1 BLACKBOX: A New Approach to the Application of Theorem Proving to Problem Solving Bart Selman Cornell University Joint work with Henry Kautz AT&T Labs

2 Planning as (Specialized) Deduction Situation calculus (McCarthy & Hayes 1969) Conceptually elegant Planning as first-order theorem proving (Green 1969) Computationally infeasible STRIPS: A New Approach to the Application of Theorem Proving to Problem Solving (Fikes & Nilsson 1971) More efficient, but still poor scaling Partial-order planning (Tate 1977, Chapman 1985, McAllester 1991) Planning as specialized implementation of a particular theorem - the modal truth criteria Can be more efficient, but still hard (Minton, Bresina, & Drummond 1994) SATPLAN (Kautz and Selman 1992, 1996) Planning as (simply) propositional reasoning

3 SATPLAN axiom schemas instantiated propositional clauses satisfying model plan mapping length problem description SAT engine(s) instantiate interpret

4 What SATPLAN Showed A general propositional theorem prover can be competitive with specialized planning (specialized theorem proving) systems Good representations –propositional, not first-order –can combine expressiveness with good computational properties Fast SAT engines –stochastic search - walksat –large SAT/CSP community sharing ideas and code +compare: general CPU's versus Lisp-chip +specialized engines can catch up, but by then, new general techniques

5 Graphplan Planning as graph search (Blum & Furst 1995) Set new paradigm for planning Like SATPLAN... Two phases: instantiation of propositional structure, followed by search Possible to translate planning graph into CNF (Kautz & Selman 1996) Unlike SATPLAN... Interleaves instantiation and pruning of plan graph Employs specialized search engine Graphplan - better instantiation SATPLAN - better search

6 Bridging Paradigms Both SATPLAN and Graphplan are disjunctive planners (Kambhampati 1996) avoid splitting - more compact representations, faster search IJCAI Challenge in Bridging Plan Synthesis Paradigms (Kambhampati 1997) study tradeoffs between refinement (instantiation) and extraction (SAT) techniques Our response: blackbox efficient planning system and highly- configurable testbed

7 Blackbox Architecture 1. Front end: turn STRIPS input into planning graph interleaves instantiation and pruning (specialized limited deduction) 2. Translation to CNF 3. Simplification (general limited deduction) critical for hardest problems –unit propagation, failed literal, binary failed literal, SAT solvers can schedule series of different solvers: walksat, satz, rel_sat, etc. new: randomized systematic solvers (more later)

8 Specialized Inference SATPLAN was proof of concept that general deduction is feasible When do specialized inference techniques have the greatest impact? Two kinds of specialized deduction Planning specific - graphplan's pruning rules General polytime inference - simplification performed after CNF generation –apply to all CNF formulas, may or may not be designed with planning in mind

9 Graph Pruning Graphplan instantiates in a forward direction, pruning nodes with incompatible preconditions incomplete pair-wise mutex relations computed by incremental constraint propagation –(see Kambhampati 1997, Euro CP) In logical terms: limited application of negative binary propagation given:  P V  Q, P V R V S V... infer:  Q V R V S V...

10 Translation to CNF Fact  Act1 V Act2 Act1  Pre1 & Pre2 Wff is entailed by original SATPLAN hand encodings, but more compact, easier to solve compare with Medic (Weld 1997), which translated STRIPS  CNF directly Act1 Act2 Fact Pre1 Pre2

11 Simplification Generated wff can be further simplified by additional consistency propagation techniques Because is in CNF, can take advantage of many fast, powerful simplifiers Compact (Crawford & Auton 1996) failed literal rule: is Wff + { P } unsat by unit propagation? binary failed literal rule: is Wff + { P V Q } unsat by unit propagation? Generally reduces number of variables and clauses by 30%

12 Randomized Sytematic Solvers Stochastic local search solvers (walksat) when they work, scale well cannot show unsat fail on some domains must use very simple (fast) heuristics Systematic solvers (Davis Putnam) complete fail on (often different) domains might use more sophisticated (costly) heuristics seem to scale badly Can we combine best features of each approach?

13 Heavy Tails Bad scaling of systematic solvers can be caused by heavy tailed distributions Deterministic algorithms get stuck on particular instances but that same instance might be easy for a different deterministic algorithm! Expected (mean) solution time increases without limit over large distributions

14 Heavy Tailed Cost Distribution

15 Randomized Restarts Solution: randomize the systematic solver Add noise to the heuristic branching (variable choice) function Cutoff and restart search after a fixed number of backtracks Eliminates heavy tails In practice: rapid restarts with low cutoff can dramatically improve performance

16 Rapid Restart Speedup

17 Blackbox as Experimental Testbed All components of blackbox are parameterized Can experiment with different schedules for instantiating, simplifying, and solving problems blackbox -solver -maxsec 20 graphplan -then compact -l -then satz -cutoff 20 -restart 100 -then walksat -cutoff restart 10

18 Blackbox Results

19 blackbox version 9B command line: blackbox -o logistics.pddl -f logistics_prob_d_len.pddl -solver compact -l -then satz -cutoff 25 -restart Converting graph to wff 6151 variables clauses Invoking simplifier compact Variables undetermined: 4633 Non-unary clauses output: Invoking solver satz version satz-rand-2.1 Wff loaded [1] begin restart [1] reached cutoff back to root [2] begin restart [2] reached cutoff back to root [3] begin restart [3] reached cutoff back to root [4] begin restart [4] reached cutoff back to root [5] begin restart **** the instance is satisfiable ***** **** verification of solution is OK **** total elapsed seconds = Begin plan 1 drive-truck_ny-truck_ny-central_ny-po_ny

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21 Summary Blackbox combines best features of Graphplan, SATPLAN, and new randomized systematic search engines Automatic generation of wffs from standard STRIPS input No performance penalty over hand-encodings! Testbed for bridging different planning paradigms

22 Current Research Issues Tradeoffs between planning-specific and more general limited inference (simplification) techniques More powerful / general rules than mutex computation? Incorporating explicit domain knowledge (Kautz & Selman, 1998) state invariants optimal conditions Other front-ends: causal encodings (McAllester, Selman, Kautz 1996), HTN encodings (Mali & Kambhampati 1998) Download from