On the methods of mechanical non-theorems (latest version)

0.3.4 Automated reasoning
In regards to disproof and implementations the branch of computer science called automated reasoning
is relevant. In particular the part of automated reasoning called model generation or model
building. In the “Handbook of Automated Reasoning” there is a chapter written by C.G. Fermüller,
A. Leitsch, U. Hustadt and T. Tammet, [FLHT01] which contains a survey of model generation
methods up until 2001. Put very briefly, there have been two basic approaches. One approach is
that of pruning the search tree in regards to the method of finite model search. The other approach
is that of producing a description of a model from the deduced formulae of a complete refutation
procedure, that has terminated without finding an inconsistency. Although exhibiting a model implies
consistency, we note that model building is more than disproof in that a model provides more
information than the confirmation we get from a procedure that merely terminates on consistency.
The models resulting from refutation procedures are typically finite descriptions of infinite Herbrand
models. However no examples of infinity axioms for which a model can be described is given
in the model generation survey in [FLHT01].
Further developments are reported in the 2004 book “Automated model building” by R. Caferra,
A. Leitsch and N. Peltier, which the present author has not had the opportunity to read.
However in 1997 N. Peltier, [Pel97a], [Pel97b], reports on methods capable of recognising infinity
axioms. These appear to be unimplemented methods. In 2000 R. Caferra and N. Peltier, [CP00]
report on a method which has been implemented. The implementation is reported to have succeed
on an example of an infinity axiom discovered by W. Goldfarb [Gol84]. The implementation
needed human interaction to succeed, but this is perhaps because of the complexity of the particular
infinity axiom chosen.
In 2009 N. Peltier published on transforming first-order sentences into equiconsistent firstorder
descriptions of tree-automata, together with an example of an infinity axiom of pure predicate
logic for which the description of said automaton has a finite model, see [Pel09]. Chapter 2 of the
present thesis is about a similar but distinct transformation. Neither of the two transformations
are known to result in infinity axioms recognisable in reasonable time. We do however in chapter
3 propose to use first-order descriptions of automata to compute atom-structures of polyadic set
algebras. The present author has used polyadic atom-structures, when implementing some of the
methods of the present thesis. The implementations are available on the present authors web-page
and terminate in reasonable time for several infinity axioms of pure predicate logic without human
interaction.
0.3.5 Personal perspective and acknowledgements
For my Cand. Scient. degree I worked on strategies for the method of finite model search under
the supervision of Professor S.O. Aanderaa. After obtaining the Cand. Scient. degree in 1993, I
attempted to enter a Dr. Scient program with S.O. Aanderaa as my supervisor.
I was not accepted for such a program, but had begun work on generalising a decidability result
due to S.O Aanderaa, see [Aan83]. I ended up with the generalisations of the method of finite
model search described in the present thesis. S.O. Aanderaa also found a serious flaw in an early
version of chapter 2 of the present thesis. Early means some time before 1999 in this setting.
As I was not accepted for the Dr. Scient. program I started working full time outside of
academia. In the year 2000 however, I took a part time job for the reason that I wanted go for a
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