Ordinal recursive bounds for Higman's theorem

Cichon, E. A.; Bittar, Elias Tahhan

doi:10.1016/s0304-3975(97)00009-1

Cited by 21 publications

(25 citation statements)

References 9 publications

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“…For instance, upper bounds for (N d × Q, ≤) for some finite set Q, along with the product ordering, can be found in [42,Theorem 2.34], where the norm of a pair (x, q) from N d × Q is max 1≤i≤d x(i): Proof. Let us first recall the definition of the Cichoń hierarchy of functions for indices α < ε 0 [7]:…”

Section: Fast-growing Upper Boundsmentioning

confidence: 99%

Demystifying Reachability in Vector Addition Systems

Leroux¹,

Schmitz²

2015

2015 30th Annual ACM/IEEE Symposium on Logic in Computer Science

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Abstract. More than 30 years after their inception, the decidability proofs for reachability in vector addition systems (VAS) still retain much of their mystery. These proofs rely crucially on a decomposition of runs successively refined by Mayr, Kosaraju, and Lambert, which appears rather magical, and for which no complexity upper bound is known.We first offer a justification for this decomposition technique, by showing that it computes the ideal decomposition of the set of runs, using the natural embedding relation between runs as well quasi ordering. In a second part, we apply recent results on the complexity of termination thanks to well quasi orders and well orders to obtain a cubic Ackermann upper bound for the decomposition algorithms, thus providing the first known upper bounds for general VAS reachability.

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Section: Fast-growing Upper Boundsmentioning

confidence: 99%

Demystifying Reachability in Vector Addition Systems

Leroux¹,

Schmitz²

2015

2015 30th Annual ACM/IEEE Symposium on Logic in Computer Science

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show abstract

“…As such, they still enjoy decision procedures for several verification problems, prominently safety (through the coverability problem) and termination. They share these properties with the other extensions of Petri nets with 1. for the upper bound, a controlled bad sequence can be extracted from any run of the backward coverability algorithm, and in turn the length of this sequence can be bounded using a length function theorem for the wqo at hand (e.g., Cichoń and Tahhan Bittar 1998;Figueira et al 2011;Rosa-Velardo 2014, for the mentioned results);…”

mentioning

confidence: 99%

The Complexity of Coverability in ν-Petri Nets

Lazić

Schmitz

2016

Proceedings of the 31st Annual ACM/IEEE Symposium on Logic in Computer Science

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We show that the coverability problem in ν-Petri nets is complete for 'double Ackermann' time, thus closing an open complexity gap between an Ackermann lower bound and a hyper-Ackermann upper bound. The coverability problem captures the verification of safety properties in this nominal extension of Petri nets with name management and fresh name creation. Our completeness result establishes ν-Petri nets as a model of intermediate power among the formalisms of nets enriched with data, and relies on new algorithmic insights brought by the use of well-quasi-order ideals.

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“…Fast-forwarding a bit, we get for instance a function of exponential growth H ω 2 (x) = 2 x+1 (x + 1) − 1, and later a non-elementary function H ω 3 akin to a tower of exponentials, and a non primitive-recursive function H ω ω of Ackermannian growth. In the following, we will use the following property of Hardy functions [38,8], which can be checked by induction provided α + β is in Cantor normal form (and justifies the use of superscripts):…”

Section: Subrecursive Functionsmentioning

confidence: 99%

“…Regarding the Cichoń functions, an easy induction on α shows that H α (x) = H α (x) + x for the hierarchy relative to H(x) def = x + 1. But the main interest of Cichoń functions is that they capture how many iterations are performed by Hardy functions [8]:…”

Section: Subrecursive Functionsmentioning

confidence: 99%

Bisimulation Equivalence of First-Order Grammars is ACKERMANN-Complete

Jancar¹,

Schmitz²

2019

2019 34th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS)

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Checking whether two pushdown automata with restricted silent actions are weakly bisimilar was shown decidable by Sénizergues (1998Sénizergues ( , 2005. We provide the first known complexity upper bound for this famous problem, in the equivalent setting of first-order grammars. This ACKERMANN upper bound is optimal, and we also show that strong bisimilarity is primitive-recursive when the number of states of the automata is fixed.

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Ordinal recursive bounds for Higman's theorem

Cited by 21 publications

References 9 publications

Demystifying Reachability in Vector Addition Systems

Demystifying Reachability in Vector Addition Systems

The Complexity of Coverability in ν-Petri Nets

Bisimulation Equivalence of First-Order Grammars is ACKERMANN-Complete

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