Abstract
The simulation of mathematical reasoning has been a driving force throughout the history of Artificial Intelligence research. However, despite significant successes in computer mathematics, computers are not widely used by mathematicians apart from their quotidian applications. An oft-cited reason for this is that current computational systems cannot do mathematics in the way that humans do. We draw on two areas in which Automated Theorem Proving (ATP) is currently unlike human mathematics: firstly in a focus on soundness, rather than understandability of proof, and secondly in social aspects. Employing techniques and tools from argumentation to build a framework for mixed-initiative collaboration, we develop three complementary arcs. In the first arc – our theoretical model – we interpret the informal logic of mathematical discovery proposed by Lakatos, a philosopher of mathematics, through the lens of dialogue game theory and in particular as a dialogue game ranging over structures of argumentation. In our second arc – our abstraction level – we develop structured arguments, from which we induce abstract argumentation systems and compute the argumentation semantics to provide labelings of the acceptability status of each argument. The output from this stage corresponds to a final, or currently accepted proof artefact, which can be viewed alongside its historical development. Finally, in the third arc – our computational model – we show how each of these formal steps is available in implementation. In an appendix, we demonstrate our approach with a formal, implemented example of real-world mathematical collaboration. We conclude the paper with reflections on our mixed-initiative collaborative approach.
Highlights
The simulation of mathematical reasoning has been a driving force throughout the history of Artificial Intelligence research [98, 86, 58, 87]
Evaluation metrics in the Automated Theorem Proving (ATP) community are focused on soundness, and the power of a solver to prove a wide selection of difficult problems with specific resource limits
Since the record of the dialogue can be presented alongside this proof, the framework delivers the history of a proof attempt as well as the proof artefact
Summary
The simulation of mathematical reasoning has been a driving force throughout the history of Artificial Intelligence research [98, 86, 58, 87]. An oft-cited reason for this is that current computational systems cannot do mathematics in the way that humans do. Despite – or perhaps because of [69] – their profound rigour, machine proofs are often thought to be unclear, uninspiring and untrustworthy, as opposed to human proofs which can be deep, elegant and explanatory [21, 41]. In order to help to close the gap between machine-constructed proofs and human-constructed ones, we consider two key areas of focus: informal and social aspects of proof discovery in the human context. We propose that theories and tools from the field of argumentation can be used to more closely align AI systems with the human context in these two areas
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