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

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Lemma 1.3.1 L 33 ∪ L 32 ∪ L 31 ∪ L 30 is a sub-formula-closed subset of L 3 . Proof: L 33 ∪ L 32 ∪ L 31 ∪ L 30 is built by means of logical connectives beginning with the atomic formulae. qed The above four algebras of boolean signature are now interconnected with operations related to variable substitution and existential quantifiers. Definition L crc 3 = (L 33 , L 32 , L 31 , L 30 , r ∗ |, p ∗ |, s ∗ |, ∃ 0 |, ∃ 1 |, ∃ 2 |) where r ∗ |, p ∗ |, s ∗ | are the restrictions of r ∗ , p ∗ , s ∗ to L 33 . ∃ 2 | is the restriction of ∃ 2 to L 33 , making it an element of L 33 → L 32 ∃ 1 | is the restriction of ∃ 1 to L 32 , making it an element of L 32 → L 31 ∃ 0 | is the restriction of ∃ 0 to L 31 , making it an element of L 31 → L 30 Proposition 1.3.2 The sort L 30 of L crc 3 is a conservative reduction class. Proof: We prove that L 30 contains the sentences of the form ∀ 0 ∃ 1 ∀ 2 φ where φ is open. This class of sentences contains the reduction class of Kahr, More and Wang, which turned out to be conservative by results of Berger, Gurevich and Koriakov. For ∀ 0 ∃ 1 ∀ 2 φ is by definition ¬∃ 0 ¬∃ 1 ¬∃ 2 ¬φ. As long as φ is open ¬φ ∈ L 33 , then ¬∃ 2 ¬φ ∈ L 32 , then ¬∃ 1 ¬∃ 2 ¬φ ∈ L 31 , then ¬∃ 0 ¬∃ 1 ¬∃ 2 ¬φ ∈ L 30 . The following is virtually the same as above but works for the conservative reduction class of Büchi. Corollary 1.3.3 The sort L 30 of L crc 3 has every conjunction of prenex sentences in L 3 as an element. Proof: It can be proven as above that every sentence Q 0 Q 1 Q 2 φ where φ is open is in L 30 . The corollary then follows since L 30 has boolean signature and thus has conjunctions. qed 1.3.3 Directed many-sorted polyadic set algebras We have seen how to turn a substantial fragment of first-order language into an algebra of directed many-sorted polyadic signature. Here we define the algebras that are the core of the present paper and the basis of each disprover deviced. These algebras turn out to be finite if they are finitely generated. Definition An algebra A is a dMsPs 3 (directed many-sorted polyadic set algebra of dimension 3) if it is a C-sub algebra of directed many-sorted polyadic signature where C is a Ps 3 . qed 24
1.3.4 Interpretation We show that one can interpret conjunctions of prenex L 3 sentences in dMsPs 3 ’s much as in Ps 3 ’s, and that each such interpretation can be used to represent a tarskian interpretation. Lemma 1.3.4 Let h = (h 3 , h 2 , h 1 , h 0 ) be a directed many-sorted polyadic homomorphism from L crc 3 to an A in Ps 3 . Then h 3 ∪ h 2 ∪ h 1 ∪ h 0 is a mapping from a subset of L 3 to A that extends to a unique polyadic homomorphism h ∗ from L 3 to A. Proof: Recall that mappings formally are represented by sets of pairs. The boolean homomorphisms h 3 , h 2 , h 1 , h 0 are defined on disjoint sets, distinguished by what quantifiers occur in the elements. Thus h 3 ∪ h 2 ∪ h 1 ∪ h 0 is a mapping from L 33 ∪ L 32 ∪ L 31 ∪ L 30 . Which is easily seen to be a sub-formula-closed subset of L 3 . By lemma 1.2.4, letting X denote L 33 ∪ L 32 ∪ L 31 ∪ L 30 and Y denote L 3 , the mapping h 3 ∪ h 2 ∪ h 1 ∪ h 0 uniquely extends to a ∨, ¬, ⊥, ∃ 0 , ∃ 1 , ∃ 2 -preserving mapping h ∗ from L 3 . It remains to show that r ∗ , s ∗ , p ∗ are preserved by h ∗ to show that h ∗ is a polyadic homomorphism. Let f denote the restriction of h ∗ to the atomic formulae of the form {R, ...} × {012}. By definition, h ∗ extends f and must be the uniquely determined polyadic homomorphism f ∗ of lemma 1.2.5. qed The following says that homomorphisms are like interpretations in the usual sense in that if an interpretation of the relation-symbols is given then an interpretation of each formula in our fragment is determined. Lemma 1.3.5 If {R, ...} are the relation-symbols of L 3 and if A is a dMsPs 3 and B 3 of A has carrier-set B 3 , then each mapping f ∈ {R, ...} × {012} → B 3 extends to a unique directed manysorted polyadic homomorphism (f ∗ 3 , f ∗ 2 , f ∗ 1 , f ∗ 0 ) from L crc 3 to A. Proof: We display boolean homomorphisms f3 ∗ , f2 ∗ , f1 ∗ , f0 ∗ from L 33 , L 32 , L 31 , L 30 to the boolean reducts B 3 , B 2 , B 1 , B 0 of A respectively, we show that the extra-boolean operations of L crc 3 are preserved and that this is a unique directed many-sorted polyadic homomorphism. By the definition of dMsPs 3 there is a C in Ps 3 such that A is a C-sub-algebra of directed many-sorted signature. Lemma 1.2.5 provides a unique polyadic homomorphism f ∗ from L 3 to C that extends f. Define f3 ∗ , f2 ∗ , f1 ∗ , f0 ∗ as the restrictions of f ∗ to L 33 , L 32 , L 31 , L 30 respectively. These preserve the required operations as they are restrictions of f ∗ , which preserves them. Let (h 3 , h 2 , h 1 , h 0 ) be an arbitrary directed many-sorted polyadic homomorphism of the required form. To show uniqueness we show that h 3 ∪ h 2 ∪ h 1 ∪ h 0 and f3 ∗ ∪ f2 ∗ ∪ f1 ∗ ∪ f0 ∗ are the same. By lemma 1.2.3, the two are the same on atomic formulae. Letting X be the atomic formulae and Y = L 33 ∪ L 32 ∪ L 31 ∪ L 30 , in lemma 1.2.4 we get the desired result. Note that f3 ∗ for instance trivially preserves ∃ 0 , ∃ 1 , ∃ 2 , as there is no pair of open formulae φ and ψ such that the syntactic equality ∃ i φ = ψ holds. qed As before a homomorphism is satisfying for a sentence if it is mapped to ⊤. Definition Let L 30 be the sort of L crc 3 consisting of sentences. A directed many-sorted polyadic homomorphism (h 3 , h 2 , h 1 , h 0 ) from L crc 3 to an A in dMsPs 3 is satisfying for a sentence φ ∈ L 30 if h 0 (φ) = ⊤. 25
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 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
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
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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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