## Applying Tree Languages in Proof Theory (2012)

Venue: | In Adrian-Horia Dediu and Carlos Martín-Vide, editors, Language and Automata Theory and Applications (LATA) 2012, volume 7183 of Lecture Notes in Computer Science |

Citations: | 8 - 7 self |

### BibTeX

@INPROCEEDINGS{Hetzl12applyingtree,

author = {Stefan Hetzl},

title = {Applying Tree Languages in Proof Theory},

booktitle = {In Adrian-Horia Dediu and Carlos Martín-Vide, editors, Language and Automata Theory and Applications (LATA) 2012, volume 7183 of Lecture Notes in Computer Science},

year = {2012},

publisher = {Springer}

}

### OpenURL

### Abstract

Abstract. We introduce a new connection between formal language theory and proof theory. One of the most fundamental proof transformations in a class of formal proofs is shown to correspond exactly to the computation of the language of a certain class of tree grammars. Translations in both directions, from proofs to grammars and from grammars to proofs, are provided. This correspondence allows theoretical as well as practical applications. 1

### Citations

581 |
Untersuchungen über das logische Schließen
- Gentzen
- 1935
(Show Context)
Citation Context ...equently, the language L(G) of a totally rigid and acyclic G is finite.3 From Proofs to Tree Languages Applying Tree Languages in Proof Theory 305 We now turn to proof theory. In the seminal article =-=[8]-=-, which can be considered the founding work of structural proof theory, Gentzen introduced the sequent calculus and proved the cut-elimination theorem. A sequent is a pair of multisets of formulas wri... |

162 | Basic Proof Theory
- Troelstra, Schwichtenberg
- 2000
(Show Context)
Citation Context ...formation is, for a number of reasons, of central importance in proof theory. It essentially works by a stepwise reduction of the complexity of the cuts (lemmas), the interested reader is referred to =-=[19]-=-. In this paper we show that for a certain class of proofs, cut-elimination corresponds exactly to the computation of the language of a certain tree grammar: we give translations from proofs to gramma... |

114 |
Recherches sur la théorie de la démonstration
- Herbrand
- 1930
(Show Context)
Citation Context ...Q and xy ∈ Q”, the above proof only shows that ϕ( √ 2, √ 2) ∨ ϕ( √ √ 2, √ 2 2) is provable (from suitable basic axioms). This situation is described from a logical point of view by Herbrand’s theorem =-=[9,2]-=-. In its simplest form it states that ∃xϕ(x) for a quantifier-free formula ϕ(x) is valid iff there are terms t1,...,tn s.t. ∨ n i=1 ϕ(ti) is a tautology. Such a disjunction is therefore also called He... |

56 |
Applied Proof Theory: Proof Interpretations and their Use in Mathematics
- Kohlenbach
- 2008
(Show Context)
Citation Context ...nd incompleteness theorem, turned out to be too ambitious. However, other kinds of analyses of proofs are possible. One type of analysis that has received a lot of attention in recent years (see e.g. =-=[15]-=-) is proof mining: the extraction of additional mathematical information from existing proofs. Such additional information can often be thought of as concrete values for existential quantifiers. In th... |

53 | The lengths of proofs
- Pudlák
(Show Context)
Citation Context ... precisely: for every simple proof π of T ⊢∃xA with A quantifier-free there are t1,...,tk with k ≤|π| |π| s.t. T ⊢ A[x\t1],...,A[x\tk]. On the other hand, cut-elimination in general is non-elementary =-=[23,17,18]-=- which shows that simplicity is a necessary assumption for Theorem 22.4 From Tree Languages to Proofs Applying Tree Languages in Proof Theory 309 Already the results in [11] show that a simple proof ... |

38 |
bounds on Herbrand’s theorem
- Statman, Lower
- 1979
(Show Context)
Citation Context ... precisely: for every simple proof π of T ⊢∃xA with A quantifier-free there are t1,...,tk with k ≤|π| |π| s.t. T ⊢ A[x\t1],...,A[x\tk]. On the other hand, cut-elimination in general is non-elementary =-=[23,17,18]-=- which shows that simplicity is a necessary assumption for Theorem 22.4 From Tree Languages to Proofs Applying Tree Languages in Proof Theory 309 Already the results in [11] show that a simple proof ... |

32 |
A Compact Representation of Proofs
- Miller
- 1987
(Show Context)
Citation Context ...risations of classes of proofs by classes of grammars. In principle, such a result is conceivable for any proof system that possesses an Herbrand-like theorem, i.e. even full higher-order logic as in =-=[16]-=-. The challenge consists in finding an appropriate type of grammars. Acknowledgements. The author would like to thank the anonymous referees for helpful comments and suggestions that led to significan... |

26 |
Automaticity I: Properties of a measure of descriptional complexity
- Shallit, Breitbart
- 1996
(Show Context)
Citation Context ...c. Furthermore, these results show that the length of a proof with cut (which is notoriously difficult to control) is intimately related to measures such as automatic complexity [21] and automaticity =-=[20]-=-, more precisely: Corollary 26. Let ∃xA be a formula and k ∈ N s.t. T ⊢ A[x\t1],...,A[x\tn] implies that every totally rigid acyclic grammar G with L(G) ={t1,...,tn} has |G| ≥k, then every simple proo... |

