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Blai Bonet
Representation of state spaces
Universidad Sim´
on Bol´ıvar, Caracas, Venezuela
c 2015 Blai Bonet
Goals for the lecture
Lecture 2
The representation problem
• Learn about the representation problem for state-spaces (graphs)
• Introduce three approaches for representation: explicit, black-box
How do we represent state spaces in practical terms?
(implicit), declarative (implicit)
• Learn about different explicit representations and their space
How do we cope with the combinatorial explosion of states?
requirements
• Learn about the black-box representation and API
• Learn about declarative representations
c 2015 Blai Bonet
Lecture 2
c 2015 Blai Bonet
Lecture 2
Three approaches for representation
Explicit representation
States have no “internal” structure: they are opaque objects
We consider three representations of state spaces:
– Explicit representation
The graph can be represented either by an adjacency matrix,
by an incidence matrix, or by adjacency lists
– Implicit black-box representation
In all cases, the representation is
– Implicit declarative representation
– Flat meaning that states have no structure
– Polynomial in size in terms of number of vertices and edges
c 2015 Blai Bonet
Lecture 2
Explicit representation: example
c 2015 Blai Bonet
Lecture 2
Explicit representation: adjacency matrix
3
3
4
4
6
2
6
5
2
1
5
1
Multi-graph G = hV, Ei with vertices V = {1, 2, 3, 4, 5, 6} and edges
E = {(1, 1), (1, 2), (1, 5), (2, 3), (2, 5), (3, 2), (4, 3), (4, 6), (5, 4), (5, 4), (6, 4)}
c 2015 Blai Bonet
Lecture 2

1 1




0 0



0 1
Adjacency matrix =
0 0




 0 0


0 0
c 2015 Blai Bonet
0
1
0
1
0
0
0
0
0
0
2
1
1
1
0
0
0
0
0
0
0
1
0
0















Lecture 2
Explicit representation: incidence matrix
4
e11
e4
e2
1
e3
e10
e11
5
1
e3
e1
e1
Incidence matrix =


0 −1 −1 0
0
0
0
0
0
0
0 






0 1
0 −1 −1 1
0
0
0
0
0 






0 0
0
0
1 −1 0
0
1
0
0
0 0
0
0
0
0 −1 1 −1 −1 −1 








0
0
1
1
0
0
0
0
0
1
1 




0 0
0
0
0
0
1 −1 0
0
0
c 2015 Blai Bonet
6
e8
e4
e2
5
4
e6
2
e10
e7
e9
3
e5
6
e8
e6
2
e7
e9
3
e5
Explicit representation: adjacency lists

vertex 1 :




vertex 2 :



vertex 3 :
Adjacency lists =
vertex 4 :





vertex 5 :


