Last update: April 1, 2020 Chapter 2 Deliberation with Deterministic Models 2.1: State-Variable Representation Automated Planning 2.2: Forward Search and Acting 2.6: Planning and Acting Malik Ghallab, Dana Nau and Paolo Traverso Dana S. Nau http://www.laas.fr/planning University of Maryland Nau – Lecture slides for Automated Planning and Acting Licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License 1
Motivation and Outline ● How to model a complex environment? Outline ▸ Generally need simplifying assumptions 2.1 State-variable representation ▸ state variables, states, actions, ● Classical planning plans • Finite, static world, just one actor • No concurrent actions, no explicit time 2.2 Forward state-space search • Determinism, no uncertainty ▸ Sequence of states and actions ⟨ s 0 , a 1 , s 1 , a 2 , s 2 , … ⟩ 2.6 Incorporating planning into an actor 2.3 Heuristic functions ● Avoids many complications 2.4 Backward search ● Most real-world environments don’t satisfy the assumptions ⇒ Errors in prediction 2.5 Plan-space search ● OK if they’re infrequent and don’t have severe consequences Nau – Lecture slides for Automated Planning and Acting 2
Domain Model ● plan : State-transition system or classical planning domain : ▸ a sequence of actions π = ⟨ a 1 , …, a n ⟩ ● Σ = ( S,A, γ,cost) or ( S,A, γ) ● π is applicable in s 0 if the actions are applicable in the order given ▸ S - finite set of states γ( s 0 , a 1 ) = s 1 ▸ A - finite set of actions γ( s 1 , a 2 ) = s 2 ▸ γ: S × A → S … - prediction (or state-transition ) function γ( s n –1 , a n ) = s n • partial function ▸ In this case define γ( s 0 , π ) = s n ▸ defined only when a is applicable in s • Domain( a ) = { s ∈ S | a is applicable in s } ● Classical planning problem : = { s ∈ S | γ( s,a ) is defined} ▸ P = (Σ, s 0 , S g ) • Range( a ) = {γ( s , a ) | s ∈ Domain( a )} ▸ planning domain, initial state, set of goal states ▸ cost: S × A → R + or cost: A → R + ● Solution for P : • optional; default is cost( a ) ≡ 1 ▸ a plan π such that that γ( s 0 , π ) ∈ S g • money, time, something else Nau – Lecture slides for Automated Planning and Acting 3
Representing Σ y ● If S and A are small enough loc 2 4 loc 0 ▸ Give each state and action a name loc 6 ▸ For each s and a , store γ( s,a ) in a lookup table loc 1 3 loc 3 2 ● In larger domains, don’t represent all states loc 7 loc 4 loc 5 1 explicitly loc 8 loc 9 ▸ Language for describing properties of states x 0 1 2 3 4 5 6 ▸ Language for describing how each action changes those properties ▸ Start with initial state … a 1 s 1 = γ( s 0 ,a 1 ) ▸ Use actions to produce other states s 0 a 1 ′ … s 1 ′ = γ( s 0 ,a 1 ′) Nau – Lecture slides for Automated Planning and Acting 4
Domain-Specific Representation ● Made to order for a specific environment ● State: arbitrary data structure ● Action: (head, preconditions, effects, cost) ▸ head : name and parameter list • Get actions by instantiating the parameters ▸ preconditions : • Computational tests to predict whether an action can be performed • Should be necessary/sufficient for the action to run without error ▸ effects : • Procedures that modify the current state ▸ cost : procedure that returns a number • Can be omitted, default is cost ≡ 1 Nau – Lecture slides for Automated Planning and Acting 5
Example ● Drilling holes in a metal workpiece ● Name and parameters: ▸ A state ▸ drill-hole ( machine, drill-bit, workpiece, geometry, machining-tolerances ) • annotated geometric model of the workpiece ● Preconditions • capabilities and status of ▸ Capabilities : can the machine and drill bit drilling machine and drill bit produce a hole with the desired geometry and ▸ Several actions machining tolerances? • put workpiece onto the drilling machine ▸ Current state : Is the drill bit installed? Is the • clamp it workpiece clamped onto the table? Etc. • load a drill bit ● Effects • drill ▸ annotated geometric model of modified workpiece ● Cost ▸ estimate of time or monetary cost Nau – Lecture slides for Automated Planning and Acting 6
