Reasoning about State Constraints in the Situation Calculus

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1 Reasoning about State Constraints in the Situation Calculus Joint work with Naiqi Li and Yi Fan Yongmei Liu Dept. of Computer Science Sun Yat-sen University Guangzhou, China Presented at IRIT June 26, 2013 Y. Liu Reasoning about State Constraints 1 / 24

2 Motivation Two important properties for dynamic systems State invariants: formulas that if true in a state, will be true in all successor states a first-order reasoning task State constraints: formulas which hold in any reachable state a second-order reasoning task Can be used to greatly reduce the planning search space Can also be used to debug domain axiomatization Y. Liu Reasoning about State Constraints 2 / 24

3 Existing Works [The planning community] restricted to formulas and propositional domains mature and very efficient [Lin, KR-04] discover state invariants in domains of unbounded size through exhaustive search in small domains no soundness guarantee for non- formulas Y. Liu Reasoning about State Constraints 3 / 24

4 Our work A sound but incomplete method for automatic verification and discovery of state constraints for a class of action theories including many planning benchmarks. Theoretically based on Skolemization and Herbrand Theorem, Implemented with SAT solvers. Y. Liu Reasoning about State Constraints 4 / 24

5 Main idea Verify a state constraint by strengthening it in a novel way so that it becomes a state invariant. Y. Liu Reasoning about State Constraints 5 / 24

6 Outline 1 Theoretical foundations 1 The situation calculus 2 Joint constraints and derivants 3 Incomplete reasoning based on Herbrand Theorem 2 Algorithms 3 Experimental results Y. Liu Reasoning about State Constraints 6 / 24

7 The situation calculus A first-order language (with some second-order ingredients) suitable for reasoning about actions and change Intuitively, a situation is a finite sequence of actions Action functions: move(x, y) Fluents (relations or functions whose value can change from situation to situation): on(x, y, s), height(x, s) Situations: S 0, do(a, s) P oss(a, s): action a is possible in situation s s s : s is a sub-history of s Y. Liu Reasoning about State Constraints 7 / 24

8 Basic action theories (BATs) D = Σ D ap D ss D una D S0, where Σ: the foundational axioms for situations, including a second-order induction axiom D ap : action precondition axioms, one for each action function A P oss(a( x), s) Π A ( x, s) D ss : successor state axioms (SSA), one for each fluent, F ( x, do(a, s)) Φ F ( x, a, s) D una : unique names axioms for actions D S0 : the initial KB, a finite set of sentences about S 0 Y. Liu Reasoning about State Constraints 8 / 24

9 An example: blocks world Action Precondition Axioms (D ap ) P oss(move(x, y), s) clear(x, s) clear(y, s), Successor State Axioms (D ss ) on(x, y, do(a, s)) a = move(x, y) on(x, y, s) ( z)a = move(x, z), clear(x, do(a, s)) ( y)( z)a = move(y, z) on(y, x, s) clear(x, s) ( y)a = move(y, x), Initial KB (D S0 ) clear(a, S 0 ), on(a, B, S 0 ) on(a, C, S 0 ), on(d, E, S 0 ) Y. Liu Reasoning about State Constraints 9 / 24

10 State constraints and invariants A situation s is executable if it is possible to perform the actions in s one by one: Exec(s) def = a, s.do(a, s ) s P oss(a, s ) Definition A formula φ(s) is a state constraint for D if D = s.exec(s) φ(s). Definition Let D SC be a set of state constraints for D. A formula φ(s) is a state invariant for D if D ap D ss D una = fol a, s.d SC φ(s) P oss(a, s) φ(do(a, s)). Proposition If D S0 = fol φ(s 0 ), and φ is an invariant, then φ is a constraint. Y. Liu Reasoning about State Constraints 10 / 24

11 Joint Constraints Definition An invariant constraint is a constraint which is also an invariant. If φ(s) ψ(s) η(s) is an invariant constraint and ψ(s) is not entailed by φ(s), we call ψ(s) a joint constraint for φ(s). So how to obtain joint constraints for a given constraint? Y. Liu Reasoning about State Constraints 11 / 24

12 Regression Recall P oss(a( x), s) Π A ( x, s), F ( x, do(a, s)) Φ F ( x, a, s) Definition We use R D [φ] to denote the formula obtained from φ by replacing each precondition atom P oss(a( x), s) with Π A ( x, s) each fluent atom F ( t, do(a( x), s)) with Φ F ( t, A( x), s) and further simplifying it via D una. Theorem D = φ R D [φ]. Y. Liu Reasoning about State Constraints 12 / 24

