## Cyclic proofs for first-order logic with inductive definitions (2005)

Venue: | In TABLEAUX’05, volume 3702 of LNCS |

Citations: | 16 - 5 self |

### BibTeX

@INPROCEEDINGS{Brotherston05cyclicproofs,

author = {James Brotherston},

title = {Cyclic proofs for first-order logic with inductive definitions},

booktitle = {In TABLEAUX’05, volume 3702 of LNCS},

year = {2005},

pages = {78--92},

publisher = {Springer-Verlag}

}

### OpenURL

### Abstract

Abstract. We consider a cyclic approach to inductive reasoning in the setting of first-order logic with inductive definitions. We present a proof system for this language in which proofs are represented as finite, locally sound derivation trees with a “repeat function ” identifying cyclic proof sections. Soundness is guaranteed by a well-foundedness condition formulated globally in terms of traces over the proof tree, following an idea due to Sprenger and Dam. However, in contrast to their work, our proof system does not require an extension of logical syntax by ordinal variables. A fundamental question in our setting is the strength of the cyclic proof system compared to the more familiar use of a non-cyclic proof system using explicit induction rules. We show that the cyclic proof system subsumes the use of explicit induction rules. In addition, we provide machinery for manipulating and analysing the structure of cyclic proofs, based primarily on viewing them as generating regular infinite trees, and also formulate a finitary trace condition sufficient (but not necessary) for soundness, that is computationally and combinatorially simpler than the general trace condition. 1

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Citation Context ... in the relatively simple, yet expressive context of first-order logic extended with ordinary inductive definitions 1 . Similar formalisms typically form the basis of mechanised theorem proving tools =-=[2, 8, 10, 13]-=-. The contribution of this paper is twofold. Firstly, we present a sound, powerful cyclic proof system that employs only the standard syntax of firstorder logic. It seems the system is most likely to ... |

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Citation Context ...ircular, reasoning have been employed in local model checking [3], theorem proving tools and frameworks [4, 7, 9, 16], in Turchin’s supercompilation [19] and in program verification based on automata =-=[20]-=-. It has also been studied in the context of tableau-style proof systems for the µ-calculus by Sprenger and Dam [17, 18] and Schöpp and Simpson [15], following an approach proposed by Dam and Gurov [5... |

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Citation Context ... in the relatively simple, yet expressive context of first-order logic extended with ordinary inductive definitions 1 . Similar formalisms typically form the basis of mechanised theorem proving tools =-=[2, 8, 10, 13]-=-. The contribution of this paper is twofold. Firstly, we present a sound, powerful cyclic proof system that employs only the standard syntax of firstorder logic. It seems the system is most likely to ... |

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Citation Context ...y defined predicate symbol ρ(t) ∈ P M otherwise 3 Sequent Calculus Proof Rules for FOL ind We shall consider proof systems for FOLind presented in the sequent calculus style originally due to Gentzen =-=[6]-=-, which is well-established as a convenient formalism for proof-theoretic reasoning. Our rules for inductively defined predicates are essentially sequent-calculus adaptations of Martin-Löf’s natural d... |

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Citation Context ... in the relatively simple, yet expressive context of first-order logic extended with ordinary inductive definitions 1 . Similar formalisms typically form the basis of mechanised theorem proving tools =-=[2, 8, 10, 13]-=-. The contribution of this paper is twofold. Firstly, we present a sound, powerful cyclic proof system that employs only the standard syntax of firstorder logic. It seems the system is most likely to ... |

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Citation Context ...have ϕ0 Φ = ∅. The ith component of ϕα Φ is then called the αth approximant2 of Pi, written as P α i . For a full exposition of the construction of the operator ϕΦ (which is standard), see e.g. Aczel =-=[1]-=-. 2 One can show that for any predicate P defined by our schema, P α = P ω for any α > ω, so it is sufficient to index approximants by natural numbers. However, the sequence of approximants does not n... |

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Citation Context ...nderlying inductive argument. Formssof cyclic, or circular, reasoning have been employed in local model checking [3], theorem proving tools and frameworks [4, 7, 9, 16], in Turchin’s supercompilation =-=[19]-=- and in program verification based on automata [20]. It has also been studied in the context of tableau-style proof systems for the µ-calculus by Sprenger and Dam [17, 18] and Schöpp and Simpson [15],... |

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Citation Context ...urposes, as ensuring well-foundedness of an underlying inductive argument. Formssof cyclic, or circular, reasoning have been employed in local model checking [3], theorem proving tools and frameworks =-=[4, 7, 9, 16]-=-, in Turchin’s supercompilation [19] and in program verification based on automata [20]. It has also been studied in the context of tableau-style proof systems for the µ-calculus by Sprenger and Dam [... |

