game description

Game Description

General Game Playing Lecture 3

Michael Genesereth / Nat Love Spring 2006

2

Finite Synchronous Games

Finite environment Environment with finitely many states One initial state and one or more terminal states

Finite Players Fixed finite number of players Each with finitely many “actions” Each with one or more goal states

Synchronous Update All players move on all steps (some no ops) Environment changes only in response to moves

3

Example

s3

s2 s5

s6

s8

s9

s4 s7 s10

s1 s11

bbba

aa

bb ab

ba

aa

ab aa

ab

ab

aa aa

ab

aabb aa

bb

ba ab

4

Example Revisited

S ={s1, s2, s3, s4, s5, s6, s7, s8, s9, s10, s11}I1={a, b} I2={a, b}

u(s1,a,a,s2) u(s4,a,a,s7) u(s7,a,a,s10) u(s10,b,a,s9)u(s1,a,b,s3) u(s4,a,b,s3) u(s8,a,b,s5) u(s10,b,b,s11)u(s1,b,a,s4) u(s6,a,a,s5) u(s8,b,b,s11) u(s11,a,a,s11)u(s1,b,b,s1) u(s6,a,b,s9) u(s9,a,a,s8)u(s2,a,a,s5) u(s6,b,a,s7) u(s9,a,b,s11) u(s2,a,b,s3) u(s6,b,b,s3)

i=s1

T ={s3, s8, s11}G1={s8, s11} G2={s3, s7}

5

Games as Mathematical Structures

An n-player game is a structure with components:

S - finite set of statesI1, …, In - finite sets of actions, one for each player

u S I1 ... In S - update relation

i S - initial game stateT S - the terminal statesG1, ..., Gn - where Gi S - the goal relations

6

Direct Description

Good News: Since all of the games that we are considering are finite, it is possible in principle to communicate game information in the form of sets of objects and tuples of objects.

Problem: Size of description. Even though everything is finite, these sets can be large.

7

Relational Nets

8

States versus Propositions

In many cases, worlds are best thought of in terms of propositions, e.g. whether a particular light is on or off. Actions often affect a subset of propositions.

States represent all possible ways the world can be. As such, the number of states is exponential in the number of such propositions, and the action tables are correspondingly large.

Idea - represent propositions directly and describe actions in terms of their effects on indidvidual propositions rather than entire states.

9

Tic-Tac-Toe

X

O

X

10

Relational States

cell(1,1,x)cell(1,2,b)cell(1,3,b)cell(2,1,b)cell(2,2,o)cell(2,3,b)cell(3,1,b)cell(3,2,b)cell(3,3,x)control(black)

11

cell(1,1,x) cell(1,1,x) cell(1,2,b) cell(1,2,b) cell(1,3,b) cell(1,3,o) cell(2,1,b) cell(2,1,b) cell(2,2,o) cell(2,2,o) cell(2,3,b) cell(2,3,b) cell(3,1,b) cell(3,1,b) cell(3,2,b) cell(3,2,b) cell(3,3,x) cell(3,3,x) control(black) control(white)

Transitions

noop

mark(1,3)

12

init(cell(1,1,b))init(cell(1,2,b))init(cell(1,3,b))init(cell(2,1,b))init(cell(2,2,b))init(cell(2,3,b))init(cell(3,1,b))init(cell(3,2,b))init(cell(3,3,b))init(control(x))

legal(W,mark(X,Y)) :- true(cell(X,Y,b)) & true(control(W))

legal(white,noop) :- true(cell(X,Y,b)) & true(control(black))

legal(black,noop) :- true(cell(X,Y,b)) & true(control(white))

…

Logical Encoding

13

Syntax of Relational Logic

14


Vocabulary: Object Variables: X, Y, Z Object Constants: a, b, c Function Constants: f, g, h Relation Constants: p, q, r Logical Operators: ~, &, |, :-, #

Terms: Variables: X, Y, Z Object Constants: a, b, c Functional Terms: f(a), g(a,b), h(a,b,c)

Sentences: Simple Sentences: p(a,g(a,b),c) Logical Sentences: r(X,Y) :- p(X,Y) & ~q(Y)

15

SafetyA rule is safe if and only if every variable in the head appears in some positive subgoal in the body.

