1. CS344 : Introduction to Artificial Intelligence
Pushpak Bhattacharyya CSE Dept., IIT Bombay
Lecture 22- Forward probability and Robot Plan

2. Robotic Blocks World on(B, table) on(A, table) on(C, A) hand empty
Robot hand
Robot hand A C B A B C
START
GOAL
on(B, table)
on(A, table)
on(C, A)
hand empty
clear(C)
clear(B)
on(C, table)
on(B, C)
on(A, B)
hand empty
clear(A)

3. Mapping the problem to probabilistic framework
Exhaustively enumerate the states
Enumerate the operators
Define probabilities of transition
P(Ok,sj|si) {probability of going from state si to sj with the output Ok which can be a robotic action}

4. State Transition C A B A B C C B B A C A unstack(C), putdown(C) START
Robot hand
Robot hand C A B
unstack(C),
putdown(C) A B C
START
pickup(B),
stack(B,A)
pickup(C),
stack(C,B) C
Robot hand B B A C A
GOAL

5. States for Blocks world problem
Total 22 states
Hand – empty
No column (1 state)
2-blk-column (6 states)
3-blk-column (6 states)
Hand – holding
Block A in Hand – no column (1 state), 2-blk-column (2 states)
Block B in Hand – no column (1 state), 2-blk-column(2 states)
Block C in Hand – no column (1 state), 2-blk-column(2 states)

6. State space and operators
State space = {s1,s2, … , s22}
Operators
pick up A (PA), pick up A (PB), pick up A (PC)
put down(DA), put down(DB), put down(DC)
stack(x , y) – total 6 operators i.e. TAB, TAB, TCB, TBC, TCA, TAC
unstack (x) – UA, UB, UC

7. Probabilistic Automaton
This gives a probabilistic automaton where probability values are specified between every states for each operator.
We need to learn total 22C2 (states) * 15 (operators) different probability values, e.g., P(PA, s2 | s1) = 0.3, P(DC, s5| s2), …

8. Formula for Operator Sequence Probability
Forward Algorithm to calculate operator sequence probability.
e.g. seq = UC DC PB TBA PC TCB
P(seq) = P(UC DC PB TBA PC TCB )
(marginalization, probability of seq
with 6th state = si )
= ∑ P(UC DC PB TBA PC TCB, s6 = si) 21
i = 0

9. Back to HMM

10. A Simple HMM r q a: 0.2 a: 0.3 b: 0.2 b: 0.1 a: 0.2 b: 0.1 b: 0.5

11. The forward probabilities of “bbba”
Time Ticks 1 2
INPUT ε b
bb bbb bbba
1.0
0.2
0.0
0.1
P(w,t)
0.3