Into the Infinite - Theory Exploration for Coinduction

Einarsdóttir, Sólrún Halla; Johansson, Moa; Pohjola, Johannes AAman

doi:10.1007/978-3-319-99957-9_5

Cited by 5 publications

(4 citation statements)

References 21 publications

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“…Finally, the use of transfinite lists used in this article are reminiscent of streams defined by coinduction. It could be interesting investigate to which extent the techniques employed for the analysis of AITP systems can be transferred to systems that automate the coinduction principle such as [11,26].…”

Section: Discussionmentioning

confidence: 99%

Quantifier-free induction for lists

Hetzl,

Vierling

2024

Arch. Math. Logic

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We investigate quantifier-free induction for Lisp-like lists constructed inductively from the empty list $$ nil $$ nil and the operation $${\textit{cons}}$$ cons , that adds an element to the front of a list. First we show that, for $$m \ge 1$$ m ≥ 1 , quantifier-free $$m$$ m -step induction does not simulate quantifier-free $$(m + 1)$$ ( m + 1 ) -step induction. Secondly, we show that for all $$m \ge 1$$ m ≥ 1 , quantifier-free $$m$$ m -step induction does not prove the right cancellation property of the concatenation operation on lists defined by left-recursion.

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Section: Discussionmentioning

confidence: 99%

Quantifier-free induction for lists

Hetzl,

Vierling

2024

Arch. Math. Logic

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“…exponentiation in Peano arithmetic) can drastically slow down testing if a large test case is generated. Similarly, co-recursive functions can lead to non-terminating values [10]. One solution is to extend QuickSpec with observational equivalence checking, similarly as shown in the Music example in section 2.3.…”

Section: Conclusion and Further Workmentioning

confidence: 99%

Conjectures, Tests and Proofs: An Overview of Theory Exploration

Johansson

Smallbone

2021

Electron. Proc. Theor. Comput. Sci.

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A key component of mathematical reasoning is the ability to formulate interesting conjectures about a problem domain at hand. In this paper, we give a brief overview of a theory exploration system called QuickSpec, which is able to automatically discover interesting conjectures about a given set of functions. QuickSpec works by interleaving term generation with random testing to form candidate conjectures. This is made tractable by starting from small sizes and ensuring that only terms that are irreducible with respect to already discovered conjectures are considered. QuickSpec has been successfully applied to generate lemmas for automated inductive theorem proving as well as to generate specifications of functional programs. We give an overview of typical use-cases of QuickSpec, as well as demonstrating how to easily connect it to a theorem prover of the user's choice.

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“…As opposed to using proof goal as the basis for discovering sub-goals, when eagerly generating lemmas there is a vast space of possible lemmas to consider. Currently, two main approaches exist for filtering candidate conjectures, counterexample-based and observational equivalencebased [18,22,23,43]. These filtering techniques are all based on testing and therefore require automatic creation of concrete examples.…”

Section: Introductionmentioning

confidence: 99%

“…-An evaluation of our theory exploration system on a set of benchmarks for induction proofs taken from CVC4 [37] and TIP 2015 [11], specifically the IsaPlanner benchmarks [21]. We compare our implementation with a current leading theory exploration system, Hipster [18], using a novel metric. This metric is insensitive to the amount of found lemmas, but rather measures their usefulness in the context of theorem proving.…”

Section: Introductionmentioning

confidence: 99%

Theory Exploration Powered by Deductive Synthesis

Singher

Itzhaky

2021

Lecture Notes in Computer Science

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This paper presents a symbolic method for automatic theorem generation based on deductive inference. Many software verification and reasoning tasks require proving complex logical properties; coping with this complexity is generally done by declaring and proving relevant sub-properties. This gives rise to the challenge of discovering useful sub-properties that can assist the automated proof process. This is known as the theory exploration problem, and so far, predominant solutions that emerged rely on evaluation using concrete values. This limits the applicability of these theory exploration techniques to complex programs and properties.In this work, we introduce a new symbolic technique for theory exploration, capable of (offline) generation of a library of lemmas from a base set of inductive data types and recursive definitions. Our approach introduces a new method for using abstraction to overcome the above limitations, combining it with deductive synthesis to reason about abstract values. Our implementation has shown to find more lemmas than prior art, avoiding redundant lemmas (in terms of provability), while being faster in most cases. This new abstraction-based theory exploration method is a step toward applying theory exploration to software verification and synthesis.

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Into the Infinite - Theory Exploration for Coinduction

Cited by 5 publications

References 21 publications

Quantifier-free induction for lists

Quantifier-free induction for lists

Conjectures, Tests and Proofs: An Overview of Theory Exploration

Theory Exploration Powered by Deductive Synthesis

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