LocModGen – Model Generation by Local Search
Manfred Kerber
http://www.cs.bham.ac.uk/~mmk
Computer Science, The University of Birmingham
Birmingham B15 2TT, England
Humans use semantic information in their proof search, and typically they know important models of mathematical concepts. Models can be used to direct proof search in
systems such as Scott [4] or the semantically guided clause graph procedure [1]. Model
generation itself can be computationally very hard and typically complete methods are
used, which are built on variants of the Davis-Putnam-Loveland-Logemann method. Such
model generators are Mace (Models And Counter-Examples) [2] and Finder [3]. In this
abstract we propose a method using a local search approach. Such approaches have proved
successful in constraint satisfaction and propositional logic model generation. We adopt
the method for the generation of finite models of first-order clauses.
The approach we are proposing takes as input a first-order clause set, FCLAUSE, and the
intended cardinality of the model, n. In a first step the first-order clause set is translated
to a propositional logic one, C. Then a random initial interpretation I is generated and
the number of clauses satisfied in C is calculated. This interpretation is modified until the
process is either stuck and restarted, or a model is found (see Figure 1).
Compared to the actual PHP-implementation, which can be run for some few examples
from www.cs.bham.ac.uk/~mmk/demos/LocModGen/, there are some simplifications in the
abstract description. Typically the procedure tries to make small positive steps (S+), by
modifying one single element in the interpretation. The change is taken over only if it
increases the number of satisfied propositional logic clauses. If the procedure does not
make any progress for a while – measured by a counter j – it tries to make a big positive
step (B+), in which the change is taken over only if it increases the number of satisfied
propositional logic clauses. If neither is successful for a number of steps (e.g., 100 steps)
it makes a big arbitrary step (B*). If all this is not successful after a number of steps
(e.g., 150 steps), the procedure restarts (RESTART). There are two resource constraints,
the maximal number of steps allowed and the maximum time spent on the search. If the
process runs out of resources, it will return the best interpretation found so far.
For transforming first-order clauses to propositional logical ones, it is necessary to fix
the cardinality n of the universe. It is assumed that the universe is given in the form of
the first n natural numbers, {1, . . . , n}. We add constant symbols 1̄, . . . , n̄ to the syntax,
which stand for 1, . . . , n, respectively, (in order to simplify notation we will write simply
1, . . . , n). All clauses are replaced by their variable-free counterparts, that is, each variable
is systematically replaced by each of its possible values in {1, . . . , n}. For instance the
1st order clauses
cardinality n
PP
q
P
m prop clauses C
?
random interpretation I; j := 0
k := card({c ∈ C | I |= c})
?
,l Yes
- RETURN I
,k = ml
,
l
l,
No
?
0
I := modify(I)
k 0 := card({c ∈ C | I 0 |= c})
?
l
,
Yes - I := I 0 ; k := k 0
,k 0 > kl
,
l
j := 0
l,
No
?
j := j + 1
No
6
,l
, j <l Yes
lmax,
l,
Figure 1: Flow diagram
clause P (x, a), Q(y, f (x)) with n = 3 will be replaced by the following nine clauses
• P (1, a), Q(1, f (1))
• P (2, a), Q(1, f (2))
• P (3, a), Q(1, f (3))
• P (1, a), Q(2, f (1))
• P (2, a), Q(2, f (2))
• P (3, a), Q(2, f (3))
• P (1, a), Q(3, f (1))
• P (2, a), Q(3, f (2))
• P (3, a), Q(3, f (3))
While we know how to interpret the elements 1, 2, 3, . . . , n, and the binary predicate
symbol equal, which has a fixed interpretation to model equality, the other symbols, a, f ,
P , and Q in the example, can be interpreted in an arbitrary way.
While the method presented here has the disadvantage that we cannot use it to prove
that there is no model, we can use it to define a notion of “near-model”, which tells that
an interpretation satisfies k out of m propositional logical clauses. Such near models may
be used as a heuristic in adaptations of the methods described in [4] and [1]. The method
presented here can also be directly adapted to deal with higher-order formulae.
References
[1] Seungyeob Choi and Manfred Kerber. Semantic selection for resolution in clause graphs.
In B. McKay, J. Slaney, eds., AI-2002: 15th Australian Joint Conference on Artificial
Intelligence, pages 83–94. Springer, 2002.
[2] William McCune. Mace4: Models and counter-examples. Argonne National Laboratory,
Illinois, USA, 2003. http://www.mcs.anl.gov/AR/mace4/.
[3] John Slaney. Finder: Finite domain enumerator–system description. In A. Bundy, ed.,
12th CADE, pages 798–801. Springer, 1994.
[4] John Slaney, Ewing Lusk, William McCune. SCOTT: Semantically Constraint Otter
– System Description. In A. Bundy, ed., 12th CADE, pages 764–768, Springer, 1994.