Examples

1. Consider the following program P taken from [NNS, 1995; Example 6]. The program is also available in a separate file for download.

p(a) | q(a) | r(a).
r(b) :- r(a).
p(b) | q(b).
r(c) | p(c).
r(d) :- p(c).


The ⟨{p,q},{r},∅⟩-minimal models of P can be computed as follows (predicates p and q are subject to minimization while the interpretation of r may vary freely):

$ lparse --dlp example.lp > example.sm
$ circ2dlp example.sm -m "p* q*" -v "r*" | gnt 0
Answer set: 1
Stable Model: r(b) r(a) r(d) r(c) p(b)
Answer set: 2
Stable Model: r(b) r(a) r(d) r(c) q(b)
Answer set: 3
Stable Model: r(b) r(a) r(c) q(b)
Answer set: 4
Stable Model: r(b) r(a) r(c) p(b)
False
Number of isminimal calls: 43
Duration: 0.020


2. The models of the prioritized circumscription Circ(P,{p,r}>{q},∅,∅) can be computed as follows:

$ circ2dlp example.sm -m "p* r*: q*" | gnt 0
Answer set: 1
Stable Model: r(d) p(c) q(a) q(b)
Answer set: 2
Stable Model: q(a) r(c) q(b)
False
Number of isminimal calls: 17
Duration: 0.006


$ circ2dlp example.sm -m "* : q*" | gnt 0


3. The models for Circ(P,{p}>{q}>{r},∅,∅) are obtained similarly:

$ circ2dlp example.sm -m "p* : q* : r*" | gnt 0
Answer set: 1
Stable Model: r(b) r(a) r(c) q(b)
False
Number of isminimal calls: 15
Duration: 0.007


4. Reiter-style minimal diagnoses [Reiter, 1987] for a digital circuit can be easily captured using parallel circumscription. We provide a set of such encodings whose ⟨{ab},{high},∅⟩-minimal models correspond to minimal diagnoses. The encodings are in the internal format of smodels and textual format can be produced using the program lplist with options --gnt or --dlv. These instances have been produced as follows.

• A random tree is first generated using the inverse Pruefer algorithm.
• The leaves of the tree, corresponding to the inputs of the digital circuit, are assigned random boolean values
• The intermediate nodes are assigned random logical operations corresponding to the gates of the circuit.
• The gate at root node of the tree produces the output of the circuit.
• The value of the output is calculated and swapped in order to obtain faulty behaviour for the circuit.