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### Deciding whether a finite set of words has rank at most two

Jean Néraud
<span title="">1993</span> <i title="Elsevier BV"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/elaf5sq7lfdxfdejhkqbtz6qoq" style="color: black;">Theoretical Computer Science</a> </i> &nbsp;
., Deciding whether a finite set of words has rank at most two, Theoretical Computer Science 112 (1993) 311-337.  ...  Given a finite subset X of a free monoid A*, we define the rank of X as r(X)=min{ 1 YI: XE Y*}. The problem we study here is to decide whether or not r(X)<2.  ...  The case of sets of rank 1 Given a finite set of words X, it is easy to decide whether or not r(X) = 1. Indeed, we shall apply the following algorithm.  ...
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1016/0304-3975(93)90023-m">doi:10.1016/0304-3975(93)90023-m</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/eomtdmad5bb7hg5ho5d2awupre">fatcat:eomtdmad5bb7hg5ho5d2awupre</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20190323003539/https://core.ac.uk/download/pdf/81988252.pdf" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/6c/b3/6cb31e8528526ed2451b50830a53c189642824eb.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1016/0304-3975(93)90023-m"> <button class="ui left aligned compact blue labeled icon button serp-button"> <i class="external alternate icon"></i> elsevier.com </button> </a>

### Unsolvable Problems About Small Cancellation and Word Hyperbolic Groups

G. Baumslag, C. F. Miller, H. Short
<span title="">1994</span> <i title="Oxford University Press (OUP)"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/djmawcsftbaaxk2nkcppkone3i" style="color: black;">Bulletin of the London Mathematical Society</a> </i> &nbsp;
There is a small cancellation group E such that there is no algorithm to determine whether or not any finitely generated subgroup of E is all of E, or is finitely presented, or has a finitely generated  ...  We apply a construction of Rips to show that a number of algorithmic problems concerning certain small cancellation groups and, in particular, word hyperbolic groups, are recursively unsolvable.  ...  Hence at most two of the differentials d 2 can be non-zero maps. One of these is d 2 : H 2 (L, Z) → H 0 (L, H 1 (K P , Z)). Notice that this map has finitely generated cokernel.  ...
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1112/blms/26.1.97">doi:10.1112/blms/26.1.97</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/5hettjmfq5aqrmxg6wtmedftgy">fatcat:5hettjmfq5aqrmxg6wtmedftgy</a> </span>
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### On an algorithm to decide whether a free group is a free factor of another

Pedro V. Silva, Pascal Weil
<span title="2007-12-20">2007</span> <i title="EDP Sciences"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/foywj2stzfcwdln5izx2cxkroe" style="color: black;">RAIRO - Theoretical Informatics and Applications</a> </i> &nbsp;
We revisit the problem of deciding whether a finitely generated subgroup H is a free factor of a given free group F .  ...  We show that the latter dependency can be made exponential in the rank difference rank(F ) − rank(H), which often makes a significant change. Résumé.  ...  If A is such an automaton, then A has at most 1 2 (n 2 − n) pairs of distinct vertices, and hence at most 1 2 (n 2 − n) children, each of which has at most n − 1 states.  ...
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1051/ita:2007040">doi:10.1051/ita:2007040</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/774ztdcjmvam5dzcttj2qy5phu">fatcat:774ztdcjmvam5dzcttj2qy5phu</a> </span>
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### Hierarchical Information Patterns and Distributed Strategy Synthesis [chapter]

Dietmar Berwanger, Anup Basil Mathew, Marie van den Bogaard
<span title="">2015</span> <i title="Springer International Publishing"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/2w3awgokqne6te4nvlofavy5a4" style="color: black;">Lecture Notes in Computer Science</a> </i> &nbsp;
Peterson and Reif [13] showed that for games in this setting, it is decidable whether distributed winning strategies exist -although, with nonelementary complexity -, and if yes, finite-state winning strategies  ...  One fundamental decidable case occurs when there is a total ordering among players, such that each player has access to all the information that the following ones receive.  ...  In other words, a game has dynamic hierarchical information, if there is no history at which the information sets of two players are incomparable.  ...
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1007/978-3-319-24953-7_28">doi:10.1007/978-3-319-24953-7_28</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/axov2fthorhidaovqhr65ao7xe">fatcat:axov2fthorhidaovqhr65ao7xe</a> </span>
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### On an algorithm to decide whether a free group is a free factor of another [article]

