Unprovability results for clause set cycles

Hetzl, Stefan; Vierling, Jannik

doi:10.1016/j.tcs.2022.07.003

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2024

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(2 citation statements)

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“…This gives rise to the question whether right-decomposition can be proved by ∀ 1 induction and more generally to the more general question how ∃ 1 induction and ∀ 1 induction over lists with concatenation are related. This question is relevant for AITP, since there are systems such as [21] that are based on ∃ 1 induction [18] and systems such as [10] that are based on ∀ 1 induction [31, Chapter 5]. We plan to investigate this question separately in the future.…”

Section: Definition 48mentioning

confidence: 99%

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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: Definition 48mentioning

confidence: 99%

“…So far the work in this research program [17][18][19][20]31] has concentrated on induction for natural numbers only. However, since lists and other inductive data types are fundamental structures of computer science, it is of paramount importance for the subject of AITP to analyze the mechanical properties of these inductive datatypes.…”

Section: Introductionmentioning

confidence: 99%