Discussion ● Advantage of domain-specific representation: ● State-variable representation ▸ use whatever works best for that particular ▸ Simple formats for describing states and domain actions ▸ Limited representational capability ● Disadvantage: ▸ for each new domain, need new • But easy to compute, easy to reason representation and deliberation algorithms about ▸ Domain-independent search algorithms and ● Alternative: domain-independent representation heuristic functions that can be used in all state-variable planning problems ▸ Try to create a “standard format” that can be used for many different planning domains ▸ Deliberation algorithms that work for anything in this format Nau – Lecture slides for Automated Planning and Acting 7
State-Variable Representation ● E : environment that we want to represent ● B : set of symbols called objects ▸ names for objects in E , mathematical constants, … d3 ● Example r1 ▸ B = Robots ∪ Containers ∪ Locs ∪ { nil } c1 c2 d1 d2 • Robots = { r1 } • Containers = { c1, c2 } • Locs = { d1, d2, d3 } ● B only needs to include objects that matter at the current level of abstraction ● Can omit lots of details ▸ physical characteristics of robots, containers, loading docks, roads, … Nau – Lecture slides for Automated Planning and Acting 8
Properties of Objects ● Define ways to represent properties of objects ● Terminology from first-order logic: ▸ Two kinds of properties: rigid and varying ▸ ground : fully instantiated, no variable symbols ▸ atom ≡ atomic formula ≡ positive literal ● Rigid property: stays the same in every state ≡ predicate symbol with list of arguments ▸ Two equivalent notations: ▸ negative literal ≡ negated atom ≡ atom with • A mathematical relation a negation sign in front of it adjacent = {( d1,d2 ), ( d2,d1 ), ( d1,d3 ), ( d3,d1 )} • A set of ground atoms adjacent(d1,d2), adjacent(d2,d1), d3 adjacent(d1,d3), adjacent(d3,d1) r1 c1 c2 d1 d2 Nau – Lecture slides for Automated Planning and Acting 9
Varying Properties ● Varying property (or fluent ): may differ in different states ▸ Represent it using a state variable that we can assign a value to d3 r1 ● Set of state variables c1 c2 d1 d2 X = { loc(r1) , loc(c1) , loc(c2) , cargo(r1) } ● Each state variable x ∈ X has a range = {all values that can be assigned to x } ▸ Range( loc ( r1 )) = Locs Instead of “domain”, to avoid confusion ▸ Range( loc ( c1 )) = Range( loc ( c2) ) = Robots ∪ Locs with planning domains ▸ Range( cargo ( r1 )) = Containers ∪ { nil } Nau – Lecture slides for Automated Planning and Acting 10
States as Functions ● Represent each state as a variable-assignment function ▸ Function that maps each x ∈ X to a value in Range( x ) d3 s 1 (loc(r1)) = d1 , s 1 (cargo(r1)) = nil , r1 s 1 (loc(c1)) = d1 , s 1 (loc(c2)) = d2 c1 c2 d1 d2 ● Mathematically, a function is a set of ordered pairs s 1 = { (loc(r1), d1), (cargo(r1), nil), (loc(c1), d1) , (loc(c2), d2) } ● Write it as a set of ground positive literals (or ground atoms ): s 1 = { loc(r1) = d1, cargo(r1) = nil, loc(c1) = d1, loc(c2)=d2 } Nau – Lecture slides for Automated Planning and Acting 11
Action Templates d3 ● Action template: a parameterized set of actions r1 α = (head(α), pre(α), eff(α), cost(α)) c1 c2 d1 d2 ● head( α ): name, parameters move ( r,l,m ) pre: loc ( r )= l , adjacent ( l,m ) Each parameter has a range ⊆ B eff: loc ( r ) ← m ● pre( α ): precondition literals take( r,l,c ) rel ( t 1 ,…, t k ), var ( t 1 ,…, t k ) = t 0 , pre: cargo ( r )= nil, loc ( r )= l , loc ( c )= l ¬ rel ( t 1 ,…, t k ), ¬ var ( t 1 ,…, t k ) = t 0 eff: cargo ( r ) ← c , loc ( c ) ← r ▸ Each t i is a parameter or an element of B put( r,l,c ) ● eff( α ): effect literals pre: loc ( r )= l , loc ( c )= r var ( t 1 ,…, t k ) ← t 0 eff: cargo ( r ) ← nil , loc ( c ) ← l ● cost( α ): a number Range( r ) = Robots = { r1 } ▸ Optional, default is 1 Range( l ) = Range( m ) = Locs = { d1,d2,d3 } Range( c ) = Containers = { c1,c2 } Nau – Lecture slides for Automated Planning and Acting 12
Actions ● A = set of action templates ● Action: ground instance of an α ∈ A ▸ replace each parameter with something in its range move ( r,l,m ) pre: loc ( r )= l , adjacent ( l, m ) ● A = {all actions we can get from A } eff: loc ( r ) ← m = {all ground instances of members of A } take( r,l,c ) pre: cargo ( r )= nil, loc ( r )= l , loc (c)= l move(r1,d1,d2) eff: cargo( r ) ← c , loc( c) ← r pre: loc(r1) = d1 , adjacent(d1,d2) eff: loc(r1) ← d2 put( r,l,c ) d3 pre: loc ( r )= l , loc ( c )= r r1 eff: cargo( r ) ← nil , loc( c) ← l c1 c2 d1 d2 Range( r ) = Robots = { r1 } Range( l ) = Range( m ) = Locs = { d1,d2,d3 } Range( c ) = Containers = { c1,c2 } Nau – Lecture slides for Automated Planning and Acting 13
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