Mathematics in the age of the Turing machine

Hales, Thomas C.

doi:10.1017/cbo9781107338579.008

Cited by 8 publications

(5 citation statements)

References 85 publications

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“…Both sets of proofs are already minimized as described in Section 4.3. The proof dependencies were extracted 33 and the number of dependencies was compared. The complete results of this comparison are available online, 34 sorted by the difference between the length of the Original proof and the Advised proof.…”

Section: Initial Comparison Of the Advised And Original Proofsmentioning

confidence: 99%

Learning-Assisted Automated Reasoning with Flyspeck

2014

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The considerable mathematical knowledge encoded by the Flyspeck project is combined with external automated theorem provers (ATPs) and machine-learning premise selection methods trained on the Flyspeck proofs, producing an AI system capable of proving a wide range of mathematical conjectures automatically. The performance of this architecture is evaluated in a bootstrapping scenario emulating the development of Flyspeck from axioms to the last theorem, each time using only the previous theorems and proofs. It is shown that 39 % of the 14185 theorems could be proved in a push-button mode (without any high-level advice and user interaction) in 30 seconds of real time on a fourteen-CPU workstation. The necessary work involves: (i) an implementation of sound translations of the HOL Light logic to ATP formalisms: untyped first-order, polymorphic typed first-order, and typed higher-order, (ii) export of the dependency information from HOL Light and ATP proofs for the machine learners, and (iii) choice of suitable representations and methods for learning from previous proofs, and their integration as advisors with HOL Light. This work is described and discussed here, and an initial analysis of the body of proofs that were found fully automatically is provided.

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Section: Initial Comparison Of the Advised And Original Proofsmentioning

confidence: 99%

Learning-Assisted Automated Reasoning with Flyspeck

2014

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“…Isabelle/HOL, Satallax, and LEO-II performed well in recent experiments related to the Flyspeck project (Hales, 2013), in which a formalized proof of the Kepler conjecture is being developed (mainly) in HOL Light; cf. the experiments reported by Kaliszyk and Urban (2012 , Table 7).…”

Section: Recent Applications Of Automated Thf0 Proversmentioning

confidence: 84%

“…These systems are all based on the LCF approach (Gordon et al, 1979), in which powerful proof tactics are iteratively built up from a small kernel of basic proof rules. Other LCF-based provers for higher-order logic are the minimalist system HOL Light (Harrison, 2009), which provides powerful automation tactics and which has recently played a key role in the verification of Kepler's conjecture (Hales, 2013), and the ProofPower system (Arthan, 2011), which provides special support for a set-theoretic specification language.…”

Section: Early Systemsmentioning

confidence: 99%

Automation of Higher-Order Logic

Benzmüller¹,

Miller²

2014

Handbook of the History of Logic

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“…P x ==> !y. P y [22] (The extra 'consider a being A such that T;' step is needed because in miz3 variables have to be introduced before they can be used. The Jaśkowski/Fitch-style natural deduction proof can use the free variable a without introducing it, but in miz3 this is not allowed.…”

Section: The Miz3 Proof Languagementioning

confidence: 99%

A Synthesis of the Procedural and Declarative Styles of Interactive Theorem Proving

Wiedijk¹

2012

Logical Methods in Computer Science

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Abstract. We propose a synthesis of the two proof styles of interactive theorem proving: the procedural style (where proofs are scripts of commands, like in Coq) and the declarative style (where proofs are texts in a controlled natural language, like in Isabelle/Isar). Our approach combines the advantages of the declarative style -the possibility to write formal proofs like normal mathematical text -and the procedural style -strong automation and help with shaping the proofs, including determining the statements of intermediate steps.Our approach is new, and differs significantly from the ways in which the procedural and declarative proof styles have been combined before in the Isabelle, Ssreflect and Matita systems. Our approach is generic and can be implemented on top of any procedural interactive theorem prover, regardless of its architecture and logical foundations.To show the viability of our proposed approach, we fully implemented it as a proof interface called miz3, on top of the HOL Light interactive theorem prover. The declarative language that this interface uses is a slight variant of the language of the Mizar system, and can be used for any interactive theorem prover regardless of its logical foundations. The miz3 interface allows easy access to the full set of tactics and formal libraries of HOL Light, and as such has 'industrial strength'.Our approach gives a way to automatically convert any procedural proof to a declarative counterpart, where the converted proof is similar in size to the original. As all declarative systems have essentially the same proof language, this gives a straightforward way to port proofs between interactive theorem provers.

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Mathematics in the age of the Turing machine

Cited by 8 publications

References 85 publications

Learning-Assisted Automated Reasoning with Flyspeck

Learning-Assisted Automated Reasoning with Flyspeck

Automation of Higher-Order Logic

A Synthesis of the Procedural and Declarative Styles of Interactive Theorem Proving

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