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Ackermannian and Primitive-Recursive Bounds with Dickson's Lemma

2011
*
2011 IEEE 26th Annual Symposium on Logic in Computer Science
*

*Dickson's*

*Lemma*is a simple yet powerful tool widely used in termination proofs, especially when dealing

*with*counters or related data structures. ... Our upper

*bounds*improve earlier results

*and*are essentially tight. ... For fixed k the complexity is

*primitive*-

*recursive*,

*and*it is

*Ackermannian*when k is part of the input-which is the case in the encoding of logical formulae of Demri

*and*Lazić [8] . C. ...

##
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Zeno, Hercules and the Hydra: Downward Rational Termination Is Ackermannian
[chapter]

2013
*
Lecture Notes in Computer Science
*

non-

doi:10.1007/978-3-642-40313-2_57
fatcat:oaprbstnfjd2zcjmh2rrrcgjbu
*primitive**recursive*problems. ... Via another equivalent problem, namely termination for a class of rational relations, we show that satisfiability for safety MTL is not*primitive**recursive*, yet is*Ackermannian*, i.e., among the simplest ... The union k F k is the class of all*primitive**recursive*functions, while F ω defined by F ω (x) = F x (x) is an Ackermann-like non-*primitive**recursive*function; we call*Ackermannian*such functions that ...##
###
The Power of Well-Structured Systems
[chapter]

2013
*
Lecture Notes in Computer Science
*

Well-structured systems, aka WSTSs, are computational models where the set of possible configurations is equipped

doi:10.1007/978-3-642-40184-8_2
fatcat:7rweytwv7rgntjrcrdygelwvca
*with*a well-quasi-ordering which is compatible*with*the transition relation between configurations ... This structure supports generic decidability results that are important in verification*and*several other fields. ... We thank Christoph Haase*and*Prateek Karandikar for their helpful comments on an earlier version of this paper. The remaining errors are of course entirely ours. ...##
###
Phase Transition Thresholds for Some Natural Subclasses of the Computable Functions
[chapter]

2006
*
Lecture Notes in Computer Science
*

Special emphasis is put on descent

doi:10.1007/11780342_57
fatcat:epkdvlcmija7tdynj37u4zutxa
*recursive*functions, witness*bounding*functions for well-partial orders*and*Ramsey functions. ... In this paper we first survey recent advances on phase transition phenomena which are related to natural subclasses of the*recursive*functions. ... Basic examples are provided by*Dickson's**Lemma**and*Higman's*Lemma*. Assume that Y, ≤ Y is a partial ordering. ...##
###
The Fixed Initial Credit Problem for Partial-Observation Energy Games is Ack-complete
[article]

2016
*
arXiv
*
pre-print

In this paper we study two-player games

arXiv:1512.04255v6
fatcat:xf4m6rhkyvgwrkmkcgfes7xk7q
*with*asymmetric partial observation*and*an energy objective. ... Such games are played on a weighted automaton by Eve, choosing actions,*and*Adam, choosing a transition labelled*with*the given action. ... More coarsely, we have that 2 (h(|G|)+1) 2*bounds*the size of S. It follows from*Lemmas*3-6 that the latter*bound*is*primitive**recursive*for all fixed G. ...##
###
Complexity Hierarchies beyond Elementary

2016
*
ACM Transactions on Computation Theory
*

ranging from simple towers of exponentials to

doi:10.1145/2858784
fatcat:qg5iacgrpzbhpb5bzaxvcfccm4
*Ackermannian**and*beyond. ... This hierarchy allows the classification of many decision problems*with*a non-elementary complexity, which occur naturally in logic, combinatorics, formal languages, verification, etc.,*with*complexities ... The whole phase can thus be performed in time polynomial in , which is*bounded*by F h,ω (f (n)) for some*primitive*-*recursive*f by*Lemma*4.5. ...##
###
Polynomial automata: Zeroness and applications

2017
*
2017 32nd Annual ACM/IEEE Symposium on Logic in Computer Science (LICS)
*

While this problem is non-

doi:10.1109/lics.2017.8005101
dblp:conf/lics/BenediktDSW17
fatcat:ssuoiuxksrf5zbp35dpsuz7lua
*primitive**recursive*in general, we highlight a subclass of polynomial automata for which the Zeroness Problem is*primitive**recursive*. ... We introduce a generalisation of weighted automata over a field, called polynomial automata,*and*we analyse the complexity of the Zeroness Problem in this model, that is, whether a given automaton outputs ... However*Dickson's**Lemma*does not yield quantitative*bounds*on the stabilisation of (2) . ...##
###
Multiply-Recursive Upper Bounds with Higman's Lemma
[chapter]

