Game AI

Game AI Prof. Christian Bauckhage

outline lecture 16 recap reactive behavior programing reduction of DFAs composition of DFAs non-deterministic FAs probabilistic FAs summary

recap hierarchy of behaviors strategic long term goals

tactical mid term goals

resources

contextual / world knowledge

information

reasoning / planning

reactive short term goals

computation time

deterministic finite state machine / automaton

⇔ a tuple A = S, Σ, δ, s0 , F




where S ≡ finite set of states Σ ≡ alphabet of input symbols δ : S × Σ → S ≡ state transition function s0 ∈ S ≡ initial state F ⊆ S ≡ set of final states

example: behavior of a guardian

 S = guard, charge, attack, retreat, flee, die, celebrate  F = die s0 = guard die

guard

charge

attack

celebrate

retreat

flee

example: behavior of a guardian

 Σ = HR, Hr, hr, aD, oD where H : agent’s health > 20% h : agent’s health < 20% R : opponent in view r : opponent in range aD : agent is dead oD : opponent is dead

note

symbols in Σ reflect current game entity attributes i.e. aspects of the current overall game state ⇔ we assume that there is a function that generates symbols from game states

example: behavior of a guardian

δ:S×Σ→S

aD aD oD

HR oD

charge

aD

HR

attack Hr

HR

HR

hr

Hr R H

guard

die

Hr

oD

hr

Hr

celebrate aD

oD

oD

retreat

flee

hr

hr

aD

observe

δ is typically implemented using look-up tables, e.g.

guard charge attack .. .

HR charge charge charge .. .

Hr error attack attack .. .

hr error retreat flee .. .

aD die die die .. .

oD guard guard celebrate .. .

note

in the context of game AI, state machines actually produce output for each state there are scripts that update NPC attributes, render animations, etc. ⇒ we should better talk about finite state transducers

reactive behavior programing

observe

DFAs for real games may become very large and complex ⇒ we might want to minimize DFAs

observe

DFAs for real games may become very large and complex ⇒ we might want to minimize DFAs

in fact, for every DFA, there exists a unique minimal DFA

observe

DFAs for real games may become very large and complex ⇒ we might want to minimize DFAs

in fact, for every DFA, there exists a unique minimal DFA two classes of states can always be removed / merged unreachable states indistinguishable / equivalent states

notation

Σ∗ ≡ set of all words over Σ  ∈ Σ ≡ empty character w = x : xs ∈ Σ∗ ≡ word where x ∈ Σ, xs ∈ Σ∗ δ∗ : S × Σ∗ → S ≡ transition function such that  δ(s, x), if xs =  ∗
δ (s, x : xs) = ∗ δ δ(s, x), xs , otherwise

unreachable state ⇔ state si is unreachable, if there is no word w ∈ Σ∗ such that si = δ∗ (s0 , w) a

s1 a

s0

b b

s2 a

s3

c

s4

equivalent states ⇔ two states si and sj are equivalent si ∼ sj , if ∀ w ∈ Σ∗ : δ∗ (si , w) ∈ F ⇔ δ∗ (sj , w) ∈ F

equivalent states ⇔ two states si and sj are equivalent si ∼ sj , if ∀ w ∈ Σ∗ : δ∗ (si , w) ∈ F ⇔ δ∗ (sj , w) ∈ F

⇔ the machine does the same when started in either state

example

aD

aD

aD

aD oD

HR

guard

oD

charge

die

Hr aD

HR

guard

attack Hr

HR

oD

HR oD

charge

attack

hr

Hr

celebrate aD

oD

hr

⇔ oD

retreat

flee

hr

hr

Hr

Hr R H

HR

celebrate

retreat aD hr

oD

oD hr

HR hr

aD

Hr

HR

oD

die

Hr HR

aD

note



there are efficient algorithms O n log n , n = |S| to identify equivalent states (Hopcroft, 1971) ⇒ DFA / FSM minimization is solved and allows for optimization of behavior programs

observe

more comprehensive behaviors may be realized by combining simple DFAs into more complex ones typical compositions include . . .

cascaded composition

A1

A2

parallel composition

A1

A2

feedback composition

A

observe

there are numerous ideas of how to generalize the idea of DFAs / DFSMs an important one is the following . . .

non-deterministic automaton / state machine (NFA / NFSM) ⇔ a FSM where δ is replaced by ∆ : S × Σ → 2S ⇔ when observing a symbol x in state s, the machine may transit to one out of several states charge a

guard a

retreat

note

“one can show” that NFAs are equivalent to DFAs

probabilistic automaton / state machine ⇔ a generalization of NFSMs

note

instead of a function ∆ : S × Σ → 2S

we may think of a relation ∆⊆S×Σ×S

example function ∆(s1 , a) = {s2 , s3 }

relation  ∆ = (s1 , a, s2 ), (s1 , a, s3 ),

∆(s1 , b) = {s1 , s2 , s3 }

(s1 , b, s1 ), (s1 , b, s2 ), (s1 , b, s3 ),

∆(s2 , a) = {s2 , s3 , s4 } .. .

(s2 , a, s2 ), (ss , a, s3 ), (s3 , a, s4 ), . . .

note

instead of a relation ∆⊆S×Σ×S we may think of an indicator function  τ : S × Σ × S → 0, 1 where τ s, x, s


0

=


1, if s, x, s 0 ∈ ∆ 0, otherwise

note

for every x ∈ Σ, we may define a (transition) matrix  
 Mx s 0 s = τ s, x, s 0 ∈ 0, 1

note

for every x ∈ Σ, we may define a (transition) matrix  
 Mx s 0 s = τ s, x, s 0 ∈ 0, 1

we may replace Mx by a column stochastic matrix Px where X      Px s 0 s ∈ 0, 1 and Px s 0 s = 1 s0

observe

states of a probabilistic automaton are stochastic vectors v where X   vs ∈ 0, 1 and vs = 1 s

observe

states of a probabilistic automaton are stochastic vectors v where X   vs ∈ 0, 1 and vs = 1 s

for instance, when starting in state v and seeing symbols a, b, c, the automaton will be in state v 0 = Pc Pb Pa v

note

PFAs ⇔ inhomogeneous discrete time Markov chains

summary

we now know about

reduction and composition of DFAs non-deterministic FAs probabilistic FAs