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Ackermannian and Primitive-Recursive Bounds with Dickson's Lemma

Diego Figueira, Santiago Figueira, Sylvain Schmitz, Philippe Schnoebelen
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.  ... 
doi:10.1109/lics.2011.39 dblp:conf/lics/FigueiraFSS11 fatcat:yw7fgfomsnf6pcdguirvg673vq

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

Ranko Lazić, Joël Ouaknine, James Worrell
2013 Lecture Notes in Computer Science  
non-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  ... 
doi:10.1007/978-3-642-40313-2_57 fatcat:oaprbstnfjd2zcjmh2rrrcgjbu

The Power of Well-Structured Systems [chapter]

Sylvain Schmitz, Philippe Schnoebelen
2013 Lecture Notes in Computer Science  
Well-structured systems, aka WSTSs, are computational models where the set of possible configurations is equipped 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.  ... 
doi:10.1007/978-3-642-40184-8_2 fatcat:7rweytwv7rgntjrcrdygelwvca

Phase Transition Thresholds for Some Natural Subclasses of the Computable Functions [chapter]

Andreas Weiermann
2006 Lecture Notes in Computer Science  
Special emphasis is put on descent 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.  ... 
doi:10.1007/11780342_57 fatcat:epkdvlcmija7tdynj37u4zutxa

The Fixed Initial Credit Problem for Partial-Observation Energy Games is Ack-complete [article]

Guillermo A. Pérez
2016 arXiv   pre-print
In this paper we study two-player games 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.  ... 
arXiv:1512.04255v6 fatcat:xf4m6rhkyvgwrkmkcgfes7xk7q

Complexity Hierarchies beyond Elementary

Sylvain Schmitz
2016 ACM Transactions on Computation Theory  
ranging from simple towers of exponentials to 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.  ... 
doi:10.1145/2858784 fatcat:qg5iacgrpzbhpb5bzaxvcfccm4

Polynomial automata: Zeroness and applications

Michael Benedikt, Timothy Duff, Aditya Sharad, James Worrell
2017 2017 32nd Annual ACM/IEEE Symposium on Logic in Computer Science (LICS)  
While this problem is non-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) .  ... 
doi:10.1109/lics.2017.8005101 dblp:conf/lics/BenediktDSW17 fatcat:ssuoiuxksrf5zbp35dpsuz7lua

Multiply-Recursive Upper Bounds with Higman's Lemma [chapter]

S. Schmitz, Ph. Schnoebelen
2011 Lecture Notes in Computer Science  
This leads to tight multiply-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.  ... 
doi:10.1007/978-3-642-22012-8_35 fatcat:ncqmdypmsrf6jm3wfgahp3lhsu

The Parametric Complexity of Lossy Counter Machines

Sylvain Schmitz, Michael Wagner
2019 International Colloquium on Automata, Languages and Programming  
We close this gap 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 .  ... 
doi:10.4230/lipics.icalp.2019.129 dblp:conf/icalp/Schmitz19 fatcat:5wf4jirquvc7xbnxa5nyrcmvga

Decidability and Complexity in Weakening and Contraction Hypersequent Substructural Logics

A. R. Balasubramanian, Timo Lang, Revantha Ramanayake
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, 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.  ... 
doi:10.1109/lics52264.2021.9470733 fatcat:b6lz7nsc7jfnxdx4q44tcs2que

Complexity Bounds for Ordinal-Based Termination [chapter]

Sylvain Schmitz
2014 Lecture Notes in Computer Science  
We examine Kreisel's question in the particular context of program termination proofs, 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.  ... 
doi:10.1007/978-3-319-11439-2_1 fatcat:amvkmpn4lrfcvdpprvnbbwjzny

Zeno, Hercules, and the Hydra

Ranko Lazić, Joël Ouaknine, James Worrell
2016 ACM Transactions on Computational Logic  
., among the easiest non-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  ... 
doi:10.1145/2874774 fatcat:ztb6l7vzsbgadavxw5tsnzm6ba

On Termination of Polynomial Programs with Equality Conditions [article]

Yangjia Li, Naijun Zhan, Mingshuai Chen, Hui Lu, Guohua Wu, Joost-Pieter Katoen
2020 arXiv   pre-print
We present an explicit 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.  ... 
arXiv:1510.05201v3 fatcat:vngdhx4y3fa6bmn7erftgkcqxu

Well Quasi-Orders in Computer Science (Dagstuhl Seminar 16031)

Jean Goubault-Larrecq, Monika Seisenberger, Victor Selivanov, Andreas Weiermann, Marc Herbstritt
2016 Dagstuhl Reports  
Wqo-Theory is a highly developed part of Combinatorics 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.  ... 
doi:10.4230/dagrep.6.1.69 dblp:journals/dagstuhl-reports/Goubault-Larrecq16 fatcat:o3uwzu5ptfavfes6kgexpi4a6q

Nonelementary Complexities for Branching VASS, MELL, and Extensions

Ranko Lazić, Sylvain Schmitz
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 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.  ... 
doi:10.1145/2733375 fatcat:wyonep56kzfnndlkru7dprmwnq
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