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sdp-simple-solver.py
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sdp-simple-solver.py
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from random import randint
from types import FunctionType
###########
# helpers #
###########
# postulate argmax : {A : Set } → (f : A → Val) → List A → A
def argmax(A: list, f: FunctionType) -> int:
if len(A) == 0: return None
b = A[0]
if len(A) == 1: return b
for a in A:
if f(a) > f(b): b = a
return b
def print_output(optPolSeq):
print('\n')
print("t".ljust(10, " "), "Policy t")
print('-'*50)
for i, p in enumerate(optPolSeq):
print(str(i).ljust(10, " "),p)
print('\n')
#####################################################################################################
# SDP class #
# - X: states -> list[state] #
# or #
# dict(key = t, value = list[state]) #
# - Y: controls -> dict(key = state, value = list[state]) #
# or #
# dict(key = t, value = dict(key = state, value = list[state])) #
# - next: transition function -> function(t, state, control) : X[t+1] #
# - reward -> dict(key = (t, x, y, next(t,x,y), value = reward_value)) #
#####################################################################################################
class SDP:
def __init__(self, X, Y, next, reward):
self.X = X
self.Y = Y
self.next = next
self.reward = reward
# allow polymorhism handling between dict and list
self.X_is_list = type(self.X) is list
# -- value of a policy sequence through Bellman's equation where
# measure = sum
# fmap = List
def val(self, t, ps, x) -> int:
if len(ps) == 0: return 0
p = ps[0] # fetch last policy
n = self.next(t, x, p[x]) # fetch next value
return self.reward[(t, x, p[x], n)] + self.val(t+1, ps[1:], n)
# optExt : {t n : Nat} → PolicySeq (suc t) n → Policy t
def optExt(self, t, ps) -> dict:
p = {}
if self.X_is_list: X = self.X
else: X = self.X[t]
for x in X:
if self.X_is_list: Y = self.Y[x]
else: Y = self.Y[t][x]
p[x] = argmax(Y, lambda y : self.reward[(t, x, y, self.next(t, x, y))] + self.val(t+1, ps, self.next(t, x, y)))
return p
# backward induction
# bi t (suc n) = let ps = bi (suc t) n in optExt ps :: ps
def bi(self, t, n) -> list[dict]:
if n == 0: return []
ps = self.bi(t + 1, n - 1)
return [self.optExt(t, ps)] + ps
#####################################################################################################################
# Generation Dilemma SDP class #
# alpha -> probability of staying in GU (1-alpha -> prob of transitioning to B) #
# beta -> probability of staying in BT (1-beta -> prob of transitioning to GS) #
#####################################################################################################################
class GenerationDilemma:
def __init__(self, alpha, beta, steps) -> None:
self.alpha = alpha
self.beta = beta
self.steps = steps
self.X = ['GU', 'GS', 'BT', 'B']
self.Y = {
'GU': ['a','b'],
'GS': [],
'BT': [],
'B' : []
}
self.reward = {}
for t in range(steps):
self.reward[(t, 'GU', 'a', 'GU')] = 3
self.reward[(t, 'GU', 'a', 'B' )] = -5
self.reward[(t, 'GU', 'b', 'BT')] = -1
self.reward[(t, 'BT', None, 'GS')] = 5
self.reward[(t, 'BT', None, 'BT')] = -1
self.reward[(t, 'B', None, 'B')] = -5
self.reward[(t, 'GS', None, 'GS')] = 5
def next(self, t, x, y) -> str:
match x,y:
case 'GU', 'a' : return {True: 'GU', False: 'B'}[randint(1,100) <= (self.alpha * 100)]
case 'GU', 'b' : return 'BT'
case 'BT', _ : return {True: 'BT', False: 'GS'}[randint(1,100) <= (self.beta * 100)]
case 'B', _ : return 'B'
case 'GS', _ : return 'GS'
case _,_ : raise Exception('invalid tuple')
#####################################################################################################################
# Cylinder SDP class #
# SDP cilinder problem from "Sequential decision problems, dependently typed solutions" #
# Account for non-viable and unreachable states in state space and decision making #
# By construction, if a tuple (t, x = X[t], y = Y[t][x]) is found, then it must be a viable and reachable one #
#####################################################################################################################
class Cylinder:
global all_states
all_states = ['a', 'b', 'c', 'd', 'e']
def __init__(self) -> None:
self.X = {
0: ['b', 'c', 'd', 'e'],
1: ['c', 'd', 'e'],
2: ['d', 'e'],
3: ['e'],
4: ['d', 'e'],
5: ['c', 'd', 'e'],
6: ['b', 'c', 'd', 'e'],
7: all_states,
}
# 'A' -> ahead, 'R' -> right, 'L' -> left
self.Y = {
0: {
'b': ['R'],
'c': ['A', 'R'],
'd': ['A', 'R', 'L'],
'e': ['A', 'L']
},
1: {
'c': ['R'],
'd': ['A', 'R'],
'e': ['A', 'L']
},
2: {
'd': ['R'],
'e': ['A']
},
3: {
'e': ['A', 'L']
},
4: {
'd': ['A', 'R', 'L'],
'e': ['A', 'L']
},
5: {
'c': ['A', 'R', 'L'],
'd': ['A', 'R', 'L'],
'e': ['A', 'L']
},
6: {
'b': ['A', 'R', 'L'],
'c': ['A', 'R', 'L'],
'd': ['A', 'R', 'L'],
'e': ['A', 'L']
},
7: {
'a': ['A', 'R'],
'b': ['A', 'R', 'L'],
'c': ['A', 'R', 'L'],
'd': ['A', 'R', 'L'],
'e': ['A', 'L']
}
}
# postulate reward : (t : Nat) → (x : X t) → Y t x → X (suc t) → Val
# in this case (t,x,y) determines a unique next element (next t x y) since we are not using probabilities (M X = list(X)).
# we fix random rewards for each tuple (t,x,y,next(t,x,y))
self.reward = {}
for t in range(len(self.Y)):
for x in self.X[t]:
for y in self.Y[t][x]:
n = self.next(t, x, y)
self.reward[(t, x, y, n)] = randint(1,100)
def next(self, t, x, y) -> str:
i = all_states.index(x)
if y == 'R': return all_states[i+1]
if y == 'L': return all_states[i-1]
return x
#########
# TESTS #
#########
# generation dilemma instance
print('# GENERATION DILEMMA')
gen_steps = 5
gen_instance = GenerationDilemma(alpha = 0.5, beta = 0.5, steps = gen_steps)
gen_sdp = SDP(gen_instance.X, gen_instance.Y, gen_instance.next, gen_instance.reward)
print_output(gen_sdp.bi(0, gen_steps))
#######################################################################################################
# cylinder SDP instance
print('# CYLINDER PROBLEM')
cylinder_instance = Cylinder()
cylinder_sdp = SDP(cylinder_instance.X, cylinder_instance.Y, cylinder_instance.next, cylinder_instance.reward)
print_output(cylinder_sdp.bi(0, 8))