The ramification problem in temporal databases: an approach with conflicting constraints

In Section 2 we explain this in details.

A preliminary version of this work appears in the Proceedings of the 19th IEEE International Conference on Tools with Artificial Intelligence (Papadakis et al. 2007).

The direct effect of its action is up(s ₁).

A more simple situation contains the value of fluents.

Quantifiers are omitted in the expression of these propositions. They are considered to be implicitly universally quantified over their temporal and non-temporal arguments.

Any change in the time intervals has as effect a new situation.

For example if \(S=\{f_1([3,8]),\neg f_2([4,9]),f_3([10,34])\}\) is temporal situation then FluentHold \((S,5)=\{f_1,\neg f_2\}\). In the specific example the FluentHold changes at time points 9, 10 and 34.

Notice that many different temporal situations could refer to the same situation at a specific time point. For example S ₁ = {f ₁([2, 10),f ₂([4, 8])} and S ₂ = {f ₁(4, 34]),f ₂([2,10])} has the same non-temporal situation at time point 5 S′ = FluentHold(S ₁,5) = FluentHold(S ₂, 5) = {f1_, f2)}.

For example \(\{f_{1}([[7,9]],\neg f_{1}([[10,\infty]]),....\}\) means that at time point 10 we have transition from one situation to another because the truth value of fluent f ₁ changes. Notice that this transition happens without an action taking place.

G _f ([a, b]) is true when the propositional combination of fluent names which consist the fluent formula is true at time interval [a, b].

In the case that we are not pruning with the use of relation I mean that we want one of them, without to have order of fancy between them. It is possible to be more pruning than the pruning of the algorithm 0 with the only restrict to be at least one pair (a, b) ∈ I for each Integrity Constraint. This is necessary as we explain in proof of Theorem 2.

A →B ≡ C ₁ ∧ .... ∧ C _m . In order A → B to be true all C _i , i = 1,...m must be true.

This allows us to prune the undesirable effects. In the example of circuit switch, there is no static rule \(\neg light \rightarrow \neg up(s_2))\). Thus the situation S ₂ cannot be possible consistent situation.

Notice that for each fluent there could be more than one causal relationships. This happens in the case that fluent f _k is present in more than one integrity constraints. At this step we integrate all this causal relationships in one. For example if f ₁ ∧ f ₂   causes  f ₃  if    f ₄ and f ₅ ∧ f ₆   causes  f ₃  if    f ₇ then the static rule False → f ₃ is firstly transformed in f ₁ ∧ f ₂ ∧ f ₄ →f ₃ and finally in (f ₅ ∧ f ₆ ∧ f ₇) ∨ (f ₁ ∧ f ₂ ∧ f ₄) →f ₃.

The fluent formula G ^m is in CNF form. Thus \(G^{m} \equiv G_{1} \vee .... \vee G_{n} \equiv (f_{11} \wedge ... \wedge f_{1k}) \vee (f_{21} \wedge ... \wedge f_{2k}) \vee ... (f_{n1} \wedge ... \wedge f_{nk})\)

The list L ^′ is the list, which contains the time intervals, in which the fluent \(\neg f\) is true.

If we determine the experience(p, 0, t′) then the update is automatic.

The set R _strict is the set of static rules which is produced by the set of the strict integrity constraints.

\(\neg f \in E\) means that there is one static rule with a higher priority which has the \(\neg f\) as result.

Here we include the effects of the set R _i which are consistent with the effects of the rules with a higher priority.

The steps 1.a–1.c do not executed at time point 11 beacuse no action takes place at this time point.

There is no executable strict or defeasible rule with higher priority.

There is an inconsistency between the direct effects of one action and the indirect effects of some other action.

By step 5 of the algorithm producing the static rules.

It is the case when e.g. f ₂([a,b])→f ₁([a,b]), \(f_{3}([c,d])\rightarrow \neg f_{1}([c,d])\) and [a,b] ∩ [c,d] ≠ {} and f ₂([a,b]),f ₃([c,d]) does not change during the execution of the static rules. Thus we have an infinite execution.

\(\neg f \in E\) means that there is one static rule with a higher priority which has as conclusion the \(\neg f\).

The case that there is an inconsistency between the indirect effects which are produced by the static rule with the same priority.

The transition from one situation to the next happens after the evolution of one or more static rules.

This means that each \(S_{t}^{i}=FluentHold(S_{m},t)\), for a temporal situation S _m .

Theorems 1, 2 and 3 ensure that such a pair exists.

Do them executable. e.g A→B with priority K trigger the B →C with priority K-1, this trigger the rule C →D with priority K-2,.....