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

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1.2.4 Polyadic set algebras of ternary relations We have seen how to turn a fragment of first-order language into an algebra of polyadic signature. Here the ternary relations over given sets are are turned into algebras of polyadic signature. A mapping from the variables of L 3 to some set U is called a sequence and an interpretation is an assignment of each formula of L 3 to a set of sequences. This form of interpretation is due to Tarski. In contrast to other, quite viable, forms of interpretation tarskian interpretation makes Γ-equivalence coincide with equality of interpretations in given models for Γ. Since we use 3 variables, a set of sequences is a ternary relation. We refer to the relation assigned to φ by an interpretation as the relation defined by φ. By the classical Löwenheim-Skolem-Tarski theorems and the absence of a symbol for equality, satisfiable sentences each have a model over the reals, R. In such models L 3 -formulae define subsets of R 3 . This gives rise to a geometric interpretation of elements of Ps 3 ’s that will be appealed to later, instead of very detailed symbolic proof, see fig.1.1 and fig. 1.2. Definition Let U be some set. The full Ps 3 over U is an algebra (P(U 3 ), ∪, −, ∅, r U , p U , s U , c U 0 , c U 1 , c U 2 ) where (P(U 3 ), ∪, −, ∅) is the boolean algebra of sets of triples of elements of U. Operations related to variable substitutions are for V ⊆ U r U (V ) = {u ∈ U 3 : u ◦ r ∈ V } = {abc ∈ U 3 : cab ∈ V } p U (V ) = {u ∈ U 3 : u ◦ p ∈ V } = {abc ∈ U 3 : bac ∈ V } s U (V ) = {u ∈ U 3 : u ◦ s ∈ V } = {abc ∈ U 3 : bbc ∈ V } Operations for the existential quantifiers are c U 0 (V ) = {abc ∈ U 3 : there is a u ∈ U such that ubc ∈ V } c U 1 (V ) = {abc ∈ U 3 : there is a u ∈ U such that auc ∈ V } c U 2 (V ) = {abc ∈ U 3 : there is a u ∈ U such that abu ∈ V } An algebra is said to be a full Ps 3 if it is the full Ps 3 over some set U. By abuse of notation let P(U 3 ) denote the full Ps 3 over U. Figure 1.1: Cylindrification along x-axis, z-axis is orthogonal to the present paper and the dotted line marks the xy-diagonal plane which is also orthogonal to the present paper. 18
Figure 1.2: Substitution, cylindrifies that which meets the xy-diagonal plane. Rotation and permutation amount to suitably renaming the axis, then rotating or mirroring the figures respectively. Definition A Ps 3 is an algebra of polyadic signature that is embeddable into a full Ps 3 . The next property relates the defining equations for r ∗ , s ∗ , p ∗ to those of r U , s U , p U . For this the author finds geometric interpretation helpful, see fig. 1.1 and fig. 1.2. Lemma 1.2.2 For each i ∈ 3 and V, W ⊆ U r U (¬V ) = ¬(r U (V )). p U (¬V ) = ¬(p U (V )). s U (¬V ) = ¬(s U (V )). r U (V ∪ W ) = r U (V ) ∪ r U (W ). p U (V ∪ W ) = p U (V ) ∪ p U (W ). s U (V ∪ W ) = s U (V ) ∪ s U (W ). r U (c U i (V )) = c U r(i) V . p U (c U i (V )) = p U p(i) V . s U (c U 0 (V )) = c U 0 (V ). s U (c U 1 (V )) = c U 0 (p U (V ). s U (c U 2 (V )) = c U 2 (s U (V ). 1.2.5 Interpretation Here we go some length to see that polyadic homomorphisms from L 3 to Ps 3 ’s are essentially the same as interpretations of L 3 formulae as ternary relations over some set. The statement is broken down into a few extension properties, which are of use later in the present paper. Extension properties are also known as universal properties. The first extension property says that polyadic homomorphisms from L 3 to Ps 3 ’s are like interpretations in that if we interpret the relation-symbols of L 3 then an interpretation of each atomic formula of L 3 is determined. We use the notation {R, ...} × {012} for the set of atomic formulae of L 3 in which all variables of L 3 occur and in which they occur in the order specified by 012. The following is a consequence of lemma 1.2.1. 19
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 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 ⌢ · · · ⌢
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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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