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Citation Context ...urposes, as ensuring well-foundedness of an underlying inductive argument. Formssof cyclic, or circular, reasoning have been employed in local model checking [3], theorem proving tools and frameworks =-=[4, 7, 9, 16]-=-, in Turchin’s supercompilation [19] and in program verification based on automata [20]. It has also been studied in the context of tableau-style proof systems for the µ-calculus by Sprenger and Dam [... |

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Citation Context ...reasoning. Our rules for inductively defined predicates are essentially sequent-calculus adaptations of Martin-Löf’s natural deduction rules [11]; McDowell and Miller have considered a similar system =-=[12]-=-. We write sequents of the form Γ ⊢ ∆, where Γ, ∆ are finite multisets of formulas. We use the standard sequent calculus rules for the propositional connectives and quantifiers, as well as the followi... |

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Citation Context ...rise our progress and outline our aims for our ongoing work in this area. 1 However, there is no difficulty in extending our approach to more complex formalisms such as iterated inductive definitions =-=[11]-=-sDue to space constraints, the proofs of many results in this paper have been omitted and the proofs that appear are only sketched. Full proofs will appear in the author’s forthcoming PhD thesis. 2 Sy... |

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Citation Context ...on argument or, more pertinently for our purposes, as ensuring well-foundedness of an underlying inductive argument. Formssof cyclic, or circular, reasoning have been employed in local model checking =-=[3]-=-, theorem proving tools and frameworks [4, 7, 9, 16], in Turchin’s supercompilation [19] and in program verification based on automata [20]. It has also been studied in the context of tableau-style pr... |

19 | On the Structure of Inductive Reasoning: Circular and Tree-Shaped Proofs in the µ-Calculus
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Citation Context ...], in Turchin’s supercompilation [19] and in program verification based on automata [20]. It has also been studied in the context of tableau-style proof systems for the µ-calculus by Sprenger and Dam =-=[17, 18]-=- and Schöpp and Simpson [15], following an approach proposed by Dam and Gurov [5]. Our aim is to study cyclic reasoning in the relatively simple, yet expressive context of first-order logic extended w... |

12 |
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Citation Context ...urposes, as ensuring well-foundedness of an underlying inductive argument. Formssof cyclic, or circular, reasoning have been employed in local model checking [3], theorem proving tools and frameworks =-=[4, 7, 9, 16]-=-, in Turchin’s supercompilation [19] and in program verification based on automata [20]. It has also been studied in the context of tableau-style proof systems for the µ-calculus by Sprenger and Dam [... |

8 | On global induction mechanisms in a µ-calculus with explicit approximations
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Citation Context ...], in Turchin’s supercompilation [19] and in program verification based on automata [20]. It has also been studied in the context of tableau-style proof systems for the µ-calculus by Sprenger and Dam =-=[17, 18]-=- and Schöpp and Simpson [15], following an approach proposed by Dam and Gurov [5]. Our aim is to study cyclic reasoning in the relatively simple, yet expressive context of first-order logic extended w... |

5 | Verifying temporal properties using explicit approximants: Completeness for context-free processes
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(Show Context)
Citation Context ... [19] and in program verification based on automata [20]. It has also been studied in the context of tableau-style proof systems for the µ-calculus by Sprenger and Dam [17, 18] and Schöpp and Simpson =-=[15]-=-, following an approach proposed by Dam and Gurov [5]. Our aim is to study cyclic reasoning in the relatively simple, yet expressive context of first-order logic extended with ordinary inductive defin... |

3 |
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Citation Context |

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and Dilian Gurov. µ-calculus with explicit points and approximations
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Citation Context ...0]. It has also been studied in the context of tableau-style proof systems for the µ-calculus by Sprenger and Dam [17, 18] and Schöpp and Simpson [15], following an approach proposed by Dam and Gurov =-=[5]-=-. Our aim is to study cyclic reasoning in the relatively simple, yet expressive context of first-order logic extended with ordinary inductive definitions 1 . Similar formalisms typically form the basi... |

2 | Formal verification of processes
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Citation Context ...istence of a so-called trace manifold for a pre-proof — which is apparently less general than Definition 4.12 and formulated with respect to a so-called induction order (a notion introduced by Schöpp =-=[14]-=- and crucially employed by Sprenger and Dam [18]). A trace manifold consists of finite trace segments together with conditions ensuring that for any infinite path, the segments can be “glued together”... |