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Every Computably Enumerable Random Real Is Provably Computably Enumerable Random
, 2009
"... We prove that every computably enumerable (c.e.) random real is provable in Peano Arithmetic (PA) to be c.e. random. A major step in the proof is to show that the theorem stating that “a real is c.e. and random iff it is the halting probability of a universal prefixfree Turing machine ” can be prov ..."
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We prove that every computably enumerable (c.e.) random real is provable in Peano Arithmetic (PA) to be c.e. random. A major step in the proof is to show that the theorem stating that “a real is c.e. and random iff it is the halting probability of a universal prefixfree Turing machine ” can be proven in PA. Our proof, which is simpler than the standard one, can also be used for the original theorem. Our positive result can be contrasted with the case of computable functions, where not every computable function is provably computable in PA, or even more interestingly, with the fact that almost all random finite strings are not provably random in PA. We also prove two negative results: a) there exists a universal machine whose universality cannot be proved in PA, b) there exists a universal machine U such that, based on U, PA cannot prove the randomness of its halting probability. The paper also includes a sharper form of the KraftChaitin Theorem, as well as a formal proof of this theorem written with the proof assistant Isabelle.
The Complexity of the Four Colour Theorem
, 2009
"... The four colour theorem states that the vertices of every planar graph can be coloured with at most four colours so that no two adjacent vertices receive the same colour. This theorem is famous for many reasons, including the fact that its original 1977 proof includes a nontrivial computer verifica ..."
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The four colour theorem states that the vertices of every planar graph can be coloured with at most four colours so that no two adjacent vertices receive the same colour. This theorem is famous for many reasons, including the fact that its original 1977 proof includes a nontrivial computer verification. Recently, a formal proof of the theorem was obtained with the equational logic program Coq. In this paper we use the computational method for evaluating (in a uniform way) the complexity of mathematical problems presented in [8, 6] to evaluate the complexity of the four colour theorem. Our method uses a Diophantine equational representation of the theorem. We show that the four colour theorem has roughly the same complexity as the Riemann hypothesis and almost four times the complexity of Fermat’s last theorem. 1
Evaluating the Complexity of Mathematical Problems. Part 1
, 2009
"... In this paper we provide a computational method for evaluating in a uniform way the complexity of a large class of mathematical problems. The method, which is inspired by NKS1, is based on the possibility to completely describe complex mathematical problems, like the Riemann hypothesis, in terms of ..."
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In this paper we provide a computational method for evaluating in a uniform way the complexity of a large class of mathematical problems. The method, which is inspired by NKS1, is based on the possibility to completely describe complex mathematical problems, like the Riemann hypothesis, in terms of (very) simple programs. The method is illustrated on a variety of examples coming from different areas of mathematics and its power and limits are studied.
Formal Proof: Reconciling Correctness and Understanding
, 2009
"... Hilbert’s concept of formal proof is an ideal of rigour for mathematics which has important applications in mathematical logic, but seems irrelevant for the practice of mathematics. The advent, in the last twenty years, of proof assistants was followed by an impressive record of deep mathematical th ..."
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Hilbert’s concept of formal proof is an ideal of rigour for mathematics which has important applications in mathematical logic, but seems irrelevant for the practice of mathematics. The advent, in the last twenty years, of proof assistants was followed by an impressive record of deep mathematical theorems formally proved. Formal proof is practically achievable. With formal proof, correctness reaches a standard that no penandpaper proof can match, but an essential component of mathematics — the insight and understanding — seems to be in short supply. So, what makes a proof understandable? To answer this question we first suggest a list of symptoms of understanding. We then propose a vision of an environment in which users can write and check formal proofs as well as query them with reference to the symptoms of understanding. In this way, the environment reconciles the main features of proof: correctness and understanding.
Mathematical Problems. Part 1 ∗
, 2008
"... In this paper we provide a computational method for evaluating in a uniform way the complexity of a large class of mathematical problems. The method is illustrated on a variety of examples coming from different areas of mathematics and its power and limits are studied. 1 ..."
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In this paper we provide a computational method for evaluating in a uniform way the complexity of a large class of mathematical problems. The method is illustrated on a variety of examples coming from different areas of mathematics and its power and limits are studied. 1