13 Derivants Definition Let φ(s) be a formula. For the class of action theories we consider, the following can be put into a formula. R D [ x.p oss(a( x), s) φ(do(a( x), s))] We convert the quantifer-free part into a CNF. We call each conjunct a derivant of φ(s). Theorem If φ(s) η(s) is an invariant constraint, and C(s) is a derivant of φ(s) not implied by φ(s), then C(s) is a joint constraint for φ(s). Y. Liu Reasoning about State Constraints 13 / 24

Checking first-order entailment of sentences By Skolemization, reduce it to checking satisfiability of a set Φ of sentences By Herbrand Theorem, reduce it to checking satisfiability of grounding of

14 Checking first-order entailment of sentences By Skolemization, reduce it to checking satisfiability of a set Φ of sentences By Herbrand Theorem, reduce it to checking satisfiability of grounding of sentences in Φ using an infinite Herbrand domain Use a sequence of finite partial Herbarnd domains for grounding until unsatisfiability is confirmed or some bound is reached Y. Liu Reasoning about State Constraints 14 / 24

15 Algorithm 1: Verifying constraints Given a formula φ 1 Test if D S0 = fol φ(s 0 ) 2 If not, return false 3 Test if φ is a state invariant 4 If yes, return φ 5 Pick a derivant C of φ not entailed by φ 6 Continue this procedure with φ C until a bound is reached Also check if φ can be strengthened by some subset of C in order to obtain more intuitive and succinct joint constraints Y. Liu Reasoning about State Constraints 15 / 24

16 An example 1 We were to test whether φ 1 = holding(x) ontable(x) is a state constraint. 2 φ 1 is not proved to be a state invariant, so the system strengthened it with φ 2 = on(x 1, x 2 ) ontable(x 1 ). 3 Again, φ 1 φ 2 is not a state invariant so the system strengthened it with φ 3 = on(x 1, x 2 ) holding(x 1 ). 4 After getting φ 4 = on(x 1, x 2 ) on(x 1, x 3 ) x 2 = x 3, φ 1... φ 4 was proved to be a state invariant, and thus, φ 1,..., φ 4 were all proved to be state constraints. Y. Liu Reasoning about State Constraints 16 / 24

17 Algorithm 2: Verifying constraints Given a formula φ If D S0 = fol φ(s 0 ) and φ is a state invariant using the set of verified constraints, then return true, else return false Y. Liu Reasoning about State Constraints 17 / 24

18 Algorithm 3: Enumerating unary and binary constraints 1 Enumerate all formulas containing one or two literals 2 Collect those which are verified to be constraints, together with their joint constraints Y. Liu Reasoning about State Constraints 18 / 24

19 Experimental results Using 3 IPC (International Planning Competition) domains: blocks world, logistics, satellite Almost all known state constraints can be verified in a reasonable amount of time (the max time is 23min) Meanwhile succinct and intuitive joint state constraints are discovered Y. Liu Reasoning about State Constraints 19 / 24

20 State constraints verified or discovered in Blocks World Constraints on(y, x) clear(x), holding(x) on(y, x), holding(x) on(x, y), holding(x) clear(x), holding(x) handempty, on(x, y) on(x, z) y = z, on(x, x), on(x, y) on(y, x) (not considered by Lin) Constraints holding(x) clear(x) ( y)on(y, x), handempty ( x)holding(x), ontable(x) holding(x) ( y)on(x, y) Y. Liu Reasoning about State Constraints 20 / 24

21 Time performance Domain Enum Formulas Formulas Binary Min Max Min Max B.W. 15min 1.8s 23min 0.5s 0.5s Logistics 17min 19.2s 68.8s 6.3s 7.0s Satellite 10min 2.5s 3.9s 5.7s 7.8s max for formulas is orders greater than that for ones, because we do not strengthen formulas max in Blocks World is significantly greater than that in the other domains, because B.W. is the most structured Y. Liu Reasoning about State Constraints 21 / 24

22 Related work from model checking A. R. Bradley and Z. Manna. Checking safety by inductive generalization of counterexamples to induction. In FMCAD, based on finite-state boolean transition system strengthen a constraint to be an invariant, or return a counterexample trace use minimum subclause generalized from counterexamples to accelerate convergence Y. Liu Reasoning about State Constraints 22 / 24

23 Conclusions A sound but incomplete method for automatic verification and discovery of constraints Verify a state constraint by strengthening it so that it becomes a state invariant Can be applied to infinite domains Experimental results show the effectiveness of our method Y. Liu Reasoning about State Constraints 23 / 24

24 Future Work Completeness issue of our method Theoretical formulation of related works in planning Cross-fertilization between related works in AI and formal methods Our paper is available at ymliu Y. Liu Reasoning about State Constraints 24 / 24

Reasoning about State Constraints in the Situation Calculus

Proceedings of the Twenty-Third International Joint Conference on Artificial Intelligence Reasoning about State Constraints in the Situation Calculus Naiqi Li 1 Yi Fan 1,2 Yongmei Liu 1 1 Department of