vertex 6 :
Lecture 2
e1 , e2 , e3
e4 , e5
e6
e7 , e9
e10 , e11
e8
c 2015 Blai Bonet
Explicit representation: trade offs
Lecture 2
Black-box representation
Adjacency matrix:
Structure of the graph is wired into functions that provide the
necessary information to explore the state space
– space = O(V 2 )
– time for scan of vertex successors = O(V )
Incidence matrix:
This information is:
– space = O(V E)
– initial state sinit
– time for scan of vertex successors = O(E)
– goal states SG
Adjacency lists:
– applicable actions and successor states for given state
– space = O(V + E)
– time for scan of vertex successors = O(δ(v))
A succinct and efficient representation is feasible when states
are structures that store information
All are infeasible for big problems such as the 15-puzzle!
c 2015 Blai Bonet
Lecture 2
c 2015 Blai Bonet
Lecture 2
Black-box representation
Black-box representation: API
Structure of the graph is wired into functions that provide the
necessary information to explore the state space
1
typedef unsigned Action
2
3
4
This information is:
5
6
–
init():
provides the initial state
7
tells if state is goal
9
8
–
is-goal():
–
successors():
struct AbstractState {
bool is-goal()
list<pair<State, Action> > successors()
}
struct DerivedState : AbstractState {
% ... [internal representation of concrete state]
10
bool is-goal()
list<pair<State, Action> > successors()
11
provides successor states and actions
12
13
}
14
15
DerivedState init() % returns initial state
A succinct and efficient representation is feasible when states
are structures that store information
c 2015 Blai Bonet
Lecture 2
Black-box representation: example for 15-puzzle
1
2
3
struct State15Puzzle : AbstractState {
char pos[16] % pos[i] contains the ’tile’ in ith-position
char blank
% contains position of blank
c 2015 Blai Bonet
Lecture 2
Black-box representation: space vs. time trade-offs
Typically, states can be represented in two manners:
4
bool is-goal() {
for i = 0 to 15
if pos[i] != i return false
return true
}
5
6
7
8
9
– Space efficient: compact form that permits to store more states
in available memory. However, operations need to ‘decode’ and
‘encode’ states and thus consume time
10
list<pair<State, Action> > successors()
11
12
– Time efficient: operations are efficient since there is explicit
representation, but states consume more space
}
This is a space-consuming representation because it requires
17 × 8 = 136 bits per state
Best representation depends on the chosen algorithm
A state for the 15-puzzle can be stored in 64 bits (16 positions
requiring 4 bits each) . . .
c 2015 Blai Bonet
Lecture 2
c 2015 Blai Bonet
Lecture 2
Black-box representation: example for 15-puzzle
1
2
Declarative representation
State spaces are described using a high-level and general
representation languages
struct State15Puzzle2 : AbstractState {
uint32 tiles[2] % tiles occupying positions (in order)
3
bool is-goal() {
return (tiles[0] == 0x01234567) & (tiles[1] == 0x89ABCDEF)
}
4
5
6
– generality and flexibility
7
list<pair<State, Action> > successors()
8
9
Representation languages provide:
}
– abstraction that hide low-level implementation
Space-efficient representation requiring 64 bits per state
(not the best though...)
Representation languages lack:
Space efficiency doesn’t directly translate into time performance
Active research topic is to reduce gap in efficiency between
black-box and declarative representions
c 2015 Blai Bonet
– state-of-the-art efficiency on selected problems
Lecture 2
c 2015 Blai Bonet
Declarative representation: examples
Lecture 2
Propositional representation of states
States represent situations or configuration of the environment
– PSVN: production systems on fixed-length vectors of labels
A fixed collection of propositions (atoms) can be used to describe
all states in the model
– STRIPS: propositional representation of planning problems
– SAS+ : multi-valued variable representation of planning problems
