7.4.3 Sistema continuo de extracción en dos fases.

In this section, we show how to construct a pseudometric on the class of scenarios. This will give us a notion of distance between scenarios.

We start by observing that, given a scenarios, there are clauses that are more likely to extendsthan other clauses. For example, consider the scenario that describes me sitting at a desk and writing these lines. Now consider two possible extensions of that scenario: One extending it by the fact that outside it is raining, and one by the fact that outside there is no gravity anymore. Then, clearly, the former extension is more common, similar, plausible, or likely than the latter. (We deliberately leave open the exact interpretation of this ordering.) We capture this ordering by assigning those clauses that are very common small numbers and those that are less common bigger numbers—and clauses that are either equally common or cannot be compared in terms of commonness get the same number.

To do so, we fix for the whole chapter a bijective enumerationeof all the clauses: so, writingClsfor the set of all general logic programming clauses,e :Cls→Nis

a bijection. To simplify notion, we often identify clauses and their corresponding numbers. Then we define:

Definition 4.1.2(Clause extension orderingρ).Aclause extension orderingρis a func- tionρ:Σ→ _R>1Nthat assigns each scenariosa sequenceρ(s)of real numbers

>1 (i.e.ρ(s) :N→R>1).

Thus, all the possible extensions of sby one clause are of the forms∪{A} whereA∈Cls\s(and we identifyAwith its corresponding number), and they are ordered by their “plausibility” as given byρ(s).

Now, we describe the tree structure ofΣ. This is obtained by identifying scenarios by their program (s≡s0iffPs=Ps0), and then consider the tree of program extensions:

∅

. . . Ai . . . Aj . . .

. . . Ak . . . Al . . . .._.

For example, the program consisting of the clausesAiandAlcan either be realized

by the sequence of extensionshAi,AliorhAl,Aii. So now we also take the order into

necessary.97 This is welcomed because it might well be that the plausibility of a program depends on the order in which it was constructed: A program that was constructed by adding clauses each being a bit less plausible than the previous might in sum be more plausible, than obtaining the same program by first adding the last and very implausible clause, then a more plausible one, then again an implausible one, and so on. Thus, with our simplifications we now regard a program as a finite sequence of natural numbers:s=hs(1),s(2), . . . ,s(n)i.

Definition 4.1.3(Tree-structure of(Σ,ρ)).Given a clause extension orderingρ, the tree structure of(Σ,ρ)is given by the treeTρwhich is constructed as follows:

• The nodes in the tree are finite injective sequences of natural numbers.98

• The root ofT is the empty sequenceh i.

• Given a nodes, its successors are the nodess__nforn∈

N\s,99and they

are ordered from left to right by

s_m < s_niff

ρ(s)(m)< ρ(s)(n) , or ρ(s)(m) =ρ(s)(n)andm < n .

So the further we go to the top of the program extension tree Tρ, the more

implausible the programs become (since this corresponds to making more and more assumptions). And the further we go to the right in the tree, the more implausible the programs become (since we add more implausible clauses). Thus, we define the plausibility orweightof a program by

µ∗_ρ(s) :=

length_X(s)

i=1

ρsi−1 s(i) ,

wheres 0 is the empty sequence. Soµ∗ρ(s)just adds up theρ-values of the

nodes along the path to the programs.

We say ρ issymmetric, if for alls = hs1, . . . ,sniand all permutationsπ of {1, . . . ,n}we haveµ∗_ρ(hs1, . . . ,sni) =µ∗ρ(hsπ(1), . . . ,sπ(n)i). If we speak ofρin a context where we think of the programs of scenarios as sets (and not as finite sequences) we tacitly assume thatρis symmetric.

Finally, we can define a metric onΣgiven a clause extension orderingρ.

Definition 4.1.4(Induced metricdρ).Fix a clause extension orderingρand define the

pseudometricdρ:Σ×Σ→R>0bydρ(s,s0) :=|µ∗(s) −µ∗(s0)|.

This is indeed a pseudometric because the properties of non-negativity, symmetry, and triangle inequality100_{are directly inherited from the properties of the absolute}

97_{We could have done that from the start (and then identify programs with the same clauses but}

different order on every occasion we used them so far), but that would have resulted in an unnecessarily complicated notation.

98_{Where a sequence is injective if no number occurs twice.}

99_Wheres__{nis the sequence obtained fromsby addingnas the last number. And}

N\sis short for N\ {s(i)|16i6length(s)}.

100_{Formally, these properties are, respectively,∀s,s}0_∈Σ:dρ(s

,s0)>0,∀s,s0∈Σ:dρ(s,s0) = dρ(s0_{,s), and∀s,s}0_,s00_{∈Σ:dρ(s,s}00_)6dρ(s,s0_{) +dρ(s}0_,s00_).

value function|·|. In general, it is not a metric (proper) because it doesn’t satisfy dρ(s,s0) =0⇔s=s0since different scenarios might have the same weight.

Moreover, this pseudometric induces a system of spheres in the sense of Lewis (1973, 13f.): Given a scenarioi∈Σ, the set $iof spheres aroundiis given by $i:=

Bρ(i)|∈

R>0 whereBρ(i)are the-balls aroundi, that is,Bρ(i) :={s∈Σ0|dρ(i,s)6}.101