Dynamic Rippling, Middle-Out Reasoning and Lemma Discovery

Johansson, Moa; Dixon, Lucas; Bundy, Alan

doi:10.1007/978-3-642-17172-7_6

Cited by 7 publications

(5 citation statements)

References 22 publications

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“…We note that learning mathematical abstractions—either in the form of problem-solving strategies or useful lemmas—has been extensively explored in the past work. However, the prior work has typically focused on either extracting lemmas or strategies from human-generated proofs and examples [22,23], or in producing potentially useful lemmas interactively in the context of a particular inductive proof [24,25]. To the best of our knowledge, our work is the first to demonstrate the learning of mathematical problem-solving and abstractions in a general computational foundation solely from exploration, without human examples.…”

Section: Related Workmentioning

confidence: 99%

Peano: learning formal mathematical reasoning

Poesia

Goodman

2023

Phil. Trans. R. Soc. A.

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General mathematical reasoning is computationally undecidable, but humans routinely solve new problems. Moreover, discoveries developed over centuries are taught to subsequent generations quickly. What structure enables this, and how might that inform automated mathematical reasoning? We posit that central to both puzzles is the structure of procedural abstractions underlying mathematics. We explore this idea in a case study on five sections of beginning algebra on the Khan Academy platform. To define a computational foundation, we introduce Peano, a theorem-proving environment where the set of valid actions at any point is finite. We use Peano to formalize introductory algebra problems and axioms, obtaining well-defined search problems. We observe existing reinforcement learning methods for symbolic reasoning to be insufficient to solve harder problems. Adding the ability to induce reusable abstractions (‘tactics’) from its own solutions allows an agent to make steady progress, solving all problems. Furthermore, these abstractions induce an order to the problems, seen at random during training. The recovered order has significant agreement with the expert-designed Khan Academy curriculum, and second-generation agents trained on the recovered curriculum learn significantly faster. These results illustrate the synergistic role of abstractions and curricula in the cultural transmission of mathematics. This article is part of a discussion meeting issue ‘Cognitive artificial intelligence’.

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Section: Related Workmentioning

confidence: 99%

Peano: learning formal mathematical reasoning

Poesia

Goodman

2023

Phil. Trans. R. Soc. A.

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“…85, Table 1 includes the time taken to complete when successful over an average of 10 runs. We omit problems 5,16,26,27,59,60,62,63,70,71,76,77 which are out of scope since they concern conditional equations. The remaining problems not listed in the table were in scope but our tool could not solve them.…”

Section: Implementation and Empirical Evaluationmentioning

confidence: 99%

“…Proof planning was developed as a way to better control the heuristic in automated reasoning tools [11]. It gave rise to Clam system and IsaPlanner [13,20,27]. A lemma discovery strategy based on "rippling", a form of rewriting used in proof planning, was to construct a lemma from a failed proof [24].…”

Section: Related Work and Conclusionmentioning

confidence: 99%

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CycleQ: An Efficient Basis for Cyclic Equational Reasoning

Jones¹,

Ong²,

Ramsay³

2021

Preprint

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We propose a new cyclic proof system for automated, equational reasoning about the behaviour of pure functional programs. The key to the system is the way in which cyclic proof and equational reasoning are mediated by the use of contextual substitution as a cut rule. We show that our system, although simple, already subsumes several of the approaches to implicit induction variously known as "inductionless induction", "rewriting induction", and "proof by consistency". By restricting the form of the traces, we show that global correctness in our system can be verified incrementally, taking advantage of the well-known size-change principle, which leads to an efficient implementation of proof search. Our CycleQ tool, accessible as a GHC plugin, shows promising results on a number of standard benchmarks.

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“…The proof planning approach suggests using middle-out reasoning here -the missing ingredients do not have to be immediately provided, but place-holders in the form of meta-variables are used to allow planning to continue; the meta-variables are then incrementally instantiated as verification conditions are satisfied. For the present example, this step is currently done on paper -see [12,11] for examples of middle-out reasoning in proof planning.…”

Section: From Plan To Proofmentioning

confidence: 99%

Theory Exploration: a role for Model Theory?

Smaill¹

EPiC Series in Computing

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There is an empirical claim that, when exploring a mathematical theory, after a succession of key results have been obtained, a point of equilibrium is reached where any query of interest can be resolved by routine reasoning from the results already established. Here is suggested some ways of thinking about the situation, in general. There are at least some situations where we can establish that all results (of a certain shape) will follow by routine reasoning from a small number of key properties. An example is described, and the significance for automated theory exploration discussed.

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Dynamic Rippling, Middle-Out Reasoning and Lemma Discovery

Cited by 7 publications

References 22 publications

Peano: learning formal mathematical reasoning

Peano: learning formal mathematical reasoning

CycleQ: An Efficient Basis for Cyclic Equational Reasoning

Theory Exploration: a role for Model Theory?

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