Each state is described by a valuation of the atoms
c 2015 Blai Bonet
Lecture 2
c 2015 Blai Bonet
Lecture 2
Propositional representation of blocksworld
Propositional representation of blocksworld
D
A
B C
D
A
B C
Atoms:
Atoms:
Atoms:
Atoms:
Atoms:
Valuation:
Valuation:
Valuation:
Valuation:
Valuation:
Clear(A), Clear(B), Clear(C),
On(A,A), On(A,B), On(A,C),
On(B,A), On(B,B), On(B,C),
On(C,A), On(C,B), On(C,C),
OnTable(A), OnTable(B), OnTable(C)
Clear(A), Clear(B), Clear(C),
On(A,A), On(A,B), On(A,C),
On(B,A), On(B,B), On(B,C),
On(C,A), On(C,B), On(C,C),
OnTable(A), OnTable(B), OnTable(C)
State s = { Clear(A), Clear(C), On(A,B), OnTable(B), OnTable(C) }
c 2015 Blai Bonet
Lecture 2
c 2015 Blai Bonet
Propositional representation of blocksworld
Lecture 2
STRIPS
Language based on propositional variables:
B
C
A
– finite set F of propositional variables (atoms)
– an initial state I ⊆ F
– a goal description G ⊆ F
– finite set A of operators; each operator a ∈ A given by
Valuation:
Valuation:
Valuation:
Valuation:
Valuation:
Clear(A), Clear(B), Clear(C),
On(A,A), On(A,B), On(A,C),
On(B,A), On(B,B), On(B,C),
On(C,A), On(C,B), On(C,C),
OnTable(A), OnTable(B), OnTable(C)
I
precondition pre ⊆ F tells when action is applicable
I
positive effects add ⊆ F tells what becomes true
I
negative effects del ⊆ F tells what becomes false
– non-negative costs c(a) for applying actions a ∈ A
State s = { Clear(B), On(B,C), On(C,A), OnTable(A) }
c 2015 Blai Bonet
Lecture 2
c 2015 Blai Bonet
Lecture 2
STRIPS: blocksworld
STRIPS: blocksworld
A
B C
B
C
A
A
B C
B
C
A
initial state
goal state
initial state
goal state
Atoms: Clear(?x), On(?x,?y), OnTable(?x)
I = { Clear(A), Clear(C), On(A,B), OnTable(B), OnTable(C) }
Actions: Move(?x,?y,?z), MoveToTable(?x), MoveFromTable(?x,?y)
G = { On(B,C), On(C,A) }
c 2015 Blai Bonet
Lecture 2
c 2015 Blai Bonet
STRIPS: blocksworld
A
B C
initial state
Lecture 2
STRIPS: state space for blocksworld
Init
B
C
A
A
B C
C
A
B
A B C
···
goal state
A
B C
Move(A,B,C):
A
B C
···
A
B C
B
A C
C
A B
B
A C
C
A B
– pre = { Clear(A), Clear(C), On(A,B) }
– add = { Clear(B), On(A,C) }
··· ··· ···
– del = { Clear(C), On(A,B) }
B
C
A
C
A B
···
···
A B C
Goal
c 2015 Blai Bonet
Lecture 2
c 2015 Blai Bonet
Lecture 2
Example: Blocksworld in STRIPS (PDDL)
STRIPS: semantics
(define (domain Blocksworld)
(:types block)
(:predicates (clear ?x - block) (on ?x ?y - block) (ontable ?x - block))
(:action move
:parameters (?x ?y ?z - block)
:precondition (and (clear ?x) (clear ?z) (on ?x ?y))
:effect (and (not (clear ?z)) (not (on ?x ?y)) (on ?x ?z) (clear ?y)))
(:action move_to_table
:parameters (?x ?y - block)
:precondition (and (clear ?x) (on ?x ?y))
:effect (and (not (on ?x ?y)) (clear ?y) (ontable ?x)))
Problem P = hF, A, I, G, ci defines model hS, A, sinit , SG , f, ci:
– states S are all the 2n valuations to atoms in F , |F | = n
– initial state sinit assigns true to all p ∈ I and false to all p ∈
/I
– goal states are assignments satisfying the goals in G
– executable actions at state s are A(s) = {a : pre ⊆ s}
(:action move_from_table
:parameters (?x ?y - block)
:precondition (and (ontable ?x) (clear ?x) (clear ?y))
:effect (and (not (ontable ?x)) (not (clear ?y)) (on ?x ?y)))
– outcome f (s, a) defined by effects: f (s, a) = (s \ del) ∪ add
)
– non-negative action costs c(a)
(define (problem bw_3_1)
(:domain Blocksworld)
(:objects A B C - block)
(:init (clear A) (clear C) (on A B) (ontable B) (ontable C))
(:goal (and (on B C) (on C A)))
)
Model is exponential in size of problem P (e.g. blocksworld)
c 2015 Blai Bonet
Lecture 2
Summary
• Approaches for representation: explicit, black-box, declarative
• Explicit representation not generally feasible for state-space models
• Black-box reperesentation is efficient and implemented directly into
programming language
• Required API for black-box representation
• STRIPS example on the blocksworld
c 2015 Blai Bonet
Lecture 2
c 2015 Blai Bonet
Lecture 2