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## The Limits of Decidability for First Order Logic on CPDA Graphs (2012)

Venue: | In STACS 2012, volume 14 of LIPIcs |

Citations: | 3 - 0 self |

### Citations

169 |
A variant of a recursively unsolvable problem.
- Post
- 1946
(Show Context)
Citation Context ...omponents of a single order-3 stack rather than just two order-3 stacks in their entirety. This is the core of the undecidability result, which goes via a reduction from Post’s Correspondence Problem =-=[16]-=-. Additionally order-3 links provide edges in the graph that are ‘non-local’ in nature, allowing ɛ-closure to be eliminated as a requirement for undecidability. At order-4 we get that even the Σ1-theo... |

103 | On infinite transition graphs having a decidable monadic theory.
- Caucal
- 2003
(Show Context)
Citation Context ...y consider the ‘ɛ-closures’ of such graphs, whose edges consist of an unbounded number of transitions rather than just single steps. The ɛ-closures of n-PDA graphs form precisely the Caucal Hierarchy =-=[6, 3, 5]-=-, which is defined independently in terms of graph transformations. This deep result has as a consequence that every n-PDA graph has decidable MSO theory. ∗ The author is funded by La Fondation Scienc... |

66 | The Caucal hierarchy of infinite graphs in terms of logic and higher-order pushdown automata,
- Carayol, Wöhrle
- 2003
(Show Context)
Citation Context ...y consider the ‘ɛ-closures’ of such graphs, whose edges consist of an unbounded number of transitions rather than just single steps. The ɛ-closures of n-PDA graphs form precisely the Caucal Hierarchy =-=[6, 3, 5]-=-, which is defined independently in terms of graph transformations. This deep result has as a consequence that every n-PDA graph has decidable MSO theory. ∗ The author is funded by La Fondation Scienc... |

63 | Higher-order pushdown trees are easy
- Knapik, Niwiński, et al.
(Show Context)
Citation Context ...mise as a model for verifying higher-order functional programs [12, 13]. Unfortunately n-PDA expressively coincide with order-n recursion schemes only when the latter satisfy a property called safety =-=[10]-=-, unsafe recursion schemes are strictly more expressive [15, 14]. Hence a more powerful automaton is needed, which motivates order-n collapsible pushdown automata (n-CPDA) [7]. Inspired by panic autom... |

63 | Types and higher-order recursion schemes for verification of higher-order programs.
- Kobayashi
- 2009
(Show Context)
Citation Context ...rate a single infinite tree. Enjoying decidable µ-calculus theories, the class of trees generated by recursion schemes shows a lot of promise as a model for verifying higher-order functional programs =-=[12, 13]-=-. Unfortunately n-PDA expressively coincide with order-n recursion schemes only when the latter satisfy a property called safety [10], unsafe recursion schemes are strictly more expressive [15, 14]. H... |

54 | Collapsible pushdown automata and recursion schemes. In: LICS.
- Hague, Murawski, et al.
- 2008
(Show Context)
Citation Context ... property called safety [10], unsafe recursion schemes are strictly more expressive [15, 14]. Hence a more powerful automaton is needed, which motivates order-n collapsible pushdown automata (n-CPDA) =-=[7]-=-. Inspired by panic automata [11], which can be viewed as the special case at order-2, atomic elements in collapsible stacks eminate ‘links’ that target a component of the stack further below. We conc... |

31 |
Higher order pushdown automata, the caucal hierarchy of graphs and parity games
- Cachat
(Show Context)
Citation Context ...y consider the ‘ɛ-closures’ of such graphs, whose edges consist of an unbounded number of transitions rather than just single steps. The ɛ-closures of n-PDA graphs form precisely the Caucal Hierarchy =-=[6, 3, 5]-=-, which is defined independently in terms of graph transformations. This deep result has as a consequence that every n-PDA graph has decidable MSO theory. ∗ The author is funded by La Fondation Scienc... |

