Example 4.2 or the blocks world e ample, we produce the encoding for plans with two states. The schemata .      yield the following formulae for e assume that
.

the facts ontableA, clearA and onAB are observable.

| 8 ontable A    | 1 1 1          | 1 2      |     |
|----------------|----------------|----------|-----|
| 8 ontable A    | 1 2  ↑         | l l      |     |
| 8 clccrA       | 8 ontable A    | 2 1 1    | 2 2 |
| 8 clccrA t     | 8 on A B       | 22       | 2 l |
| S ON A B T     | 8 clccrA       | 11111111 | l 2 |
| 8 anAB →       | 1 2  →         | l l      |     |
| 2 1 1          | 2 2            |          |     |
| S clearA       | 8, on to ble A | 22       | 2 1 |

1 1 l 2

 $\begin{array}{ccc}2&1&&2&2\\ \\ \\ 1&1&&1&2\\ \\ 2&1&&2&2\end{array}$  . 
Truth value assignments that satisfy these formulae represent transition functions of con ditional plans. The remaining formulae describe how the plan determines which operators are e ecuted. or schemata . and . yield the following formulae.

0

l ,0 1,t →     2,t      2,t → l, t
chemata . and . that describe state transitions yield for the following formulae.

' Cl and 0
s t s t

| s t   |
|-------|
| s t   |

s t s t

s,clearA   clearAt     s  s'    →   s',t+1 s,clearA     clearAt     s  s'     →   s',t+1 δ, on table λ = ontable At = s = s' = → s', t+l δ, on table λ  s, om AB e e' e - - - s', t +1 8 on AB on AB f
chemata .  and  .  yield for

$$\mathbf{\Sigma}_{0}$$

and the following formulae.

l , t 0, 2 2 , t st on A Be clear At → o,t 0 t →
0, 1 0, s onBA   clearB 
1 t l, s s t 2 t 2.s

| s,t ontableA clearB e →  2,t   |
|--------------------------------|
| ontableBe clearAe →  3,t       |
| s t                            |

3 3, s l t 1, 2 2 t 2, 2

| 2, 1   | l t   |
|--------|-------|
| 3, 1   | l , t |

2 , t 3,2 2, t

$$\begin{array}{l}{\lceil\bot\rceil}\end{array}$$

n this formali ation conditional plans are determined by the valuation of variables i i i i i i i i i i i and - i i . To make e plicit the meaning of plans there are at least two possibilities.   ive a formal definition of conditional plans for e ample as programs in a simple programming language with conditionals and iteration, or give a mechanism for e ecuting the plans implicitly represented by the valuations of the afore mentioned variables.

e choose the latter alternative because it is more straightforward.

o assume we have a valuation 
→
that assigns truth values to the set of propositional variables that implicitly represent a conditional plan, and the parameter man that is the length of the plan e  ecution. The procedure in   igure   e  ecutes the implicitly