1. 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 s2 s5 s8 s1 s3 s6 s9 s11 s4 s7 s10 aa ab aa bb ab aa aa ab bb

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}
Why is the transition table formalized as a relation instead of a function?

5. Games as Mathematical Structures
An n-player game is a structure with components:
S - finite set of states
I1, …, In - finite sets of actions, one for each player
u  S  I1 ... In  S - update relation
i  S - initial game state
T  S - the terminal states
G1, ..., 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.
Chess, for example, has approximately 1030 states.

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

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. Transitions 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)
noop
mark(1,3)

12. Logical Encoding init(cell(1,1,b)) init(cell(1,2,b)) init(cell(1,3,b))
init(control(x))
legal(W,mark(X,Y)) :-
true(cell(X,Y,b)) &
true(control(W))
legal(white,noop) :-
true(control(black))
legal(black,noop) :-
true(control(white)) …

13. Syntax of Relational Logic

14. Syntax of Relational Logic
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
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. Safety
A 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.

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. t s r p q
A set of rules is recursive if it contains a cycle. Otherwise, it is non-recursive.

17. Recursion
A set of rules is recursive if it contains a cycle. Otherwise, it is non-recursive. t s r p q

18. Stratified Negation
The 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:

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:

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:

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:

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:

27. 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(control(x))

33. Legality legal(W,mark(X,Y)) :- true(cell(X,Y,b)) & true(control(W))
legal(white,noop) :-
true(control(black))
legal(black,noop) :-
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) & ~open
goal(white,0) :- line(o)
goal(black,100) :- line(o)
goal(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
Propositions represent conditions that are true or false in each state of the world. Connectives express logical relationships among truth values - negation, conjunction, disjunction, and so forth. Transitions express truth in one state as a function of truth in the preceding state.

44. Relational States p p q r p q

45. State Machine p p q r p q

46. Logical Encoding 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)

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. Buttons and Lights Formalization
P ={p, q, r}
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}

49. Transitions 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)
noop
mark(1,3)

50. Buttons and Lights p q r s a b c d
Propositions represent conditions that are true or false in each state of the world. Connectives express logical relationships among truth values - negation, conjunction, disjunction, and so forth. Transitions express truth in one state as a function of truth in the preceding state.

51. Buttons and Lights Formalization
S ={s000, s001, s010, s011, s100, s101, s110, s111}
I = {a, b, c}
u(s000,a,s100) u(s000,b,s010) u(s000,c,s001)
u(s001,a,s001) u(s000,b,s010) u(s000,c,s001)
u(s010,a,s010) u(s000,b,s010) u(s000,c,s001)
u(s011,a,s011) u(s000,b,s010) u(s000,c,s001)
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. Propositional Net Components
Propositions
Connectives
Transitions p q r
Propositions represent conditions that are true or false in each state of the world. Connectives express logical relationships among truth values - negation, comjunction, disjunction, and so forth. Transitions express truth in one state as a function of truth in the preceding state.

54. Propositional Net
Propositions represent conditions that are true or false in each state of the world. Connectives express logical relationships among truth values - negation, conjunction, disjunction, and so forth. Transitions express truth in one state as a function of truth in the preceding state.

55. Markings
Propositions represent conditions that are true or false in each state of the world. Connectives express logical relationships among truth values - negation, conjunction, disjunction, and so forth. Transitions express truth in one state as a function of truth in the preceding state.

56. Inputs
Propositions represent conditions that are true or false in each state of the world. Connectives express logical relationships among truth values - negation, conjunction, disjunction, and so forth. Transitions express truth in one state as a function of truth in the preceding state.

57. Enablement ?
Propositions represent conditions that are true or false in each state of the world. Connectives express logical relationships among truth values - negation, conjunction, disjunction, and so forth. Transitions express truth in one state as a function of truth in the preceding state.

58. Update
Propositions represent conditions that are true or false in each state of the world. Connectives express logical relationships among truth values - negation, conjunction, disjunction, and so forth. Transitions express truth in one state as a function of truth in the preceding state.

59. Buttons and Lights Pressing button a toggles p.
Pressing button b interchanges p and q. p q a b
Propositions represent conditions that are true or false in each state of the world. Connectives express logical relationships among truth values - negation, comjunction, disjunction, and so forth. Transitions express truth in one state as a function of truth in the preceding state.

60. Propositional Net for Buttons and Lightsq p a
Propositions represent conditions that are true or false in each state of the world. Connectives express logical relationships among truth values - negation, conjunction, disjunction, and so forth. Transitions express truth in one state as a function of truth in the preceding state.

61. Propositional Nets and State Machines
Propositional Nets as State Machines
State Machines as Propositional Nets
One proposition per state
Only one proposition is true at each point in time
s110
Decompose states into boolean propositions, with the idea that different states correspond to different subsets of propositions. In this case, the system is in state s1 if and only if p, q, and r are true.
Propositions represent conditions that are true or false in each state of the world. Connectives express logical relationships among truth values - negation, conjunction, disjunction, and so forth. Transitions express truth in one state as a function of truth in the preceding state.
Propositions with no input connectives or transitions are called extrinsic propositions / input variables; all others are referred to as intrinsic propositions / state variables.

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
The idea of propositional net is basically a combination of logi nets and petri nets. Closely related to old notion of causal nets and similar to dynamic bayes nets with categorical probabilities.

63. Object Nets p q r
2,1
1.3
Decompose states into relations with the idea that each state corresponds to a particular combination of extensions for the relations.
We capture behavior by connecting relations with relational operators, with or without one time step of delay.