Safe Rule:

Unsafe Rule:

In GDL, we require all rules to be safe.

r(x,y):−p(x,y)∧q(y,z)

r(x,z) :−p(x,y)∧q(y,x)

16

Dependency Graph

The dependency graph for a set of rules is a directed graph in which (1) the nodes are the relations mentioned in the head and bodies of the rules and (2) there is an arc from a node p to a node q whenever p occurs with the body of a rule in which q is in the head.

r(x,y):−p(x,y),q(x,y)

s(x,y):−r(x,y)

s(x,z):−r(x,y),t(y,z)

t(x,z) :−s(x,y),s(y,x)

A set of rules is recursive if it contains a cycle. Otherwise, it is non-recursive.

r

p q

s

t

17

Recursion

A set of rules is recursive if it contains a cycle. Otherwise, it is non-recursive.

r(x,y):−p(x,y),q(x,y)

s(x,y):−r(x,y)

s(x,z):−r(x,y),t(y,z)

t(x,z) :−s(x,y),s(y,x)

r

p q

s

t

18

Stratified NegationThe negation in a set of rules is said to be stratified if there is no recursive cycle in the dependency graph involving a negation.

Stratified Negation:

Negation that is not stratified:

In GDL, we require that all negations be stratified.

19

Extensional and Intensional Relations

Database applications start with a partial database, i.e. sentences for some relations (extensional relations) and not others (intensional relations). Rules are then written to define the intensional relations in terms of the extensional relations.

rules

Extensional Intensional

Given an extensional database and a set of rules, we can obtain the database’s closure as follows.

20

Example

Database applications start with a partial database, i.e. sentences for some relations (extensional relations) and not others (intensional relations). Rules are then written to define the intensional relations in terms of the extensional relations.

Given an extensional database and a set of rules, we can obtain the database’s closure as follows.

21

Single Rule

The value of a single non-recursive rule on a database D is the set of all rule heads obtained by consistently substituting ground terms from D for variables in such a way that the substituted subgoals are all in D.

Sample Rule:

Database:

Extension:

q(x,z):−p(x,y),p(y,z)

{p(a,b),p(b,c),p(c,d)}

{q(a,c),q(b,d)}

22

Multiple Rules

The value of a set of rules with a common relation on a database D is the union of the values on the individual rules.

Sample Rules:

Sample Database:

Value:

q(x,y) :−p(x,y)

q(x,z):−p(x,y),p(y,z)


{q(a,b),q(b,c),q(c,d),q(a,c),q(b,d)}

23

Multiple Relations

The value of a set of non-recursive rules with different head relations is obtained by evaluating rules in order in which their head relations appear in the corresponding dependency graph.

Sample Rules:

Value Computation:

s(x,y):−p(x,y),q(x,y)

t(x,z) :−s(x,y),r(y,z)

{p(a,b),p(b,c),q(b,c),r(c,d)}

{s(b,c)}

{t(a,c)}

24

Recursion

To compute the value of a recursive rule, start with the empty relation. Compute the value using multiple rule computation. Iterate till no new tuples are added.

Sample Rules:

Value Computation:

q(x,y) :−p(x,y)

q(x,z):−q(x,y),q(y,z)


{q(a,b),q(b,c),q(c,d)}

{q(a,c),q(b,d)}

{q(a,d}

25

Negation

There are various ways to compute the value of negative rules.

In classical negation, a negation is true only if the negated sentence is known to be false (i.e. there must be rules concluding negated sentences). This is the norm in computational logic systems. In GDL, we do not have such rules.

In negation as failure, a negation is true if and only if the negated sentence is not known to be true. This is the norm in database systems.

26

Negation as Failure Example

Definition:

Value Computation:

childless(x):−person(x),¬father(x),¬mother(x)

{person( joe),person(bill), father(joe)}

{childless(bill)}

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Game Description Language

28

Game-Independent Vocabulary

Object Constants: 0, 1, 2, 3, … - numbers

Relation Constants: role(player) init(proposition) true(proposition) next(proposition) legal(player,action) does(player,action) goal(proposition) terminal

29

Tic-Tac-Toe Vocabulary

Object constants: white, black - players x, o, b - marks

Function Constants: mark(number,number) --> action cell(number,number,mark) --> proposition control(player) --> proposition

RelationConstants: row(number,player) column(number,player) diagonal(player) line(player) open

30

Extensional and Intensional Relations

Extensional Relations: does(player,action) true(proposition)

Intensional Relations: role(player) init(proposition) legal(player,action) next(proposition) goal(proposition,score) terminal

31

Roles

role(white)role(black)

