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## 4. An Efficient Representation And Inference Procedure For Event Logic

One might try to implement event logic using only closed intervals of the form [q; r℄, where q  r.

Such a closed interval would represent the set fpjq  p  rg of real numbers. With such closed intervals, one would define Allen's relations as follows:

```
[q1; r1℄ < [q2; r2℄
                      4
                        
                      = r1 < q2
                                    

[q1; r1℄ > [q2; r2℄
                      4
                        
                      = q1 > r2
                                    

```

[q1; r1℄ m [q2; r2℄4= r1 = q2

```
[q1; r1℄ = [q2; r2℄
                      4
                        
                      = (q1 = q2) ^ (r1 = r2)

```

| [q1; r1℄ mi [q2; r2℄   | 4 =   | q1 = r2               |
|------------------------|-------|-----------------------|
| [q1; r1℄ o [q2; r2℄    | 4 =   | q1 < q2 < r1 < r2     |
| [q1; r1℄ oi [q2; r2℄   | 4 =   | r1 > r2 > q1 > q2     |
| [q1; r1℄ s [q2; r2℄    | 4 =   | (q1 = q2) ^ (r1 < r2) |
| [q1; r1℄ si [q2; r2℄   | 4 =   | (q1 = q2) ^ (r1 > r2) |
| [q1; r1℄ f [q2; r2℄    | 4 =   | (q1 > q2) ^ (r1 = r2) |
| [q1; r1℄ fi [q2; r2℄   | 4 =   | (q1 < q2) ^ (r1 = r2) |
| [q1; r1℄ d [q2; r2℄    | 4 =   | (q1 > q2) ^ (r1 < r2) |