29 |
Ong – Safety is not a restriction at level 2 for string languages
- Aehlig, Miranda, et al.
- 2004
(Show Context)
Citation Context ...allows us to eliminate them, thereby producing an n-PDA with decidable MSO theory, and leading to the decidability of the Σ1 theory of the original n-CPDA. This is a graph analogue of Aehlig et al.’s =-=[8]-=- in which links are eliminated from 2-CPDA to produce an equivalent word-generating 2-PDA. This decidability result is the first about a fragment of first-order logic on n-CPDA at all orders and shows... |

22 |
Verifying higher-order functional programs with pattern-matching algebraic data types
- Ong, Ramsay
- 2011
(Show Context)
Citation Context ...rate a single infinite tree. Enjoying decidable µ-calculus theories, the class of trees generated by recursion schemes shows a lot of promise as a model for verifying higher-order functional programs =-=[12, 13]-=-. Unfortunately n-PDA expressively coincide with order-n recursion schemes only when the latter satisfy a property called safety [10], unsafe recursion schemes are strictly more expressive [15, 14]. H... |

18 | Regular sets of higher-order pushdown stacks
- Carayol
(Show Context)
Citation Context ...achable configurations without ever destroying an order-(n − 1) stack. This has some parallels with Carayol’s work on canonical sequences of operations witnessing the constructibility of n-PDA stacks =-=[4]-=-. If an n-CPDA only has ‘order-n links’, monotonicity allows us to eliminate them, thereby producing an n-PDA with decidable MSO theory, and leading to the decidability of the Σ1 theory of the origina... |

7 | Collapsible pushdown graphs of level 2 are treeautomatic
- Kartzow
- 2010
(Show Context)
Citation Context ...guration graphs of HOPDA are well understood, relatively little is known about the CPDA graphs. The order-2 CPDA graphs already have undecidable MSO theories but it was only recently shown by Kartzow =-=[9]-=- that first-order logic is decidable at the second level. In this paper we show the surprising result that first-order logic ceases to be decidable at order-3 and above. We delimit the fragments of th... |

3 | Collapse Operation Increases Expressive Power of Deterministic Higher Order Pushdown Automata
- Parys
(Show Context)
Citation Context ...ms [12, 13]. Unfortunately n-PDA expressively coincide with order-n recursion schemes only when the latter satisfy a property called safety [10], unsafe recursion schemes are strictly more expressive =-=[15, 14]-=-. Hence a more powerful automaton is needed, which motivates order-n collapsible pushdown automata (n-CPDA) [7]. Inspired by panic automata [11], which can be viewed as the special case at order-2, at... |

1 |
On Collapsible Pushdown Automata, their Graphs and the Power of Links
- Broadbent
- 2011
(Show Context)
Citation Context ... Order Logic on CPDA Graphs 2-CPDA 32-CPDA 32-CPDA nn-CPDA nn,(n−1)-CPDA nm-CPDA n-PDA /w ɛ-cl. /w ɛ-cl. (/w ɛ-cl.) n ≥ 4 /wo ɛ-cl. n ≥ 4, /w ɛ-cl. n ≥ 3 n ≥ 3 /w ɛ-cl. m ≤ n − 2 all n Σ1 Dec [9] Dec =-=[1]-=- Dec [1] Dec ? Und Dec [5] Σ2 Dec [9] Dec [1] Und Und Und Und Dec [5] FO Dec [9] Dec [1] Und Und Und Und Dec [5] MSO Und [7] Und [7] Und [7] Und [7] Und [7] Und [7] Dec [5] µ-c. Dec [11] Dec [7] Dec [... |

1 |
Collapse Operation Cannot Be Simulated Even By Using Higher Levels (Unpublished
- Parys
(Show Context)
Citation Context ...ms [12, 13]. Unfortunately n-PDA expressively coincide with order-n recursion schemes only when the latter satisfy a property called safety [10], unsafe recursion schemes are strictly more expressive =-=[15, 14]-=-. Hence a more powerful automaton is needed, which motivates order-n collapsible pushdown automata (n-CPDA) [7]. Inspired by panic automata [11], which can be viewed as the special case at order-2, at... |