Pedro Silva
<span title="2006-11-16">2006</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
We revisit the problem of deciding whether a finitely generated subgroup H is a free factor of a given free group F.  ...  We show that the latter dependency can be made exponential in the rank difference rank(F) - rank(H), which often makes a significant change.  ...  If A is such an automaton, then A has at most 1 2 (n 2 − n) pairs of distinct vertices, and hence at most 1 2 (n 2 − n) children, each of which has at most n − 1 states.  ...
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/math/0609552v2">arXiv:math/0609552v2</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/syh2tlz3anakbj2qkd72vhdjnq">fatcat:syh2tlz3anakbj2qkd72vhdjnq</a> </span>
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### Deciding Equivalence of Finite Tree Automata

Helmut Seidl
<span title="">1990</span> <i title="Society for Industrial &amp; Applied Mathematics (SIAM)"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/7dys7zoberdktmxyjciuy5bnse" style="color: black;">SIAM journal on computing (Print)</a> </i> &nbsp;
We s h o w : f o r mJe~j constant m i t can be decided i n p o l y n o m i a l time w h e t h e r or not two m-ambiguous finite tree automata are equivalent.  ...  FUrthermore, for every constant m we show that it can be decided in polynomial time whether or not a given finite tree automaton is m-ambiguous.  ...  Acknowledgement I thank Andreas Weber for many fruitful discussions and carefully reading of an earlier version of this paper.  ...
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1137/0219027">doi:10.1137/0219027</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/pfsympzzy5anbi2hkmddi6kpu4">fatcat:pfsympzzy5anbi2hkmddi6kpu4</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20170809012443/http://www2.in.tum.de/bib/files/Seidl90Deciding.pdf" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/25/f7/25f7528649427d87efc65a2d29cef151104517ac.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1137/0219027"> <button class="ui left aligned compact blue labeled icon button serp-button"> <i class="external alternate icon"></i> Publisher / doi.org </button> </a>

### Page 392 of Mathematical Reviews Vol. 20, Issue 4 [page]

<span title="">1959</span> <i title="American Mathematical Society"> <a target="_blank" rel="noopener" href="https://archive.org/details/pub_mathematical-reviews" style="color: black;">Mathematical Reviews </a> </i> &nbsp;
This theorem has the following consequence: not only is the word problem for groups insoluble, but so is the “meta-problem” of setting up an algorithm to decide whether a given finitely presented group  ...  algorithm to decide for an arbitrary finitely presented group F whether or not it has the property «.  ...
<span class="external-identifiers"> </span>
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### The monomorphism problem in free groups [article]

Laura Ciobanu, Abderezak Ould Houcine
<span title="2009-10-10">2009</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
Let F be a free group of finite rank.  ...  We say that the monomorphism problem in F is decidable if for any two elements u and v in F, there is an algorithm that determines whether there exists a monomorphism of F that sends u to v.  ...  The next lemma provides some basic information about the set T op(g). Lemma 2.2. The set T op(g) is finite, and the number of edges in a topological graph of rank g is at most 3g − 1. Proof.  ...
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/0910.1899v1">arXiv:0910.1899v1</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/daxv2f7tqvh6rjbeezr3iqqq6a">fatcat:daxv2f7tqvh6rjbeezr3iqqq6a</a> </span>
<a target="_blank" rel="noopener" href="https://archive.org/download/arxiv-0910.1899/0910.1899.pdf" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> File Archive [PDF] </button> </a> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/0910.1899v1" title="arxiv.org access"> <button class="ui compact blue labeled icon button serp-button"> <i class="file alternate outline icon"></i> arxiv.org </button> </a>