2011
*
Lecture Notes in Computer Science
*

This leads to tight multiply-

doi:10.1007/978-3-642-22012-8_35
fatcat:ncqmdypmsrf6jm3wfgahp3lhsu
*recursive*upper*bounds*that readily apply to several verification algorithms for well-structured systems. ... We develop a new analysis for the length of controlled bad sequences in well-quasi-orderings based on Higman's*Lemma*. ... .,*and*Schnoebelen, Ph., 2011.*Ackermannian**and**primitive*-*recursive**bounds**with**Dickson's**Lemma*. In LICS 2011 , 26th Annual IEEE Symposium on Logic in Computer Science. IEEE. To appear. ...##
###
The Parametric Complexity of Lossy Counter Machines

2019
*
International Colloquium on Automata, Languages and Programming
*

We close this gap

doi:10.4230/lipics.icalp.2019.129
dblp:conf/icalp/Schmitz19
fatcat:5wf4jirquvc7xbnxa5nyrcmvga
*and*prove F d -completeness for machines*with*d counters, which provides the first known uncontrived problems complete for the fast-growing complexity classes at levels 3 < d < ω. ... We develop for this an approach through antichain factorisations of bad sequences*and*analysing the length of controlled antichains. ... an "*Ackermannian*" non*primitive*-*recursive*function H ω ω . ...##
###
Decidability and Complexity in Weakening and Contraction Hypersequent Substructural Logics

2021
*
2021 36th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS)
*

Take-down policy If you believe that this document breaches copyright please contact us providing details,

doi:10.1109/lics52264.2021.9470733
fatcat:b6lz7nsc7jfnxdx4q44tcs2que
*and*we will remove access to the work immediately*and*investigate your claim. ...*Dickson's**lemma*) that ď is a well-quasi ordering. ... By the same argument as in the previous subsection, we conclude that ř N i"0 |S i | ď |th 1 | xh 1 y ď g N pf pnqqu| can be upper-*bounded*by a*primitive**recursive*function of N*and*n. ...##
###
Complexity Bounds for Ordinal-Based Termination
[chapter]

2014
*
Lecture Notes in Computer Science
*

We examine Kreisel's question in the particular context of program termination proofs,

doi:10.1007/978-3-319-11439-2_1
fatcat:amvkmpn4lrfcvdpprvnbbwjzny
*with*an eye to deriving complexity*bounds*on program running times. ... Our main tool for this are length function theorems, which provide complexity*bounds*on the use of well quasi orders. ... The author thanks Christoph Haase, Georg Moser,*and*Philippe Schnoebelen for helpful discussions. ...##
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Zeno, Hercules, and the Hydra

2016
*
ACM Transactions on Computational Logic
*

., among the easiest non-

doi:10.1145/2874774
fatcat:ztb6l7vzsbgadavxw5tsnzm6ba
*primitive**recursive*problems. ... However, hitherto its precise computational complexity has remained elusive,*with*only a non-elementary lower*bound*. ... problems whose complexity is*bounded*by F ω (g(n)) for*primitive**recursive*g,*and*such a problem is complete if*and*only if there exist*primitive**recursive*reductions to it from all problems in ACKERMANN ...##
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On Termination of Polynomial Programs with Equality Conditions
[article]

2020
*
arXiv
*
pre-print

We present an explicit

arXiv:1510.05201v3
fatcat:vngdhx4y3fa6bmn7erftgkcqxu
*recursive*function which is essentially*Ackermannian*, to compute the maximal length of ascending chains of polynomial ideals under a control function,*and*thereby obtain a complete ... We extend our method to programs*with*polynomial guarded commands*and*show how an incomplete procedure for MPPs*with*inequality guards can be obtained. ... As a direct consequence of the expression, the maximal length is shown to be*Ackermannian*, i.e., not*primitive**recursive*. ...##
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Well Quasi-Orders in Computer Science (Dagstuhl Seminar 16031)

2016
*
Dagstuhl Reports
*

Wqo-Theory is a highly developed part of Combinatorics

doi:10.4230/dagrep.6.1.69
dblp:journals/dagstuhl-reports/Goubault-Larrecq16
fatcat:o3uwzu5ptfavfes6kgexpi4a6q
*with*ever-growing number of applications in Mathematics*and*Computer Science,*and*Well quasiorders are going to become an important unifying concept ... established*and*younger researchers*and*thus to push forward the interaction between Wqo-Theory*and*Computer Science. ...*Ackermannian**and**Primitive*-*Recursive**Bounds**with**Dickson's**Lemma*. ...##
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Nonelementary Complexities for Branching VASS, MELL, and Extensions

2015
*
ACM Transactions on Computational Logic
*

We study the complexity of reachability problems on branching extensions of vector addition systems, which allows us to derive new non-elementary complexity

doi:10.1145/2733375
fatcat:wyonep56kzfnndlkru7dprmwnq
*bounds*for fragments*and*variants of propositional ... We show that provability in the multiplicative exponential fragment is Tower-hard already in the affine case---*and*hence non-elementary. ...*bounded*) in*Ackermannian*time. ...
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