Situations from events to proofs 1 Introduction

(P 1 ) e < e ′ implies not e ′ < e
(P 2 ) e < e ′ < e ′′ implies e < e ′′
(P 3 ) e ○ e
(P 4 ) e ○ e ′ implies e ′ ○ e
(P 5 ) e < e ′ implies not e ○ e ′
(P 6 ) e 1 < e 2 ○ e 3 < e 4 implies e 1 < e 4
(P 7 ) e < e ′ or e ○ e ′ or e ′ < e
Table 2: Postulates for event structures ([KR93], page 667)
in Table 2. 3
equivalence
By (P 7 ), (P 5 ) and (P 4 ), we can define ○ from < through the
e ○ e ′ iff neither e < e ′ nor e ′ < e .
In cases where ○ is = on E, < is a linear order. But in general, we
(i)
form sets of pairwise overlapping events, collecting them in
O(○)
def
= {i ⊆ E | (∀e, e ′ ∈ i) e ○ e ′ }
(ii)
lift < existentially to O(○), defining for i, i ′ ∈ O(○),
i < O i ′
def
⇔ (∃e ∈ i)(∃e ′ ∈ i ′ ) e < e ′
(iii)
equate temporal instants with ⊆-maximal elements of O(○)
I(○)
def
= {i ∈ O(○) | (∀i ′ ∈ O(○)) i ⊆ i ′ implies i = i ′ }
and say e occurs at i if e ∈ i.
Theorem (Kamp). < O linearly orders I(○) and for each e ∈ E,
{i ∈ I(○) | e ∈ i}
is a < O -interval (i.e. for all i, j, k ∈ I(○), if e ∈ j, e ∈ k and j < O i < O k then
e ∈ i).
How do event structures and Kamp’s theorem above relate to Φ-points?
Given a Φ-point p, let us form the triple 〈E p , < p , ○ p 〉 by collecting fluents with
interval temporal profiles in
E p
def
= {ϕ ∈ Φ | p({ϕ}) = ϕ }
3 The first two postulates are superfluous. (P 1 ) is derivable from (P 2 ), (P 3 ) and (P 5 ); and
(P 2) from (P 3) and (P 6).
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