32

Initial State

init(cell(1,1,b))init(cell(1,2,b))init(cell(1,3,b))init(cell(2,1,b))init(cell(2,2,b))init(cell(2,3,b))init(cell(3,1,b))init(cell(3,2,b))init(cell(3,3,b))init(control(x))

33

Legality

legal(W,mark(X,Y)) :- true(cell(X,Y,b)) & true(control(W))

legal(white,noop) :- true(cell(X,Y,b)) & true(control(black))

legal(black,noop) :- true(cell(X,Y,b)) & true(control(white))

34

Physics

next(cell(M,N,x)) :- does(white,mark(M,N))

next(cell(M,N,o)) :- does(black,mark(M,N))

next(cell(M,N,Z)) :- does(W,mark(M,N)) & true(cell(M,N,Z)) & Z#b

next(cell(M,N,b)) :- does(W,mark(J,K)) & true(cell(M,N,b)) & (M#J | N#K)

next(control(white)) :- true(control(black))

next(control(black)) :- true(control(white))

35

Supporting Concepts

row(M,W) :- diagonal(W) :- true(cell(M,1,W)) & true(cell(1,1,W)) & true(cell(M,2,W)) & true(cell(2,2,W)) & true(cell(M,3,W)) true(cell(3,3,W))

column(N,W) :- diagonal(W) :- true(cell(1,N,W)) & true(cell(1,3,W)) & true(cell(2,N,W)) & true(cell(2,2,W)) & true(cell(3,N,W)) true(cell(3,1,W))

line(W) :- row(M,W)line(W) :- column(N,W)line(W) :- diagonal(W)

open :- true(cell(M,N,b))

36

Goals and Termination

goal(white,100) :- line(x)goal(white,50) :- ~line(x) & ~line(o) & ~opengoal(white,0) :- line(o)

goal(black,100) :- line(o)goal(white,50) :- ~line(x) & ~line(o) & ~opengoal(white,0) :- line(x)

terminal :- line(W)terminal :- ~open

37

More Tedious Details

38

No Built-in Assumptions

What we see:

next(cell(M,N,x)) :- does(white,mark(M,N)) & true(cell(M,N,b))

What the player sees:

next(welcoul(M,N,himenoing)) :- does(himenoing,dukepse(M,N)) & true(welcoul(M,N,lorenchise))

39

Knowledge Interchange Format

Knowledge Interchange Format is a standard for programmatic exchange of knowledge represented in relational logic.

Syntax is prefix version of standard syntax.Some operators are renamed: not, and, or.Case-independent. Variables are prefixed with ?.

r(X,Y) <= p(X,Y) & ~q(Y)

(<= (r ?x ?y) (and (p ?x ?y) (not (q ?y))))(<= (r ?x ?y) (p ?x ?y) (not (q ?y)))

Semantics is the same.

40

Agent Communication Language

Start Message

(start id role (s1 … sn) startclock playclock)

Play Message

(play id (a1 ... ak))

Stop Message

(stop id (a1 ... ak))

42

Propositional Nets

43

Buttons and Lights

p q

a b

r

c

44

Relational States

pqr

pq

p

45

State Machine

pqr

pq

p

46

init(q)

legal(robot,a)legal(robot,b)legal(robot,c)

next(p) :- does(robot,a) & -true(p)next(p) :- does(robot,b) & true(q)next(p) :- does(robot,c) & true(p)next(q) :- does(robot,a) & true(q)next(q) :- does(robot,b) & true(p)next(q) :- does(robot,c) & true(q)next(r) :- does(robot,a) & true(r)next(r) :- does(robot,b) & true(r)next(r) :- does(robot,c) & true(q)

goal :- true(p) & -true(q) & true(r)term :- true(p) & -true(q) & true(r)

Logical Encoding

47

Buttons and Lights Formalization

S ={s1, s2, s3, s4, s5, s6, s7, s8}I = {a, b, c}

u(s1,a,s5) u(s000,b,s010) u(s000,c,s001)u(s2,a,s2) u(s000,b,s010) u(s000,c,s001)u(s3,a,s3) u(s000,b,s010) u(s000,c,s001)u(s4,a,s4) u(s000,b,s010) u(s000,c,s001)

I = s1

T = {s8}G = {s8}

48


P ={p, q, r}I = {a, b, c}


I = s1

T = {s8}G = {s8}

49


Transitions

noop

mark(1,3)

50

Buttons and Lights

p q

a b

r s

c d

51


S ={s000, s001, s010, s011, s100, s101, s110, s111}I = {a, b, c}


I = s0

T = {sF}G = {sF}

52

States versus Features

In many cases, worlds are best thought of in terms of features, e.g. red or green, left or right, high or low. Actions often affect subset of features.

