B2.1 Regression. Planning and Optimization. Planning and Optimization. B2.1 Regression. B2.2 Regression Example. B2.3 Regression for STRIPS Tasks

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1 Planning and Optimization October 13, 2016 B2. Regression: ntroduction & STRPS Case Planning and Optimization B2. Regression: ntroduction & STRPS Case Malte Helmert and Gabriele Röger Universität Basel October 13, 2016 B2.1 Regression B2.2 Regression Example B2.3 Regression for STRPS Tasks B2.4 Summary M. Helmert, G. Röger (Universität Basel) Planning and Optimization October 13, / 20 M. Helmert, G. Röger (Universität Basel) Planning and Optimization October 13, / 20 Forward Search vs. Backward Search B2.1 Regression Searching planning tasks in forward vs. backward direction is not symmetric: forward search starts from a single initial state; backward search starts from a set of goal states when applying an operator o in a state s in forward direction, there is a unique successor state s ; if we just applied operator o and ended up in state s, there can be several possible predecessor states s in most natural representation for backward search in planning, each search state corresponds to a set of world states M. Helmert, G. Röger (Universität Basel) Planning and Optimization October 13, / 20 M. Helmert, G. Röger (Universität Basel) Planning and Optimization October 13, / 20

2 Planning by Backward Search: Regression Search Space Representation in Regression Planners Regression: Computing the possible predecessor states regr o (S ) of a set of states S ( subgoal ) given the last operator o that was applied. formal definition in next chapter Regression planners find solutions by backward search: start from set of goal states iteratively pick a previously generated subgoal (state set) and regress it through an operator, generating a new subgoal solution found when a generated subgoal includes initial state identify state sets with logical formulas (again): each search state corresponds to a set of world states ( subgoal ) each search state is represented by a logical formula: ϕ represents {s S s = ϕ} many basic search operations like detecting duplicates are NP-complete or conp-complete pro: can handle many states simultaneously con: basic operations complicated and expensive M. Helmert, G. Röger (Universität Basel) Planning and Optimization October 13, / 20 M. Helmert, G. Röger (Universität Basel) Planning and Optimization October 13, / 20 Search Space for Regression Example Search Space for Regression search space for regression in a planning task Π = V,, O, (search states are formulas ϕ describing sets of world states; actions of search space are operators o O) init() is goal(ϕ) succ(ϕ) cost(o) returns tests if = ϕ returns all pairs o, regr o (ϕ) where o O and regr o (ϕ) is defined returns cost(o) as defined in Π h(ϕ) estimates cost from to ϕ ( Parts C and D) B2.2 Regression Example M. Helmert, G. Röger (Universität Basel) Planning and Optimization October 13, / 20 M. Helmert, G. Röger (Universität Basel) Planning and Optimization October 13, / 20

3 Example Example M. Helmert, G. Röger (Universität Basel) Planning and Optimization October 13, / 20 M. Helmert, G. Röger (Universität Basel) Planning and Optimization October 13, / 20 Example Example = regr () = regr () ϕ 2 ϕ 2 = regr ( ) M. Helmert, G. Röger (Universität Basel) Planning and Optimization October 13, / 20 M. Helmert, G. Röger (Universität Basel) Planning and Optimization October 13, / 20

4 Example for STRPS Tasks ϕ 3 = regr () ϕ 2 ϕ 2 = regr ( ) ϕ 3 = regr (ϕ 2 ), = ϕ 3 B2.3 Regression for STRPS Tasks M. Helmert, G. Röger (Universität Basel) Planning and Optimization October 13, / 20 M. Helmert, G. Röger (Universität Basel) Planning and Optimization October 13, / 20 for STRPS Tasks Regression for STRPS Planning Tasks for STRPS Tasks STRPS Regression Regression for STRPS planning tasks is much simpler than the general case: Consider subgoal ϕ that is conjunction of atoms a 1 a n (e.g., the original goal of the planning task). First step: Choose an operator o that deletes no a i. Second step: Remove any atoms added by o from ϕ. Third step: Conjoin pre(o) to ϕ. Outcome of this is regression of ϕ w.r.t. o. t is again a conjunction of atoms. optimization: only consider operators adding at least one a i Definition (STRPS Regression) Let ϕ = ϕ n be a conjunction of atoms, and let o be a STRPS operator which adds the atoms a 1,..., a k and deletes the atoms d 1,..., d l. (W.l.o.g., a i d j for all i, j.) The STRPS regression of ϕ with respect to o is { if ϕ i = d j for some i, j sregr o (ϕ) := pre(o) ({,..., ϕ n } \ {a 1,..., a k }) Note: sregr o (ϕ) is again a conjunction of atoms, or. otherwise M. Helmert, G. Röger (Universität Basel) Planning and Optimization October 13, / 20 M. Helmert, G. Röger (Universität Basel) Planning and Optimization October 13, / 20

5 for STRPS Tasks Does this Capture the dea of Regression? for STRPS Tasks STRPS Regression Example For our definition to capture the concept of regression, it should satisfy the following property: Regression Property For all sets of states described by a conjunction of atoms ϕ, all states s and all STRPS operators o, s = sregr o (ϕ) iff s o = ϕ. This is indeed true. We do not prove it now because we prove this property for general regression (not just STRPS) later. o 1 o 2 o 3 Note: Predecessor states are in general not unique. This picture is just for illustration purposes. o 1 = on clr, o 2 = on clr clr, o 3 = ont clr clr, on ont clr clr on on clr clr ont on = on on = sregr o3 () = ont clr clr on ϕ 2 = sregr o2 ( ) = on clr clr ont ϕ 3 = sregr o1 (ϕ 2 ) = on clr on ont M. Helmert, G. Röger (Universität Basel) Planning and Optimization October 13, / 20 M. Helmert, G. Röger (Universität Basel) Planning and Optimization October 13, / 20 B2. Regression: ntroduction & STRPS Case Summary B2. Regression: ntroduction & STRPS Case Summary Summary B2.4 Summary Regression search proceeds backwards from the goal. Each search state corresponds to a set of world states, for example represented by a formula. Regression is simple for STRPS operators. The theory for general regression is more complex. This is the topic of the following chapters. M. Helmert, G. Röger (Universität Basel) Planning and Optimization October 13, / 20 M. Helmert, G. Röger (Universität Basel) Planning and Optimization October 13, / 20