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

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(⊇) Let ˆσ(A 0 ) ⌢ · · · ⌢ˆσ(A k−1 ) represent a tuple (x σ(0) , . . . , x σ(n−1) ) ∈ L(W Rv , h). By definition of acceptance and recognition this translates to δ ∗ h(0, ˆσ(A 0 ) ⌢ · · · ⌢ˆσ(A k−1 )) ∈ T Rv . By part one of lemma 2.6.6 we get δ ∗ h(0, A 0 ⌢ · · · ⌢A k−1 ) ⊲ˆσ δ ∗ h(0, ˆσ(A 0 ) ⌢ · · · ⌢ˆσ(A k−1 )). By the assumed equivalence we conclude that δ ∗ h(0, A 0 ⌢ · · · ⌢A k−1 ) ∈ T Rσv and hence that (x 0 , . . . , x n−1 ) ∈ L(W Rσv , h). (only if) For this direction we assume that L(W R(σv) , h) = {(x 0 , . . . , x n−1 ) : (x σ(0) , . . . , x σ(n−1) ) ∈ L(W Rv , h)}. Then we show coherence, i.e. that W |= ∀q ′ q ′′ ∈ R[q ′ ⋄ˆσ q ′′ → (T R(v0 ,...,v n−1 )(q ′ ) ↔ T R(v0 ,...,v n−1 )(q ′′ ))] and finally the rest. Our assumption we translate into a statement about infinite tapes as follows: A ⌢ 0 n×N ∈ L(W Rσv , h) iff ˆσ(A) ⌢ 0 n×N ∈ L(W Rv , h). To show coherence we proceed as follows. Let q be an arbitrary reachable state, and q ′ , q ′′ such that q ′ ⋄ˆσ q ′′ . By def of ⋄ there must be symbols A and B such that δ(0, hA) = δ(0, hB) = q and such that δ(0, ˆσhA) = q ′ and δ(0, ˆσhB) = q ′′ . Now q ′ ∈ T Rv iff δ(0, ˆσhA) ∈ T Rv by choosing δ(0, ˆσhA) = q ′ iff δ ∗ h(0, ˆσA) ∈ T Rv iff ˆσA ⌢ 0 n×N ∈ L(W Rv , h) iff A ⌢ 0 n×N ∈ L(W Rσv , h) by the translated assumption iff δ ∗ h(0, A) ∈ T Rσv iff δ ∗ h(0, B) ∈ T Rσv since by choice of A and B we have δ(0, hA) = δ(0, hB) = q iff B ⌢ 0 n×N ∈ L(W Rσv , h) iff ˆσB ⌢ 0 n×N ∈ L(W Rv , h) iff δ ∗ h(0, ˆσB) ∈ T Rv iff δ(0, ˆσhB) ∈ T Rv iff q ′′ ∈ T Rv by choosing δ(0, ˆσhB) = q ′′ Now we show that each of the directions in the equivalence in W |= ∀q ∈ R[T Rσv (q) ↔ ∃q ′ [q ⊲ˆσ q ′ ∧ T Rv (q ′ )]] holds. (→) Let q be a reachable state such that q ∈ T Rσv . Let A ∈ M p (n, m) be a symbol by which q is reached. This means that δ(0, h(A)) = q. From q ∈ T Rσv we infer that the number represented by A is in L(W Rσv , h). We now define the q ′ that fulfils the right side of ↔ by q ′ = δ(0, h(ˆσ(A))). By definition of ⊲ˆσ we have q ⊲ˆσ q ′ . By the assumed equality the tuple of natural numbers represented by ˆσ(A) is a member of L(W Rv , h). Therefore δ(0, h(ˆσ(A))) ∈ T Rv . Since q ′ = δ(0, h(ˆσ(A))) we get q ′ ∈ T Rv . 70
(←) Again we let q be reachable and A ∈ M p (n, m) such that δ(0, h(A)) = q. For this direction we assume that there exists a q ′ such that q ⊲ˆσ q ′ and q ′ ∈ T Rv . By the definition of ⋄ˆσ we get δ(0, h(ˆσ(A))) ⋄ˆσ q ′ and by the recently shown coherence it follows that δ(0, h(ˆσ(A))) ∈ T Rv . Therefore the tuple of numbers which ˆσ(A) represents is in L(W Rv , h). By the assumed equality this means that the tuple of numbers which A represents is in L(W Rσv , h). Hence δ(0, h(A)) ∈ T Rσv wich by definition of A means that q ∈ T Rσv . 4. Existential quantification: (If): Assume W |= ∀q ∈ R[T ∃vi φ(q) ↔ ∃q ′ , x[q ˆπ i ∼ q ′ ∧ 0 = ˆπ i (x) ∧ T φ (δ(q ′ , x))]]. (T ∃vi φ not too big): Assume that there are A 0 , . . . , A k−1 ∈ M p (n, m) such that δh(0, ∗ A ⌢ 0 · · · A k−1 ) ∈ T ∃vi φ. We now wish to display l ≥ k and B 0 , . . . , B l−1 ∈ M p (n, m) such that A ⌢ 0 · · · ⌢A ⌢ k−1 0 ⌢ n×m · · · ⌢0 n×m and B ⌢ 0 · · · ⌢B l−1 are equal except possibly on the i’th row and such that δh(0, ∗ B ⌢ 0 · · · ⌢B l−1 ) ∈ T φ . To do this let l = k + 1. By the first of the current two assumptions there exists a q ′ such that δh(0, ∗ A ⌢ 0 · · · ⌢A k−1 ) ˆπ i ∼ q ′ . Using this equivalence we do by lemma 2.6.8 get B 0 , . . . , B k−1 ∈ M p (n, m) with the property that that A ⌢ 0 · · · ⌢A k−1 and B ⌢ 0 · · · ⌢B k−1 are equal except possibly on the i’th row. Moreover δh(0, ∗ B ⌢ 0 · · · ⌢B k−1 ) = q ′ . By the firs assumption there also is an x such that δ(q ′ , x) ∈ T φ . Since h is an isomorphism we can define B l−1 = h −1 (x) and we get the B 0 , . . . , B l−1 we sought to display. (T ∃vi φ big enough): Assume that δh(0, ∗ B ⌢ 0 · · · ⌢B k−1 ) ∈ T φ . By the definition of onesorted n-tape p-multi-automaton this implies that δh(0, ∗ B ⌢ 0 · · · ⌢B ⌢ k−1 0 n×m ) ∈ T φ . If we now let both q and q ′ equal δh(0, ∗ B ⌢ 0 · · · ⌢B k−1 ) and let x = 0 we can from the first assumption infer (from the right to the left side) that δh(0, ∗ B ⌢ 0 · · · ⌢B k−1 ) ∈ T ∃vi φ. (Only if): Assume now that we have a PTPS automaton with two sets of terminal states that incidentally are indexed like T ∃vi φ and T φ . Assume also that (V, δ, T ∃vi φ, h) recognises the set set of tapes A ⌢ 0 · · · ⌢A ⌢ k−1 0 n×N such that there exists a B ⌢ 0 · · · ⌢B ⌢ l−1 0 n×N different from A ⌢ 0 · · · ⌢A ⌢ k−1 0 n×N in at most the i’th row and such that δh(0, ∗ B ⌢ 0 · · · ⌢B l−1 ) ∈ T φ . Note that if l < k + 1 we can extend B ⌢ 0 · · · ⌢B l−1 by adding 0 n×m ’s at the most significant end obtaining a word whose membership status in T φ is equivalent. Also if l > k + 1 we use the fact that our automaton is projectively transitive as in lemma 2.6.7 and shorten B ⌢ 0 · · · ⌢B l−1 so as to obtain a word whose membership status in T φ is equivalent and that differs from A ⌢ 0 · · · ⌢A ⌢ k−1 0 n×N in at most the i’th row. Without loss of generality we therefore assume that l = k + 1. Recall that ˆπ i ∈ P L(M p (n, m)) is the projection that replaces every entry on the i’th row with a 0. Using ˆπ i we translate the present assumptions to the following equivalence. For all states of the form δ ∗ h(0, A 0 ⌢ · · · ⌢A k−1 ) it is the case that δ ∗ h(0, A 0 ⌢ · · · ⌢A k−1 ) ∈ T ∃vi φ iff there exists a state of the form δ ∗ h(0, B 0 ⌢ · · · ⌢B k ) such that the conjunction of (a) ˆπ i (A 0 ) ⌢ · · · ⌢ˆπ i (A k−1 ) = ˆπ i (B 0 ) ⌢ · · · ⌢ˆπ i (B k−1 ) (b) 0 n×m = ˆπ i (B k ) 71
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On the methods of mechanical non-th
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1.4 A construction of Ps 3 ’s and
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0.1 Introduction The present thesis
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can be done on automata computation
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Parallel to this mathematicians wor
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For an algebra to soundly replace a
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Dr. Philos. degree. As life is not
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1.1 Introduction In this setting a
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either φ ∈ Γ or ¬φ ∈ Γ, an
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satisfaction in Ps 3 ’s in such a
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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
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: 2. Disjunction: Let {φ, ψ, φ ∨
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
Page 92 and 93: 3.3 The h-complex algebra of a mult
Page 94 and 95: Similarly we define two relations o
Page 96 and 97: 3.3.2 Properly partitioned automata
Page 98 and 99: 3.3.5 Representability of the h-com
Page 100 and 101: then there exists a B ∈ V k+1 s.t
Page 102 and 103: By well known results from automata
Page 104 and 105: A ′ is an n-dimensional polyadic
Page 106 and 107: We now define a class of automata t
Page 108 and 109: over finite sets. Again we avoid ex
Page 110 and 111: [Büc62] J.R. Büchi. Turing machin
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