The elementary "initial case" applies (2c) upon (3a) where β = γ = δ, together with a simplification (3c) since α → β **here, getting:**
If α causes β and W 6|= ¬α **then** α explains β bec poss {α}.

These rules are intended as a compromise between expressive power, naturalness of description and relatively efficiency.

Transitivity of explanations **occurs (gathering conditions). The simplification rule (3c) is powerful and costly, so the ASP translation implements the**
following weaker rule (also, it never removes {α} **from Φ): [3c'] If** W |=VΦ − {ϕ} → V**Φ, and** α explains β bec poss **Φ then** α explains β bec poss Φ − {ϕ}.

An atom α explains β bec poss **Φ is** optimal **if there is no explanation atom**
α explains β bec poss Ψ where W |=VΨ →V**Φ an not conversely. Keeping**
only these weakest sets of conditions is useful when the derivation **is made only**
thanks to the part of W **coming from Points 1 and 2a above. This keeps all the**
relevant explanation atoms and is easier to read.

## 2.3 A Generic Diagram

The following diagram summarizes many patterns of inferred explanations:

α causes β, α causes β0, β2 causes γ, β1 **causes γ**, β3 causes ǫ, γ1 causes δ, γ3 causes δ, ǫ3 **causes γ**3;

β →IS−A β2, β1 →IS−A β, β3 →IS−A β0, β3 →IS−A β1,

γ1 →IS−A γ, γ2 →IS−A γ, γ2 →IS−A γ3, γ2 →IS−A ǫ, ǫ1 →IS−A ǫ, ǫ2 →IS−A ǫ, ǫ1 →IS−A ǫ3, ǫ2 →IS−A ǫ3.

This example shows various different "explaining paths" from a few given

causal and ontological atoms. As a first "explaining path" from α to δ **we get successively (path (1a):** α explains β2 bec poss {α}, α explains γ1 bec poss {**α, γ**1},

and α explains δ bec poss {α, γ1}**, giving** α explains δ bec poss {α, γ1} **(1a). The** four optimal paths from α to δ **are depicted, and non optimal paths (e.g. going** through β1**) exist also.**

## 3 An Asp Translation Of The Formalism 3.1 Presentation

We describe a program in DLV-Complex. A first version [8] used pure DLV, and was much slower and harder to read. We have successfully tested an example with more than a hundred symbols and more than 10 different explanation atoms for some (I, J**) (made from two copies of the example of the diagram,**
linked through a few more data). We have encountered a problem, not listed in the "three problems" evoked above. The full program, including "optimization of the result", did not work on our computer. The simplification step **and the** verification step are clearly separated from the first generating step, thus we have split the program in three parts: The first one generates various explanation atoms (including all the optimal ones). The second program keeps only the optimal explanation atoms, in order to help reading the set of the solutions,