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

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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 8
Dr. Philos. degree. As life is not always predictable I have had several breaks in my project and my non-academic career. After one such break I had the opportunity to join an automated reasoning project with Professor A. Waaler at the Department of Informatics in Oslo for half a year in 2007. I have also had the opportunity to follow the weekly mathematical logic seminar at the University of Oslo, where besides Professor S.O. Aanderaa, the Professors D. Normann and H.R. Jervell have been able to follow my progress and to give advice. I hereby thank all of the above for the said opportunities. I also thank my personal friend Dr. F.A. Dahl for helpful comments. 9
Page 1: On the methods of mechanical non-th
Page 4 and 5: 1.4 A construction of Ps 3 ’s and
Page 7 and 8: 0.1 Introduction The present thesis
Page 9 and 10: can be done on automata computation
Page 11 and 12: Parallel to this mathematicians wor
Page 13: For an algebra to soundly replace a
Page 18 and 19: 1.1 Introduction In this setting a
Page 20 and 21: either φ ∈ Γ or ¬φ ∈ Γ, an
Page 22 and 23: satisfaction in Ps 3 ’s in such a
Page 24 and 25: 1.2.4 Polyadic set algebras of tern
Page 26 and 27: Lemma 1.2.3 If {R, ...} are the rel
Page 28 and 29: 1.2.8 Exhaustive search for satisfy
Page 30 and 31: Lemma 1.3.1 L 33 ∪ L 32 ∪ L 31
Page 32 and 33: The following allows us to use homo
Page 34 and 35: Corollary 1.3.10 Let |= be a recurs
Page 36 and 37: the full Ps 3 over 3 has 27 atoms.
Page 38 and 39: set. In the example, the initial so
Page 41 and 42: Chapter 2 Automata for mechanising
Page 43 and 44: 3. The finite automata of the class
Page 45 and 46: theorem [BHMV94], the p-recognisabl
Page 47 and 48: Using this theorem contrapositively
Page 49 and 50: 4. associative, i.e. V |= ∀x[0(0(
Page 51 and 52: Again the axioms with a (*) behind
Page 53 and 54: Let m be equal to the dimension of
Page 55 and 56: Definition Let W = (V, K, δ, ι, T
Page 57 and 58: states reachable in i steps from th
Page 59 and 60: Proof: Left to the reader. qed Defi
Page 61 and 62: 2.5 Reachability refined Here we in
Page 63 and 64: Definition For each n-fold vector-s
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Definition Let W = (V, K, δ, ι, .
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δ ∗ h(ι, ˆσ(A)) = q ′ Also
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1. Immediate from the definition of
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there exist x, y ∈ V such that t(
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1. δ ∗ h(0, A 0 ⌢ · · · ⌢
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2. Disjunction: Let {φ, ψ, φ ∨
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(←) Again we let q be reachable a
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1. If Γ has an automatic model the
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3.1 Introduction This is the second
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We write R −1 (B) for the inverse
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Definition Let U be a set. The full
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Proof: We turn a fragment of first-
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3.2.2 The complex algebra tailored
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3.3 The h-complex algebra of a mult
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Similarly we define two relations o
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3.3.2 Properly partitioned automata
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3.3.5 Representability of the h-com
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then there exists a B ∈ V k+1 s.t
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By well known results from automata
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A ′ is an n-dimensional polyadic
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We now define a class of automata t
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over finite sets. Again we avoid ex
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[Büc62] J.R. Büchi. Turing machin
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