1. Siddharth Srivastava, Shlomo Zilberstein, Neil Immerman University of Massachusetts Amherst Hector Geffner Universitat Pompeu Fabra

2. Finite sets of states & registers Actions with unit increments/decrements [Lambek, 61] Abacus Programs

3. The reachability problem for abacus programs as a method for reasoning about cyclic control flows But reachability is equivalent to the halting problem for Turing machines …. Undecidable Approach: identify subclasses or less expressive frameworks … cannot capture TM, but still useful Approach: identify subclasses or less expressive frameworks … cannot capture TM, but still useful [Srivastava et al., ICAPS-10]

4. Consider situations where Actions increase or decrease numeric variables by unpredictable amounts Propositional variables can be added Plans require cyclic control E.g., delivery problem with unknown Fuel Distances Quantities of deliverables Driving will use unpredictable amount of fuel

7. But how do we express policies? Cannot map all possible states (real- valued assignments to variables) But how do we express policies? Cannot map all possible states (real- valued assignments to variables)

11. A quantitative policy is essentially qualitative iff: Maps all states represented by a qualitative state to the same action Very useful: Cannot have explicit policy representations over quantitative states anyway Theorem A non-deterministic quantitative planning problem P has a solution policy iff P has a policy that is essentially qualitative Theorem A non-deterministic quantitative planning problem P has a solution policy iff P has a policy that is essentially qualitative

12. Transition Graph

14. For every SCC: Identify edges that cannot be executed infinitely often Remove them, signifying stage when there executions have been exhausted Recurse on each resulting SCC Finally: terminating iff no SCC left on fixed point

15. Completeness: if Sieve algorithm returns non-terminating, an infinite execution is possible Surprising because of similarity to abacus programs

16. Enumerate all possible policies (yes, this is impractical in general!) But computable! Check for 1.Goal-closed (any terminal nodes in transition graph must be goal nodes) 2.Termination using sieve algorithm

17. Problems Nested variables Snow plow: using snow blower spills snow onto the driveway Delivery with fuel, unknown number of objects and truck capacities Trash-collection Solution Time (s)

18. Improve generate and test: Start with strong cyclic qualitative policies Introduce constant landmarks/intervals of values Identify limits of sieve algorithms applicability in abacus programs

19. QNP gives the first framework for planning with loops where termination and correctness are decidable properties For any class of loops Any number of unbounded variables