![](bg8.png)
of a program DB, is recursively enumerable (Kunen,
1987). Van Gelder proposed well-founded seman-
tics (Van Gelder et al., 1991). The denotation of a
program P, well-founded partial model, is defined as
the least fixed point of some monotonic operator W
P
associated with P, which gives a 3-valued model of
comp(P). However, since W
P
is asymmetric on the
treatment of positive/negative occurrence of atoms in
the clause body, well-founded semantics differs from
our semantics; for example, p in a program {p ← p}
receives f in well-founded semantics.
Supported models of a program DB are 2-valued
models of the completed program comp(DB) (Apt
et al., 1988; Marek and V.S.Subrahmanian, 1992).
They can represent solutions of a quite large class
of combinatorial problems, and hence their efficient
computation is of practical interest. Also it is well-
known that stable models used in answer set program-
ming (ASP) are a subclass of supported models and
when propositional programs are finite and tight, they
are identical (Erdem and Lifschitz, 2003). Our pro-
posal to use 3-valued model computation as a prepro-
cessing step to compute supported models looks new
and is applicable to stable model computation as well.
It eliminates, as the experiment in Section 5 shows,
the extraneous need for finding the right assignment
of {t, f} to the deterministic part of supported mod-
els. On the other hand, in ASP, stable models (or
supported models) are computed by highly developed
SAT technologies as in clingo (Gebser et al., 2019). It
is an interesting future topic to merge our matricized
approach with existing ASP computation mechanism.
7 CONCLUSION
We proposed to compute the least 3-valued comple-
tion model of a finite normal logic program DB in a
vector space by first converting DB to an equivalent
definite clause program DB
d
, the dualized version of
DB, and then computing its least 2-valued model in a
vector space using a matrix representing DB
d
, which
is translated back to the least 3-valued completion
model of DB. We then applied this 3-valued model
computation to computing 2-valued completion mod-
els of DB, i.e. supported models of DB which are a
super class of stable models. We constructed them by
appropriately assigning t or f to the undefined atoms
in the least 3-valued completion model of DB while
guided by the completion form of clauses. We imple-
mented the 2-valued and 3-valued completion model
computation by matrix operations, and confirmed the
effectiveness of 3-valued computation as a prepro-
cessing step prior to 2-valued model computation.
Assigning truth vales to undefined atoms found in
this method is the next step to compute 2-valued sup-
ported models, and verification of efficiency of this
part will be reported in a full version of this paper.
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From 3-valued Semantics to Supported Model Computation for Logic Programs in Vector Spaces
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