CS344 : Introduction to Artificial Intelligence

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

2. Robotic Blocks World 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 Robot hand Robot hand C A B C A B unstack(C), putdown(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 21 = ∑ P(UC DC PB TBA PC TCB, s6 = si) i = 0 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 )

9. Back to HMM

10. r q A Simple HMM a: 0.2 a: 0.3 b: 0.2 b: 0.1 a: 0.2 b: 0.1 b: 0.5 a: 0.4

11. The forward probabilities of “bbba”