### Büchi context-free languages

Zoltán Ésik, Szabolcs Iván
<span title="">2011</span> <i title="Elsevier BV"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/elaf5sq7lfdxfdejhkqbtz6qoq" style="color: black;">Theoretical Computer Science</a> </i> &nbsp;
Then for each n < ω, the set L n of all scattered words in Σ ∞ of rank at most n is a BCFL. Indeed, L 0 = Σ * , and L n+1 = (L ω n ∪ L −ω n ) * . Corollary 6.13.  ...  Suppose that G = (N, Σ, P, S, F ) is a BCFG such that L ∞ (G) contains only scattered words. Then the rank of each word in L ∞ (G) is at most the number of nonterminals in N. Corollary 6.15.  ...  Since any finite concatenation of scattered words of rank at most n + 1 is scattered of rank at most n + 1, it follows that w is scattered of rank at most n + 1.  ...
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1016/j.tcs.2010.11.026">doi:10.1016/j.tcs.2010.11.026</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/6ruifzqwljd55fqyfznxxgx5rm">fatcat:6ruifzqwljd55fqyfznxxgx5rm</a> </span>
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### Kleinian groups and the rank problem

Ilya Kapovich, Richard Weidmann
<span title="2005-03-03">2005</span> <i title="Mathematical Sciences Publishers"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/nk6cbwjm3re3xcnnbdd2g7hpyy" style="color: black;">Geometry and Topology</a> </i> &nbsp;
We also show that every group in this class has only finitely many Nielsen equivalence classes of generating sets of a given cardinality.  ...  We prove that the rank problem is decidable in the class of torsion-free word-hyperbolic Kleinian groups.  ...  We then enumerate all p-tuples T of words over S with 1 ≤ p ≤ k , where every word has length at most C . Let T be the set of all such tuples.  ...
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.2140/gt.2005.9.375">doi:10.2140/gt.2005.9.375</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/ph7y5pyklza7tm74ouoomkwgci">fatcat:ph7y5pyklza7tm74ouoomkwgci</a> </span>
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### Cost Automata, Safe Schemes, and Downward Closures

David Barozzini, Lorenzo Clemente, Thomas Colcombet, Paweł Parys, Emanuela Merelli, Artur Czumaj, Anuj Dawar
<span title="2020-06-29">2020</span> <i title="Schloss Dagstuhl - Leibniz-Zentrum für Informatik"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/pf6v2q6ji5h7birkbrtpdc2uhy" style="color: black;">International Colloquium on Automata, Languages and Programming</a> </i> &nbsp;
In this work we prove, under a syntactical constraint called safety, decidability of the model-checking problem for recursion schemes against properties defined by alternating B-automata, an extension  ...  We then exploit this result to show how to compute downward closures of languages of finite trees recognized by safe recursion schemes.  ...  of the monadic variable X is at most N .  ...
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.4230/lipics.icalp.2020.109">doi:10.4230/lipics.icalp.2020.109</a> <a target="_blank" rel="external noopener" href="https://dblp.org/rec/conf/icalp/BarozziniCCP20.html">dblp:conf/icalp/BarozziniCCP20</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/kk2gv3niuvdl3iyq2dptfffglq">fatcat:kk2gv3niuvdl3iyq2dptfffglq</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20201211023651/https://drops.dagstuhl.de/opus/volltexte/2020/12516/pdf/LIPIcs-ICALP-2020-109.pdf" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/58/49/5849f718f5905da43a09345a2682867578ef29f4.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.4230/lipics.icalp.2020.109"> <button class="ui left aligned compact blue labeled icon button serp-button"> <i class="unlock alternate icon" style="background-color: #fb971f;"></i> Publisher / doi.org </button> </a>

### Decidable classes of documents for XPath

Vince Bárány, Mikolaj Bojanczyk, Diego Figueira, Pawel Parys, Marc Herbstritt
<span title="2012-12-10">2012</span> <i > <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/yjwtenvm3vc4tjdmag7doffpby" style="color: black;">Foundations of Software Technology and Theoretical Computer Science</a> </i> &nbsp;
We conjecture that these two parameters are equivalent, in the sense that a class of documents has bounded match width iff it has bounded braid width.  ...  We show that for all k, satisfiability for XPath restricted to bounded depth documents with match width at most k is decidable; and that XPath is undecidable on any class of documents with unbounded braid  ...  We define that a class C of data words has match width at least/at most n if the match expansion of C has match width at least/at most n.  ...
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.4230/lipics.fsttcs.2012.99">doi:10.4230/lipics.fsttcs.2012.99</a> <a target="_blank" rel="external noopener" href="https://dblp.org/rec/conf/fsttcs/BaranyBFP12.html">dblp:conf/fsttcs/BaranyBFP12</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/c7xxinkaxbe6pldc5senl3qx3m">fatcat:c7xxinkaxbe6pldc5senl3qx3m</a> </span>
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### COMBINATORICS ON WORDS – A TUTORIAL [chapter]

