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

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Contents 0.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 0.2 Summary of the chapters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 0.2.1 Summary of chapter 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 0.2.2 Summary of chapter 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 0.2.3 Summary of chapter 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 0.3 What is known . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 0.3.1 Classification of the Entscheidungsproblem based on syntax . . . . . . . 4 0.3.2 Beyond syntactically defined classes . . . . . . . . . . . . . . . . . . . . 6 0.3.3 Algebraic logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 0.3.4 Automated reasoning . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 0.3.5 Personal perspective and acknowledgements . . . . . . . . . . . . . . . . 8 1 Turning decision procedures into disprovers 11 1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 1.1.1 Sets and mappings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 1.1.2 First-order formulae . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 1.1.3 Algebras, homomorphisms, and preservation . . . . . . . . . . . . . . . 14 1.1.4 Algebras of boolean signature . . . . . . . . . . . . . . . . . . . . . . . 15 1.2 Algebras of polyadic signature . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 1.2.1 Operations related to variable substitution . . . . . . . . . . . . . . . . 16 1.2.2 Algebras of polyadic signature . . . . . . . . . . . . . . . . . . . . . . . 16 1.2.3 L 3 as an algebra of polyadic signature . . . . . . . . . . . . . . . . . . . 17 1.2.4 Polyadic set algebras of ternary relations . . . . . . . . . . . . . . . . . . 18 1.2.5 Interpretation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 1.2.6 Additivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 1.2.7 The Lindenbaum algebra of a theory Γ . . . . . . . . . . . . . . . . . . 21 1.2.8 Exhaustive search for satisfying interpretations in a Ps 3 . . . . . . . . . . 22 1.3 Algebras of directed many-sorted polyadic signature . . . . . . . . . . . . . . . . 22 1.3.1 Algebras of directed many-sorted polyadic signature . . . . . . . . . . . . 22 1.3.2 A conservative reduction class with sub-formulae as an algebra . . . . . . 23 1.3.3 Directed many-sorted polyadic set algebras . . . . . . . . . . . . . . . . 24 1.3.4 Interpretation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 1.3.5 The directed many-sorted polyadic closure . . . . . . . . . . . . . . . . 26 1.3.6 The closure as an algorithm . . . . . . . . . . . . . . . . . . . . . . . . 28 1.3.7 Exhaustive search for satisfying interpretations in a dMsPs 3 . . . . . . . 29 iii
Page 1: On the methods of mechanical non-th
Page 5: 3.2.2 The complex algebra tailored
Page 8 and 9: search. In chapter 3 we show how to
Page 10 and 11: atom-structures. Novelties of of ch
Page 12 and 13: 0.3.2 Beyond syntactically defined
Page 14 and 15: 0.3.4 Automated reasoning In regard
Page 17 and 18: Chapter 1 Turning decision procedur
Page 19 and 20: terms of computability, satisfiabil
Page 21 and 22: 1.1.4 Algebras of boolean signature
Page 23 and 24: Definition Let A = (B, r, p, s, c 0
Page 25 and 26: Figure 1.2: Substitution, cylindrif
Page 27 and 28: 1.2.7 The Lindenbaum algebra of a t
Page 29 and 30: In the following definition boolean
Page 31 and 32: 1.3.4 Interpretation We show that o
Page 33 and 34: B 1 = the sub-boolean-algebra of L
Page 35 and 36: carrier-set n → {⊥, ⊤}. This
Page 37 and 38: Figure 1.4: Cylindrification along
Page 39: ally such that each arity respectin
Page 42 and 43: 2.1 Introduction This is the first
Page 44 and 45: 2.1.2 Notation We assume familiarit
Page 46 and 47: Definition Let n, p ∈ N and let I
Page 48 and 49: Proof: Use a and b from the lemma a
Page 50 and 51: 2.2.1 3-fold vector spaces over the
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As usual we will say that two n-fol
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Definition Let (V, K, δ, ι, T ) b
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2.4.2 Transitive p-automata, the tw
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W |= ∀q[T (q) ↔ T (δ(q, 0))].
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7. It is known from classical autom
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Lemma 2.5.2 If (W, h) and (W ′ ,
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for each t ∈ P L(M p (n, m)), for
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A way of defining f is for ξ ∈ X
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By lemma 2.1.4 we obtain a k ∈ N
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2.6.1 One-sorted multi-automata Def
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Proof: To prove this we we add to t
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Proof: (If): For tapes of length 1
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(⊇) Let ˆσ(A 0 ) ⌢ · · ·
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(c) δ ∗ h(0, B 0 ⌢ · · ·
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Chapter 3 Automata for the computat
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algebra is simple if it is embeddab
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defined by the formulae v 0 = v 1 ,
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Definition An algebra A of dMsP n -
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H is a set of elements thought of a
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Proposition 3.2.2 Let {r, p, s} be
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We consider p, n, m as fixed throug
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Proposition 3.3.1 There is an effec
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⎡ ⎤ ⎡ ⎤ A 0 A n−1 A 1 ˆr
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1. for ˆσ ∈ {ˆr, ŝ, ˆp} we h
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3.3.6 The main result Combining the
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Proof: It is easy to see that the h
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Definition Let n, p ∈ N where p i
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The following corresponds to propos
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Bibliography [Aan71] [Aan83] S.O. A
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[JT51] [KMS02] [KMW62] [Men87] [MHT
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