## Representing and reformulating diagonalization methods (1994)

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@TECHREPORT{Melis94representingand,

author = {Erica Melis},

title = {Representing and reformulating diagonalization methods},

institution = {},

year = {1994}

}

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### Abstract

Abstract Finding an appropriate representation of planning operators is crucial for theorem provers that work with proof planning. We show a new representation of operators and demonstrate how diagonalization can be represented by operators. We explain how a diagonalization operator used in one proof-plan can be analogically transferred to an operator used in another proof-plan. Finally, we find an operator that is common to all the proof-plans and thus might be considered as the Diagonal Method. This research was supported by the Max-Kade Foundation Keywords: proof planning, analogy, knowledge representation 1 Introduction As pointed out by Bundy [3] and Bledsoe [1], using proof-plans is often very helpful in automated deduction. In planning, operators are needed and therefore an appropriate representation of these operators is crucial for proof planning. The operators have the same function in proof planning as mathematical methods (in the following referred to as mmethods) have in human theorem proving. Since m-methods can be adapted to different proofs, it is also desirable to have mechanisms for adapting operators. To be employed by a human-oriented theorem prover, these operators should allow for representing logical proof methods, such as Indirect Proof, and mathematical methods, such as Cantor's Diagonal method. In this paper we examine whether the presented representation actually covers mathematician's methods and how the methods can be adapted for other proof plans. We do this by analyzing the well-known Diagonal Method which is central and widely applicable in many mathematical proofs concerning computability and decidability, including G"odel's Incompleteness theorem for arithmetic, the Unsolvability of the halting problem, Rice's theorem (see [5]), and the Second Recursion theorem (see [5]). Although this m-method seems to be clearly understood, not all proofs have an obvious common proof schema, and some proofs are difficult to generate in logical detail.

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Citation Context ...s might be considered as the Diagonal Method. This research was supported by the Max-Kade Foundation Keywords: proof planning, analogy, knowledge representation 1 Introduction As pointed out by Bundy =-=[3]-=- and Bledsoe [1], using proof-plans is often very helpful in automated deduction. In planning, operators are needed and therefore an appropriate representation of these operators is crucial for proof ... |

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Citation Context ...mmon to all the proofs and which might be considered as the Diagonal Method. 2 Representation of Methods First we give a brief definition of methods that allow for reformulation (For more details see =-=[6, 8]-=-). Sequents P, written as (ass ` concl), are pairs of a set ass of formulas and a formula concl in an object language that is extended by meta-variables for formulas, sets of formulas, and terms 1 . A... |

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Citation Context ...mmon to all the proofs and which might be considered as the Diagonal Method. 2 Representation of Methods First we give a brief definition of methods that allow for reformulation (For more details see =-=[6, 8]-=-). Sequents P, written as (ass ` concl), are pairs of a set ass of formulas and a formula concl in an object language that is extended by meta-variables for formulas, sets of formulas, and terms 1 . A... |

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Citation Context ...dered as the Diagonal Method. This research was supported by the Max-Kade Foundation Keywords: proof planning, analogy, knowledge representation 1 Introduction As pointed out by Bundy [3] and Bledsoe =-=[1]-=-, using proof-plans is often very helpful in automated deduction. In planning, operators are needed and therefore an appropriate representation of these operators is crucial for proof planning. The op... |

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Citation Context ...dered as the Diagonal Method. This research was supported by the Max-Kade FoundationsKeywords: proof planning, analogy, knowledge representations1 Introduction As pointed out by Bundy [3] and Bledsoe =-=[1]-=-, using proof-plans is often very helpful in automated deduction. In planning, operators are needed and therefore an appropriate representation of these operators is crucial for proof planning. The op... |

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Citation Context ...smaller than the cardinality of the powerset PM of M , cardM ! cardPM (see [2]) 2. Uncountability of the set of real numbers (actually the interval [0 1]), which means that card IN ! card [0; 1] (see =-=[4]-=-) 3. Unsolvability of the Halting Problem for Turing machines, which means there is no algorithm (no t-computable function) to determine whether an arbitrary Turing machine in an arbitrary configurati... |

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Decomposition techniques and their applications
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Citation Context ...tructuring meta-methods. Some of these meta-methods are Deduction-Theorem-Splitting, Conjunctive-Decomposition, and Apply-Axiom-Splitting. For a more detailed motivation, description and examples see =-=[9]-=-. Table 1 shows the top-level procedure of the analogy-driven proof-plan construction. The actual analogy procedure is embedded into the planning by a basic planner. Starting with a given source proof... |

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Citation Context ... non, G enumerability of T construction of G in T f(xo)(xo) = G(xo) contradiction Figure 2: Proof Structure of the Halting Problem 3.1.4 Godel's Theorem The following mathematical proof is taken from =-=[11]-=-. A Godel numbering f of the expressions in S with one variable is assumed. Lemma: The predicate R(xy) which states that the proof with Godel number y proves the sentence which is the instantiation OE... |

1 | Representing the diagonalization methods
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Citation Context ...mmon to all the proofs and which might be considered as the Diagonal Method. 2 Representation of Methods First we give a brief definition of methods that allow for reformulation (For more details see =-=[6]-=-). Sequents P, written as (ass ` concl), are pairs of a set ass of formulas and a formula concl in an object language that is extended by meta-variables for formulas, sets of formulas, and terms. As a... |

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Citation Context ...)) = 0 . :c(F (x 0 )(x 0 )) = 0) , and the definition of c. Figure 2 shows a proof structure of the proof of the halting problem. 3.1.4 G"odel's Theorem The following mathematical proof is taken from =-=[11]-=-. A G"odel numbering f of the expressions in S with one variable is assumed. Lemma: The predicate R(xy) which states that the proof with G"odel number y proves the sentence which is the instantiation ... |