JEAN BERSTEL, JUHANI KARHUMÄKI
<span title="">2004</span> <i title="WORLD SCIENTIFIC"> Current Trends in Theoretical Computer Science </i> &nbsp;
Cassaigne, A. Lepistö, G. Richomme, P. Séébold and J. Shallit for their careful reading of parts of a preliminary version of the paper. In addition the authors are grateful to A.  ...  As other problems on finite sets of words we state Problem 15. Is it decidable whether two finite sets X and Y ⊆ A + are conjugates, i.e., there exists a set Z such that XZ = ZY ? Problem 16.  ...  Is it decidable whether a given equation with constants has a solution in the monoid of finite languages?  ...
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1142/9789812562494_0059">doi:10.1142/9789812562494_0059</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/rb3muz6tsffbrpm26j4aj2rkey">fatcat:rb3muz6tsffbrpm26j4aj2rkey</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20110613164016/http://www-igm.univ-mlv.fr/~berstel/Articles/2003TutorialCoWdec03.pdf" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/55/92/5592d83b01e56fd0df602186b43f1881bc5d54e5.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1142/9789812562494_0059"> <button class="ui left aligned compact blue labeled icon button serp-button"> <i class="external alternate icon"></i> worldscientific.com </button> </a>

### A note on the finite basis and finite rank properties for pseudovarieties of semigroups

J. Almeida, M. H. Shahzamanian
<span title="2017-12-18">2017</span> <i title="Springer Nature"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/ealgne43xjhvfetoaem5md3khq" style="color: black;">Semigroup Forum</a> </i> &nbsp;
The finite basis property is often connected with the finite rank property, which it entails. Many examples have been produced of finite rank varieties which are not finitely based.  ...  In this note, we establish a result on nilpotent pseudovarieties which yields many similar examples in the realm of pseudovarieties of semigroups.  ...  We greatly appreciate the comments of the Editor.  ...
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1007/s00233-017-9904-3">doi:10.1007/s00233-017-9904-3</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/pjss3kw2rfb2tikruspql2rvri">fatcat:pjss3kw2rfb2tikruspql2rvri</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20190429081556/https://repositorio-aberto.up.pt/bitstream/10216/110947/2/255702.pdf" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/79/72/7972de135a46df4df99e6e0b8152c58b8945a00d.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1007/s00233-017-9904-3"> <button class="ui left aligned compact blue labeled icon button serp-button"> <i class="external alternate icon"></i> springer.com </button> </a>

### Computing rank of finite algebraic structures with limited nondeterminism [article]

Jeffrey Finkelstein
<span title="2020-05-20">2020</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
Specifically, we prove that the problem of deciding whether the rank of a finite quasigroup, given as a Cayley table, is smaller than a specified number is decidable by a circuit of depth O(loglog n) augmented  ...  The rank of a finite algebraic structure with a single binary operation is the minimum number of elements needed to express every other element under the closure of the operation.  ...  Given a magma G as a Cayley table, a finite set T of magma elements, and a natural number k in unary, decide whether there is a set S of size at most k such that S ⊆ T ⊆ S .  ...
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/1406.0879v4">arXiv:1406.0879v4</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/pmnljr6a2rgitgs6lmffvtcc2u">fatcat:pmnljr6a2rgitgs6lmffvtcc2u</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20201001003546/https://arxiv.org/pdf/1406.0879v3.pdf" title="fulltext PDF download [not primary version]" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] <span style="color: #f43e3e;">&#10033;</span> <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/f9/b0/f9b08ae58449c3bdb1f9e7636b8752c644a004d2.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/1406.0879v4" title="arxiv.org access"> <button class="ui compact blue labeled icon button serp-button"> <i class="file alternate outline icon"></i> arxiv.org </button> </a>
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