States represent all possible ways the world can be. As such, the number of states is exponential in the number of “features” of the world, and the action tables are correspondingly large.

Idea - represent features directly and describe how actions change individual features rather than entire states.

53

Propositions

Connectives

Transitions

Propositional Net Components

p q r

54

Propositional Net

55

Markings

56

Inputs

57

Enablement

?

58

Update

59

Buttons and Lights

Pressing button a toggles p.Pressing button b interchanges p and q.

p q

a b

60

Propositional Net for Buttons and Lights

q

p

a

61

Propositional Nets as State Machines

State Machines as Propositional Nets One proposition per state Only one proposition is true at each point in time

Propositional Nets and State Machines

s110

62

Comparison

Propositional Nets vs State Machines Expressively equivalent and interconvertible State Machines can be exponentially larger e.g. state machine for Tic-Tac-Toe has 5478 states propositional net has 45 propositions

Propositional Nets vs Petri Nets Propositional Nets are computable (equivalent to Petri nets with finitely many

tokens) Propositional Nets are composable without revealing inner details of components

63

p

q

r2,1 1.3

Object Nets

65

Relational Nets

66

Propositional Net Fragment

o13

o11

or1o12

o23

o21

or2o22

o33

o31

or3o32

x13

x11

xr1x12

x23

x21

xr2x22

x33

x31

xr3x32

67

Decompose states into “relations”.

Use relational operators to capture behavior.

s p q r

€

a b

b c

d e

€

a

d

€

a b c

a d es1

p

q

r2,1 1.3

ec

db

da

hf

he

gd hc

gb

ga

Relational Nets

68

Comparison

Relational Nets vs Propositional Nets Expressively equivalent and interconvertible Number of Tuples = Number of Propositions Fewer Relations than propositions Fewer connectives

Relational Nets vaguely related to RMDPs

69

Logical Encoding

70

p

q

r2,1 1.3

Relational Net

71

Relational Net Fragment

Encoding

r(X,Z) :- p(X,Y) & q(Y,Z)

p

q

r2,1 1.3

Possible Relational Net Encoding

ec

db

da

hf

he

gd hc

gb

ga

72

Relational Net Fragment

Encoding without delay Encoding with delay

true(r(X,Z)) :- next(r(X,Z)) :- true(p(X,Y)) & true(p(X,Y)) & true(q(Y,Z)) true(q(Y,Z))

p

q

r2,1 1.3

Actual Relational Net Encoding

73

Tic-Tac-Toe

X

O

X

74

Partial Propositional Net for Tic-Tac-Toe

cell(1,2,b)

mark(1,2)

cell(1,2,x)

cell(1,1,b)

mark(1,1)

cell(1,1,x)

cell(1,3,b)

mark(1,3)

cell(1,3,x)

75

Logical Description

Direct encoding in relational logic:next(cell(1,1,x)) <= does(mark(1,1)) & true(cell(1,1,b))

Use of variables to compact description:next(cell(M,N,x)) <= does(mark(M,N)) & true(cell(M,N,b))

Game-specific “views” / “macros”:row(M,W) <= true(cell(M,1,W)) & true(cell(M,2,W)) & true(cell(M,3,W))

76


Object Variables: X, Y, ZObject Constants: a, b, cFunction Constants: f, g, hRelation Constants: p, q, rLogical Operators: ~, &, |, :-, distinct

Terms: X, Y, Z, a, b, c, f(a), g(a,b), h(a,b,c)Relational Sentences: p(a,b)Logical Sentences: r(X,Y) <= p(X,Y) & ~q(Y)

An expression is ground iff it contains no variables.The Herbrand base is the set of all ground relational sentences.