21 |
Lower bounds for increasing complexity of derivations after cut elimination
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- 1979
(Show Context)
Citation Context ... precisely: for every simple proof π of T ⊢∃xA with A quantifier-free there are t1,...,tk with k ≤|π| |π| s.t. T ⊢ A[x\t1],...,A[x\tk]. On the other hand, cut-elimination in general is non-elementary =-=[23,17,18]-=- which shows that simplicity is a necessary assumption for Theorem 22.4 From Tree Languages to Proofs Applying Tree Languages in Proof Theory 309 Already the results in [11] show that a simple proof ... |

17 | On Herbrand’s Theorem
- Buss
(Show Context)
Citation Context ...Q and xy ∈ Q”, the above proof only shows that ϕ( √ 2, √ 2) ∨ ϕ( √ √ 2, √ 2 2) is provable (from suitable basic axioms). This situation is described from a logical point of view by Herbrand’s theorem =-=[9, 2]-=-. In its simplest form it states that ∃x ϕ(x) for a quantifier-free formula ϕ(x) is valid iff there are terms t1, . . . , tn s.t. ∨ n i=1 ϕ(ti) is a tautology. Such a disjunction is therefore also cal... |

16 |
Cut normal forms and proof complexity
- Baaz, Leitsch
- 1999
(Show Context)
Citation Context ...a given proof ψ, write Gnd(ψ) for G(ψ) from which all productions of the form β → d for any non-terminal β have been removed. By proof-theoretic transformations – in particular the prenexification of =-=[1]-=-, see also [4, Theorem VII.4.7], applied to cut-formulas – we obtain a proof π ′ all of whose cuts are of the form ∃x B for B quantifier-free and which satisfies L(Gnd(π ′ )) = L(G). The role of the d... |

8 | Tree Automata With Global Constraints, in "International
- FILIOT, TALBOT, et al.
(Show Context)
Citation Context ...plicitly, see [3] for a survey, while others consider global constraints specified via states. An important class of the latter kind are tree automata with global equalities and disequalities (TAGED) =-=[5,6,7]-=-. For the purposes of this paper it will turn out to be natural to work with rigid tree automata that have been introduced in [13], see also [14]. They are a subclass of TAGED (characterised by having... |

8 | Rigid tree automata
- Jacquemard, Klay, et al.
- 2009
(Show Context)
Citation Context ... one can obtain concrete values for existential quantifiers by computing the language of a grammar. In Section 2 we will develop a suitable notion of tree grammar corresponding to rigid tree automata =-=[13,14]-=-, in Section 3 we describe how to translate proofs to grammars and in Section 4 how to translate grammars to proofs. Most proofs in this paper will only be sketched, the reader interested in more deta... |

6 | Automatic complexity of strings
- Shallit, Wang
(Show Context)
Citation Context ...f two given proofs, etc. Furthermore, these results show that the length of a proof with cut (which is notoriously difficult to control) is intimately related to measures such as automatic complexity =-=[21]-=- and automaticity [20], more precisely: Corollary 26. Let ∃xA be a formula and k ∈ N s.t. T ⊢ A[x\t1],...,A[x\tn] implies that every totally rigid acyclic grammar G with L(G) ={t1,...,tn} has |G| ≥k, ... |

5 | Satisfiability of a spatial logic with tree variables
- Filiot, Talbot, et al.
- 2007
(Show Context)
Citation Context ...plicitly, see [3] for a survey, while others consider global constraints specified via states. An important class of the latter kind are tree automata with global equalities and disequalities (TAGED) =-=[5,6,7]-=-. For the purposes of this paper it will turn out to be natural to work with rigid tree automata that have been introduced in [13], see also [14]. They are a subclass of TAGED (characterised by having... |

4 | On the form of witness terms
- Hetzl
(Show Context)
Citation Context ...l is non-elementary [23,17,18] which shows that simplicity is a necessary assumption for Theorem 22.4 From Tree Languages to Proofs Applying Tree Languages in Proof Theory 309 Already the results in =-=[11]-=- show that a simple proof induces an acyclic regular(!) tree grammar whose finite language is an Herbrand-disjunction. So what have we gained from strengthening this result by adding total rigidity? O... |

2 |
Rigid tree automata and applications
- Jacquemard, Klay, et al.
(Show Context)
Citation Context ... one can obtain concrete values for existential quantifiers by computing the language of a grammar. In Section 2 we will develop a suitable notion of tree grammar corresponding to rigid tree automata =-=[13,14]-=-, in Section 3 we describe how to translate proofs to grammars and in Section 4 how to translate grammars to proofs. Most proofs in this paper will only be sketched, the reader interested in more deta... |

1 |
Proofs as Tree Languages (preprint), http://hal.archives-ouvertes.fr/hal-00613713
- Hetzl
(Show Context)
Citation Context ...ate proofs to grammars and in Section 4 how to translate grammars to proofs. Most proofs in this paper will only be sketched, the reader interested in more details is referred to the technical report =-=[10]-=-. 2 Rigid Tree Languages A feature which is important for many applications of tree languages but not present in regular tree languages is the ability to carry out equality tests between subterms, for... |

1 |
D.: Towards Algorithmic Cut-Introduction (submitted
- Hetzl, Leitsch, et al.
(Show Context)
Citation Context ...rammar can be realised by cut-formulas of a simple proof (which for the case of a single cut is always the case). A first algorithm based on this approach for the case of a single cut can be found in =-=[12]-=-. 6 Conclusion We have shown that cut-elimination in proofs where each cut contains at most one quantifier corresponds exactly to the computation of the language of a totally rigid acyclic tree gramma... |