Satisfiability (SATPlan) (Pre Lecture) Fall 2018 42 / 44

Planning Algorithms Constraint-Based Planning Exercise: Transition Function (3/3) Frame Axioms for holding-b-0 ( d e f i n e ( domain b l o c k s ) ( : p r e d i c a t e s ( o n t a b l e? x ) ( c l e a r? x ) ( handempty ) ( h o l d i n g? x ) ) ( : a c t i o n pick up : parameters (? x ) : p r e c o n d i t i o n ( and ( c l e a r? x ) ( o n t a b l e? x ) ( handempty ) ) : e f f e c t ( and ( not ( o n t a b l e? x ) ) ( not ( c l e a r? x ) ) ( not ( handempty ) ) ( h o l d i n g? x ) ) ) ( : a c t i o n put down : parameters (? x ) : p r e c o n d i t i o n ( h o l d i n g? x ) : e f f e c t ( and ( not ( h o l d i n g? x ) ) ( c l e a r? x ) ( handempty ) ( o n t a b l e? x ) ) ) ) ( p [k] = p [k+1]) (o j [k]... o l [k] ) Dantam (Mines CSCI-561) Application: Planning as Satisfiability (SATPlan) (Pre Lecture) Fall 2018 43 / 44

Planning Algorithms Constraint-Based Planning SATPlan Extensions and Implementations Operator Exclusion: Relax this constraint: o i [k] = ( o 0 [k] o (i 1) [k] o (i+1) [k] o m [k] ) Blackbox: H. Kautz and B. Selman. Unifying SAT-based and graph-based planning. International Joint Conference on Artificial Intelligence 1999. Madagascar: J. Rintanen. Madagascar: Scalable planning with SAT. 8th International Planning Competition. 2014. TMKit: N. Dantam, Z. Kingston, S. Chaudhuri, and L. Kavraki. An Incremental Constraint-Based Framework for Task and Motion Planning. International Journal of Robotics Research. 2018 Dantam (Mines CSCI-561) Application: Planning as Satisfiability (SATPlan) (Pre Lecture) Fall 2018 44 / 44