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

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set. In the example, the initial sort B 3 of A, turned out to have 13 atoms, which can be verified by a geometric argument similar to the one depicted in fig. 1.3, when viewing ω as a subset of R. B 3 of 2 3 → A has 8 times that number. Multiplication with 8 proceeds, so after i iterations search is done in an algebra with 13 · 8 i atoms. By lemma 1.1.2 and the possibility of storing the extra-boolean operations of A in the form of tables (finite sets of pairs), the question of whether there exists a satisfying interpretation for a sentence with m relation-symbols, in an algebra with 13 · 8 i atoms, can be phrased as a constraint satisfaction problem with m · 13 · 8 i variables ranging over {⊥, ⊤}. If the disprover is based on the full Ps 3 over a finite set then it is a finite model search procedure, which at each step doubles the size of the set over which satisfying interpretations are sought. Various disprovers have been implemented, based on the presented method. They are publicly available with source code included. 1 . 1.5 Related constructions and algebras The construction of definition 1.4 is rather similar to what was called the cardinal multiple of theories by Feferman and Vaught [FV59] (Section 4.7), when bearing in mind that polyadic set algebras correspond to complete and consistent theories. The constructions of [FV59] were carried over to the setting of polyadic algebras and generalised by Daigneault [Dai63]. The main construction is called the tensor product of polyadic algebras. Daigneaults paper is mainly about infinite, not necessarily atomic nor complete, polyadic algebras so the definition goes via Stone-spaces. Finite polyadic algebras are all atomic and complete, so it is worth noting that this tensor product can also be constructed by finite means. It turns out to be the polyadic version of the Kronecker product, ⊗, of finite-dimensional vector spaces. This can be seen by regarding elements of a finite (directed many-sorted) polyadic set algebra as vectors of zeros and ones and using boolean operations instead of ring-operations. In this view 2 3 → A is isomorphic to P(2 3 ) ⊗ A. Under one such isomorphism the function values of an element of 2 3 → A appear as rows of the corresponding matrix in P(2 3 ) ⊗ A. This product works for arbitrary pairs of set algebras and provides a way of combining any two disprovers devised as presented. An essential property of the construction, is that the polyadic set algebras are closed under it, and that one gets more than what one had to begin with. The direct product of polyadic algebras is not such a construction, as a sentence that is satisfiable in a product, by projection, already is satisfiable in one of the factors. There are no projections of this kind for Ps 3 ’s. They are simple. By an argument involving lemma 1.3.4, dMsPs 3 ’s are also simple. The direct product does however give rise to the extensively studied representable polyadic algebras, which together with representable relation algebras and cylindric algebras, provide a potential source for dMsPs 3 ’s, besides decision procedures, and sufficiently complete finite sets of sentences. Regarding dMsPs 3 ’s, the idea of using partial and many-sorted variants of algebras for interpretation of languages with quantifiers is far from new. An early one is by Bernays [Ber59], more are mentioned by Németi [Ném91]. Connections to category theory approaches are also mentioned there. The approaches the present author is aware of, are each different from dMsPs 3 ’s in at least one of the following two respects. Firstly, classes of partial or many-sorted algebras are not gener- 1 http://flipper.berlios.de 32
ally such that each arity respecting mapping of non-logical symbols into an A extends naturally to an interpretation in A of each sentence of an entire reduction class as in lemma 1.3.5. Secondly, known many-sorted variants, which obviously can be chopped off at some finite dimension, typically allow the definition of cylindrification within each sort, which in the present vocabulary is to say that they are not directed. Undirectedness can ruin the property that finitely generated algebras are finite. As an example of how little it takes for undirected closures to be infinite, consider P(ω 2 ), the full Ps 2 over the natural numbers. A signature for this algebra is (B, s, p, c x , c y ). Both c x and s may even be left out for the following. Let y < x denote the set of pairs of numbers whose second component is less than the first. Moreover let for each natural a, a < x denote the set of pairs whose first component is greater than a, and a < y denote the set of pairs whose second component is greater than a. The claim is that for each natural a, a < x lies in the closure of y < x. Now 0 < x lies in the closure as c y (y < x) = 0 < x. Proceed by assuming that a < x lies in the closure. Consider c y (p(a < x) ∩ y < x), here p(a < x) = a < y so the term is true if there is a y such that x is greater than y and y is greater than a, which is when a + 1 < x. 1.6 Concluding remarks The author believes that the adaption of the polyadic algebras of Tarski and Halmos to the conservative reduction class of Büchi so as to form the class dMsPs 3 is a contribution to algebraic logic. Some potential in automated reasoning has also been made likely. Distinguishing features of the dMsPs 3 ’s are the naturalness property of lemma 1.3.5, their having a finite dimension higher than two and their being directed. Directedness enables the further downward Löwenheim-Skolem property (corollary 1.3.9), for each sentence of the conservative reduction class. This property makes it possible to do exhaustive search for, and to sometimes find, satisfying interpretations for infinity axioms in finite dMsPs 3 ’s. Various finite dMsPs 3 ’s can be computed by means of given decision procedures for first-order theories. The above makes finite dMsPs 3 ’s quite suitable as components of disprovers. Each finite dMsPs 3 can together with the construction of definition 1.4 be used to devise a disprover that behaves naturally and that works by a generalisation of finite model search for a substantial fragment of first-order language. The fragment is substantial since it is a conservative reduction class. The disprover behaves naturally by proposition 1.3.7. The disprover works by a generalisation of finite model search since, by proposition 2.5.3, it does search through a set of interpretations containing a representative for every interpretation over any finite set. As long as the given dMsPs 3 allows a satisfying interpretation for an infinity axiom, the generalisation is strict. 33
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 and 14: For an algebra to soundly replace a
Page 15: Dr. Philos. degree. As life is not
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 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
Page 65 and 66: Definition Let W = (V, K, δ, ι, .
Page 67 and 68: δ ∗ h(ι, ˆσ(A)) = q ′ Also
Page 69 and 70: 1. Immediate from the definition of
Page 71 and 72: there exist x, y ∈ V such that t(
Page 73 and 74: 1. δ ∗ h(0, A 0 ⌢ · · · ⌢
Page 75 and 76: 2. Disjunction: Let {φ, ψ, φ ∨
Page 77 and 78: (←) Again we let q be reachable a
Page 79: 1. If Γ has an automatic model the
Page 82 and 83: 3.1 Introduction This is the second
Page 84 and 85: We write R −1 (B) for the inverse
Page 86 and 87: Definition Let U be a set. The full
Page 88 and 89:
Proof: We turn a fragment of first-
Page 90 and 91:
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
Page 108 and 109:
over finite sets. Again we avoid ex
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[Büc62] J.R. Büchi. Turing machin
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