77

Legality

legal(W,mark(X,Y)) true(cell(X,Y,b)) true(control(W))

legal(white,noop) true(cell(X,Y,b)) true(control(o))

legal(black,noop) true(cell(X,Y,b)) true(control(x))

78

Update

next(cell(M,N,x)) does(white,mark(M,N)) true(cell(M,N,b))

next(cell(M,N,o)) does(black,mark(M,N)) true(cell(M,N,b))

next(cell(M,N,W)) true(cell(M,N,W)) distinct(W,b)

next(cell(M,N,b)) does(W,mark(J,K)) true(cell(M,N,b)) (distinct(M,J) | distinct(N,K))

79

Update (continued)

next(control(x)) true(control(o))

next(control(o)) true(control(x))

80

Goals

goal(white,100) line(x)goal(white,0) line(o)goal(black,100) line(o)goal(white,0) line(x)

line(W) row(M,W)line(W) column(N,W)line(W) diagonal(W)

81

Supporting Concepts

row(M,W) true(cell(M,1,W)) true(cell(M,2,W)) true(cell(M,3,W))

column(N,W) true(cell(1,N,W)) true(cell(2,N,W)) true(cell(3,N,W))

diagonal(W) true(cell(1,1,W)) true(cell(2,2,W)) true(cell(3,3,W))

diagonal(W) true(cell(1,3,W)) true(cell(2,2,W)) true(cell(3,1,W))

82

Termination

terminal line(W)terminal ~open

open true(cell(M,N,b))

84

Of necessity, game descriptions are logically incomplete in that they do not uniquely specify the moves of the players.

Every game description contains complete definitions for legality, termination, goalhood, and update in terms of the primitive moves and the does relation.

The upshot is that in every state every player can determine legality, termination, goalhood and, given a joint move, can update the state.

Completeness

85

A game is playable if and only if every player has at least one legal move in every non-terminal state.

Note that in chess, if a player cannot move, it is a stalemate. Fortunately, this is a terminal state.

In GGP, we guarantee that every game is playable.

Playability

86

A game is strongly winnable if and only if, for some player, there is a sequence of individual moves of that player that leads to a terminating goal state for that player.

A game is weakly winnable if and only if, for every player, there is a sequence of joint moves of the players that leads to a terminating goal state for that player.

In GGP, every game is weakly winnable, and all single player games are strongly winnable.

Winnability

87

Comparison to Extensive Normal Form

In Extensive Normal Form, a game is modeled as a tree with actions of one player at each node.

In State Machine Form, a game is modeled as a graph and players’ moves are all synchronous.

In GGP, a game must be described formally. While ENF and SMF are expressively equivalent for finite games, SMF descriptions are simpler.

Some players may create game trees from game descriptions; however, searching game graphs can be more efficient.

88

State Machines

Propositional Nets

Relational Nets

Tabular Encoding

Logical Encoding

Programme for Today

89

Game Model

An n-player game is a structure with components:

S - finite set of statesI1, …, In - finite sets of actions, one for each playerl1, ..., ln - where li Ii S - the legality relationsu S I1 ... In S - update relationi S - initial game stateT S - the terminal statesG1, ..., Gn - where Gi S - the goal relations

90

s

Define states in terms of propositions.

Use propositional connectives to capture behavior.

Propositional Nets and State Machines

q

p

r

qp

r

s110

91

Markings

A marking for a propositional net is a function from the propositions to boolean values.

m: P {true,false}

92

Acceptability

A marking is acceptable iff it obeys the logical properties of all connectives.

Negation with input x and output y: m(y)=true m(x)=false

Conjunction with inputs x and y and output z: m(z)=true m(x)=true m(y)=true

Disjunction with inputs x and y and output z: m(z)=true m(x)=true m(y)=true

93

Update

A transition is enabled by a marking m iff all of its inputs are marked true.

The update for a marking m is the partial marking m* that assigns true to the outputs of all transitions enabled by m and false to the outputs of all other transitions.

A successor m’ of a marking m is any complete, acceptable marking consistent with m*.

94

Example

q

p

r

95


Logical Encoding

noop

mark(1,3)

96

Arguing for Evaluation Function

Assume evaluation function f partitions states into n categories S1, …, Sn.

Consider probabilities p1, …, pn of winning in each category. (More generally, consider expected utilities u1, …, un.) Use these probabilities (utilities) as evaluation function values for the corresponding categories.

Choosing a move that leads to a category with maximal value maximizes chances of winning.

97

Evaluation Functions

An ideal evaluation function is one that reflects the expected utility of each state. (In the case of win-lose games, it is the probability of winning.)

For each terminal state, it is the payoff in that state.

For each nonterminal state, it is the maximum of the expected utilities of the legal actions in that state. (The expected utility of an action in a state is the sum of the expected values of the states resulting from that action weighted by probabilities of the opponents’ actions.)

98

Evaluation Functions

Choosing moves that maximize expected value

NB: Different priors possible. Random is common.

game description

Documents

b cell3

b cell2

b cell1

b truecontrolwlegalwhite

b truecontrolblacklegalblack

x cell3

x cell1

s3 us6