Since the proof of the four color theorem in 1976, computer-generated proofs have become a reality in mathematics and computer science. During the last decade, we have seen formal proofs using verified proof assistants being used to verify the validity of such proofs.
The proof of the four colour theorem from 1976 @cite_12 @cite_5 was not the first computer-assisted proof, but it was the first to generate broad awareness of a new area of mathematics, sometimes dubbed experimental'' or computational'' mathematics. Since then, numerous theorems in mathematics and computer science have been proved by computer-assisted and computer-generated proofs. Besides obvious philosophical debates about what constitutes a mathematical proof, concerns about the validity of such proofs have been raised since. In particular, proofs based on exhausting the solution space have been met with skepticism.
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The Tale of a Brainteaser Francis Guthrie certainly did it, when he coined his innocent little coloring puzzle in 1852. He managed to embarrass successively his mathematician brother, his brother’s professor, Augustus de Morgan, and all of de Morgan’s visitors, who couldn’t solve it; the Royal Society, who only realized ten years later that Alfred Kempe’s 1879 solution was wrong; and the three following generations of mathematicians who couldn’t fix it [19]. Even Appel and Haken’s 1976 triumph [2] had a hint of defeat: they’d had a computer do the proof for them! Perhaps the mathematical controversy around the proof died down with their book [3] and with the elegant 1995 revision [13] by Robertson, Saunders, Seymour, and Thomas. However something was still amiss: both proofs combined a textual argument, which could reasonably be checked by inspection, with computer code that could not. Worse, the empirical evidence provided by running code several times with the same input is weak, as it is blind to the most common cause of “computer” error: programmer error. For some thirty years, computer science has been working out a solution to this problem: formal program proofs. The idea is to write code that describes not only what the machine should do, but also why it should be doing it—a formal proof of correctness. The validity of the proof is an objective mathematical fact that can be checked by a different program, whose own validity can be ascertained empirically because it does run on many inputs. The main technical difficulty is that formal proofs are very difficult to produce, For more than three and one-half decades, beginning in the early 1960s, a heavy emphasis on proof finding has been a key component of the Argonne paradigm, whose use has directly led to significant advances in automated reasoning and important contributions to mathematics and logic. The theorems studied range from the trivial to the deep, even including some that corresponded to open questions. Often the paradigm asks for a theorem whose proof is in hand but that cannot be obtained in a fully automated manner by the program in use. The theorem whose hypothesis consists solely of the Meredith single axiom for two-valued sentential (or propositional) calculus and whose conclusion is the Lukasiewicz three-axiom system for that area of formal logic was just such a theorem. Featured in this article is the methodology that enabled the program OTTER to find the first fully automated proof of the cited theorem, a proof with the intriguing property that none of its steps contains a term of the form i>n(i>n(i>t)) for any term i>t. As evidence of the power of the new methodology, the article also discusses OTTER's success in obtaining the first known proof of a theorem concerning a single axiom of Lukasiewicz. 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. It is demonstrated that there is no nine-input sorting network of depth six. The proof was obtained by executing on a supercomputer a branch-and-bound algorithm which constructs and tests a critical subset of all possible candidates. Such proofs can be classified as experimental science, rather than mathematics. In keeping with the paradigms of experimental science, a high-level description of the experiment and analysis of the result are given.