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Presenting intuitive deductions via symmetric simplification
 In CADE10: Proceedings of the tenth international conference on Automated deduction
, 1990
"... In automated deduction systems that are intended for human use, the presentation of a proof is no less important than its discovery. For most of today’s automated theorem proving systems, this requires a nontrivial translation procedure to extract humanoriented deductions from machineoriented pro ..."
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In automated deduction systems that are intended for human use, the presentation of a proof is no less important than its discovery. For most of today’s automated theorem proving systems, this requires a nontrivial translation procedure to extract humanoriented deductions from machineoriented proofs. Previously known translation procedures, though complete, tend to produce unintuitive deductions. One of the major flaws in such procedures is that too often the rule of indirect proof is used where the introduction of a lemma would result in a shorter and more intuitive proof. We present an algorithm, symmetric simplification, for discovering useful lemmas in deductions of theorems in first and higherorder logic. This algorithm, which has been implemented in the TPS system, has the feature that resulting deductions may no longer have the weak subformula property. It is currently limited, however, in that it only generates lemmas of the form C ∨ ¬C ′ , where C and C ′ have the same negation normal form. 1
Proof Interpretations and the Computational Content of Proofs. Draft of book in preparation
, 2007
"... This survey reports on some recent developments in the project of applying proof theory to proofs in core mathematics. The historical roots, however, go back to Hilbert’s central theme in the foundations of mathematics which can be paraphrased by the following question ..."
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Cited by 14 (1 self)
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This survey reports on some recent developments in the project of applying proof theory to proofs in core mathematics. The historical roots, however, go back to Hilbert’s central theme in the foundations of mathematics which can be paraphrased by the following question
Real Computation with Least Discrete Advice: A Complexity Theory of Nonuniform Computability
, 2009
"... It is folklore particularly in numerical and computer sciences that, instead of solving some general problem f: A → B, additional structural information about the input x ∈ A (that is any kind of promise that x belongs to a certain subset A ′ ⊆ A) should be taken advantage of. Some examples from ..."
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It is folklore particularly in numerical and computer sciences that, instead of solving some general problem f: A → B, additional structural information about the input x ∈ A (that is any kind of promise that x belongs to a certain subset A ′ ⊆ A) should be taken advantage of. Some examples from real number computation show that such discrete advice can even make the difference between computability and uncomputability. We turn this into a both topological and combinatorial complexity theory of information, investigating for several practical problems how much advice is necessary and sufficient to render them computable. Specifically, finding a nontrivial solution to a homogeneous linear equation A · x = 0 for a given singular real n × nmatrix A is possible when knowing rank(A) ∈ {0, 1,..., n−1}; and we show this to be best possible. Similarly, diagonalizing (i.e. finding a basis of eigenvectors of) a given real symmetric n × nmatrix A is possible when knowing the number of distinct eigenvalues: an integer between 1 and n (the latter corresponding to the nondegenerate case). And again we show that n–fold (i.e. roughly log n bits of) additional information is indeed necessary in order to render this problem (continuous and) computable; whereas finding some single eigenvector of A requires and suffices with Θ(log n)–fold advice.
Herbrand’s theorem and extractive proof theory
, 2008
"... Extractive Proof Theory: New results by logical analysis of proofs Proof theory has its historic origin in foundational issues centered around (relative) consistency proofs (Hilbert’s program). Since the 1950’s Georg Kreisel pushed for a shift of emphasis in proof theory towards the use of proof th ..."
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Extractive Proof Theory: New results by logical analysis of proofs Proof theory has its historic origin in foundational issues centered around (relative) consistency proofs (Hilbert’s program). Since the 1950’s Georg Kreisel pushed for a shift of emphasis in proof theory towards the use of proof theoretic transformations (as developed in the course of Hilbert’s program) to analyze given proofs P e.g. of ineffectively proved ∀∃statements C with the aim to extract new information on C that could not be read off from P directly. Herbrand’s fundamental theorem plays an important role in this development. The general situation is as follows: Input: Ineffective proof P of C Goal: Additional information on C: • effective bounds (e.g. on the number of solutions of an ineffectively proven finiteness theorem, see theorem 1.9) or effective rates of convergence in nonlinear analysis (see sections 4 and 5), • algorithms for computation of actual solutions of ineffectively established existential statements, • continuous dependency or full independence from certain parameters (e.g. rates of convergence or stability for iterative processes in fixed point theory and ergodic theory that are independent
1. Extractive Proof Theory: New results by logical analysis of proofs
"... Proof theory has its historic origin in foundational issues centered around ..."
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Proof theory has its historic origin in foundational issues centered around