65. Relational Nets

66. Propositional Net Fragment
or1
o12
x13
x11
xr1
x12
o23
o21
or2
o22
x23
x21
xr2
x22
Parameterized Procedural Nets. View nodes. Get difficult to draw, and we need a linearization for transmission anyway.
o33
o31
or3
o32
x33
x31
xr3
x32

67. Relational Nets Decompose states into “relations”.
Use relational operators to capture behavior. s1 q s p r e c d b a h f g p q r
2,1
1.3
Decompose states into relations with the idea that each state corresponds to a particular combination of extensions for the relations.
We capture behavior by connecting relations with relational operators, with or without one time step of delay.

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
The idea of propositional net is basically a combination of logi nets and petri nets. Closely related to old notion of causal nets and similar to dynamic bayes nets with categorical probabilities.

69. Logical Encoding

70. Relational Net p q r
2,1
1.3
Decompose states into relations with the idea that each state corresponds to a particular combination of extensions for the relations.
We capture behavior by connecting relations with relational operators, with or without one time step of delay.

71. Possible Relational Net Encoding
Relational Net Fragment
Encoding
r(X,Z) :- p(X,Y) & q(Y,Z) p q r
2,1
1.3 e c d b a h f g
Decompose states into relations with the idea that each state corresponds to a particular combination of extensions for the relations.
We capture behavior by connecting relations with relational operators, with or without one time step of delay.

72. Actual Relational Net Encoding
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 r
2,1
1.3
Decompose states into relations with the idea that each state corresponds to a particular combination of extensions for the relations.
We capture behavior by connecting relations with relational operators, with or without one time step of delay.

73. Tic-Tac-Toe X O

74. Partial Propositional Net for Tic-Tac-Toe
mark(1,1)
cell(1,1,x)
cell(1,1,b)
mark(1,2)
cell(1,2,x)
cell(1,2,b)
Parameterized Procedural Nets. View nodes. Get difficult to draw, and we need a linearization for transmission anyway.
mark(1,3)
cell(1,3,x)
cell(1,3,b)

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. Syntax of Relational Logic
Object Variables: X, Y, Z
Object Constants: a, b, c
Function Constants: f, g, h
Relation Constants: p, q, r
Logical 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(control(o))
legal(black,noop) 
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)) 
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)
A propositional state machine is similar to an ordinary state machine except that we conceptualize states as sets of propositions rather than simple atomic entities, the idea being that each state corresponds to the set of propsitions true in that state.Formally, a propositional state machine is a structure <P,I,O,do,see,s1>, where P is a set of propositions, I is a set of input objects, O is a set of output objects, do is an update function that maps an input and a subset of P to a new subset of P, see is an output function that maps each subset of P into an output, and s1 is a subset of P.Here is Buttons and Lights in this formulation. We start with a structure <{p,q,r},{},{},do,see,{}>. S = {{},{p},...} do(a,{})={p} do(a,{r})={p,r} do(a,{q})={p,q} do(a,{q,r})={p,q,r} do(a,{p})={} do(a,{p,r})={r} do(a,{p,q})={q} do(a,{p,q,r})={q,r} etc. see({})=nil see({p})=nil see({q})=nil see({r})=nil see({p,q})=nil see({p,r})=nil see({q,r})=nil see({p,q,r})=nil s1 = {} goal({p,q,r})

81. Supporting Concepts row(M,W)  true(cell(M,1,W))  true(cell(M,2,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))
true(cell(1,3,W)) 
true(cell(3,1,W))
A propositional state machine is similar to an ordinary state machine except that we conceptualize states as sets of propositions rather than simple atomic entities, the idea being that each state corresponds to the set of propsitions true in that state.Formally, a propositional state machine is a structure <P,I,O,do,see,s1>, where P is a set of propositions, I is a set of input objects, O is a set of output objects, do is an update function that maps an input and a subset of P to a new subset of P, see is an output function that maps each subset of P into an output, and s1 is a subset of P.Here is Buttons and Lights in this formulation. We start with a structure <{p,q,r},{},{},do,see,{}>. S = {{},{p},...} do(a,{})={p} do(a,{r})={p,r} do(a,{q})={p,q} do(a,{q,r})={p,q,r} do(a,{p})={} do(a,{p,r})={r} do(a,{p,q})={q} do(a,{p,q,r})={q,r} etc. see({})=nil see({p})=nil see({q})=nil see({r})=nil see({p,q})=nil see({p,r})=nil see({q,r})=nil see({p,q,r})=nil s1 = {} goal({p,q,r})

82. Termination terminal  line(W) terminal  ~open
open  true(cell(M,N,b))

84. Completeness
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.

85. Playability
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.

86. Winnability
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.

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. Programme for Today State Machines Propositional Nets Relational Nets
Tabular Encoding
Logical Encoding
Close with remarks about the relationship of GGP and the long-range goals of Artificial Intelligence.

89. Game Model An n-player game is a structure with components:
S - finite set of states
I1, …, In - finite sets of actions, one for each player
l1, ..., ln - where li  Ii  S - the legality relations
u  S  I1 ... In  S - update relation
i  S - initial game state
T  S - the terminal states
G1, ..., Gn - where Gi  S - the goal relations

90. Propositional Nets and State Machines
Define states in terms of propositions.
Use propositional connectives to capture behavior.
s110 p q s r q p r
Decompose states into boolean propositions, with the idea that different states correspond to different subsets of propositions. In this case, the system is in state s1 if and only if p, q, and r are true.
Propositions represent conditions that are true or false in each state of the world. Connectives express logical relationships among truth values - negation, conjunction, disjunction, and so forth. Transitions express truth in one state as a function of truth in the preceding state.
Propositions with no input connectives or transitions are called extrinsic propositions / input variables; all others are referred to as intrinsic propositions / state variables.

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 p r q
Transition is not enabled. Two possible successors - one with true and one with p false. That is all.

95. Logical Encoding 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)
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.