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

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By well known results from automata theory we can, given a finite set Y of automata, effectively construct such an X. By proposition 3.3.1 there is one finite PTPS-automaton W = (V, δ, {T j } j∈J ) that recognises each relation recognised by the elements of X. Now X was carefully designed so as to make W meet the criteria of proposition 3.3.3. Therefore there is a properly partitioned automaton, W ′ = (V, δ, {T j} ′ j∈J ′), such that {T j} ′ j∈J ′ generates the same boolean algebra as does {T j } j∈J . We see that W ′ is finite since it has the same carrier set as W. Let now H = (R, . . .) be the atom-structure of W ′ , let h = (h n , . . . , h 0 ) be the n-homomorphism induced by W ′ and let H h + = (B′ n, . . . , B 0, ′ r ′ , p ′ , s ′ , c ′ 0, . . . , c ′ n−1) be the h-complex algebra of H. By proposition 3.3.5 we may define an embedding f = (f n , . . . , f 0 ) from H h + to the full dMsPs n over N as in the proof of proposition 3.2.2, i.e. for i ≤ n and q ∈ R we let f i [q] hi = γ −1 [q] hi , for Q ⊆ R let f i ( ⋃ {[q] hi |q ∈ Q}) = ⋃ {γ −1 [q] hi |q ∈ Q}. We now show that the given A is a sub-dMsPs n of the image of H h + under f, the embedding needed to prove the present proposition is then obtained by restricting f −1 to A. Since {T j} ′ j∈J ′ generates the same boolean algebra as does {T j } j∈J and since each set recognised by an automaton of Y is of the form γ −1 (T j ) we see that the sort B n is a sub-boolean algebra of the image of B n ′ under f n . Therefore the finitely generated A is a sub-dMsPs n of the image of H h + under f. qed 3.4.2 Ps n ’s Here we show that in the case of one-sorted polyadic algebras we are not generally able to obtain an embedding as in the case of dMsPs n . We use a trick and define one-sorted n-dimensional polyadic algebras as a sub-class of dMsPs n . It is left to the reader to verify that this definition coincides with n-dimensional (quasi) polyadic algebra as found in the literature, see e.g. L. Henkin, J.D. Monk and A. Tarski [MHT85] or I. Németi [Ném91]. These may also be compared to the quantifier algebras due to C.Pinter [Pin73], which are called substitution-cylindric algebras by I. Németi [Ném91] and H. Andréka and I. Sain and I. Németi [ASN01]. Definition An n-dimensional polyadic set algebra, or Ps n for short, is an A = (B, r, p, s, c 0 , . . . , c n−1 ) such that A ′ = (B, . . . , B, r, p, s, c 0 , . . . , c n−1 ) is a dMsPs n . Moreover a mapping f from A to some Ps n is a P n -embedding if (f, . . . , f) is a dMsP n - embbeding from A ′ to some dMsPs n . Example Let (N,
Proof: It is easy to see that the h-complex algebra of a properly partitioned finite automaton is finite. So to prove the proposition it suffices to display n, p ∈ N and an infinite Ps n that is generated by a finite set of first-order definable relations over (N, +, | p ). Let n = 3, p = 2 and consider the A ∈ Ps 3 with the following generator X = {(x 0 , x 1 , x 2 )|x 1 < x 0 }. The generator is definable by the formula ¬(∃v 2 (v 0 + v 2 = v 1 )), which says that it is not the case that v 0 ≤ v 1 . We will now by induction show that for each a ∈ N it is the case that the set {(x 0 , x 1 , x 2 )|a < x 0 } lies in A. Let N denote the full dMsPs 3 over N. To begin with {(x 0 , x 1 , x 2 )|0 < x 0 } lies in A, since c N 1 (X) = {(x 0 , x 1 , x 2 )|∃y(x 0 , y, x 2 ) ∈ X)}. This set is exactly the set of triples such that the second component is greater that 0. To proceed the induction, assume that A = {(x 0 , x 1 , x 2 )|a < x 0 } lies in A. Consider c 1 (p N (A) ∩ X). Here p N (A) = {(x 0 , x 1 , x 2 )|a < x 1 } therefore p N (A) ∩ X = {(x 0 , x 1 , x 2 )|a < x 1 ∧ x 1 < x 0 } furthermore c 1 (p N (A) ∩ X) = {(x 0 , x 1 , x 2 )|∃y(y ∈ N ∧ a < y ∧ y < x 0 )} which of course is {(x 0 , x 1 , x 2 )|a + 1 < x 0 }. qed 3.4.3 MsPs n ’s Definition Let A = (B n , . . . , B 0 , r, p, s, c − 0 , . . . , c − n−1) be a dMsPs n . An n-dimensional manysorted polyadic set algebra, or MsPs n for short, is an A ′ = (A, c + 0 , . . . , c + n−1) such that for each i < n the mapping c + i : B i → B i+1 has the property that for any dMsP n -embedding f from A to a full U ∈ dMsPs n and any i < n it is the case that c + i is a boolean embedding and that f(c + i (c − i (x))) = c U (f(x)). Moreover an n-homomorphism f from A ′ to some MsPs n is a MsP n -embedding if f is an embedding from A to some dMsPs n that preserves each of c + 0 , . . . , c + n−1. Corollary 3.4.3 There exists a MsPs n generated by a finite set of first-order definable relations over (N, +, | p ) that can not be embedded in the h-complex algebra of a properly partitioned finite automaton for any n-homomorphism h. Proof: As in the case of Ps n consider the Ps 3 generated by the set X = {(x 0 , x 1 , x 2 )|x 1 < x 0 }. Let A = (B n , . . . , B 0 , . . . , c + 0 , . . . , c + n−1) be the MsPs n generated by the set X. For each i < n the boolean algebra B i can be embedded in B n using c + 0 , . . . c + n−1. Therefore the Ps 3 generated by X is definable over the boolean algebra B n . Combining this proposition 3.4.2 we conclude that B n is infinite. qed 3.4.4 Polyadic equality algebras Here we will see that the situation is the same for polyadic equality algebras as for polyadic algebras using one or many sorts. Polyadic equality algebras were considered already by P. R. Halmos. In the finite-dimensional case they coincide with C.C. Pinters quantifier algebras with equality, see [Pin73]. The cylindric algebras of A. Tarski and the relation algebras of A. De Morgan and C. Peirce are reducts of polyadic equality algebras. Definition Let A = (B n , . . .) be either a dMsPs n , a Ps n or a MsPs n . Let for i, j < n the ij-diagonal d ij ∈ B n be defined by d ij = {(x 0 , . . . , x n−1 )|x i = x j }. Let A ′ = (A, {d ij } i,j
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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
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Lemma 1.2.3 If {R, ...} are the rel
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1.2.8 Exhaustive search for satisfy
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Lemma 1.3.1 L 33 ∪ L 32 ∪ L 31
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The following allows us to use homo
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Corollary 1.3.10 Let |= be a recurs
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the full Ps 3 over 3 has 27 atoms.
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set. In the example, the initial so
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Chapter 2 Automata for mechanising
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3. The finite automata of the class
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theorem [BHMV94], the p-recognisabl
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Using this theorem contrapositively
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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
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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
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
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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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