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Unrestricted State Complexity of Binary Operations on Regular and Ideal Languages [article]

Janusz Brzozowski, Corwin Sinnamon
<span title="2017-12-21">2017</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
We study the state complexity of binary operations on regular languages over different alphabets.  ...  It is known that if L'_m and L_n are languages of state complexities m and n, respectively, and restricted to the same alphabet, the state complexity of any binary boolean operation on L'_m and L_n is  ...  That paper dealt only with unrestricted product and binary boolean operations on regular languages.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/1609.04439v3">arXiv:1609.04439v3</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/6lkl7d2k2jczvn64nf2ba25bnm">fatcat:6lkl7d2k2jczvn64nf2ba25bnm</a> </span>
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Towards a Theory of Complexity of Regular Languages [article]

Janusz A. Brzozowski
<span title="2017-02-16">2017</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
We then turn to the study of the quotient/state complexity of common operations on regular languages: reversal, (Kleene) star, product (concatenation) and boolean operations.  ...  In addition to the quotient complexity of the language -- which is the number of its (left) quotients, and is the same as its state complexity -- we also consider the size of its syntactic semigroup and  ...  Acknowledgment I am very grateful to Sylvie Davies, Corwin Sinnamon, Marek Szyku la, and Hellis Tamm not only for careful proofreading, but also for their many contributions to this paper.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/1702.05024v1">arXiv:1702.05024v1</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/gzkyubh2ireizdsuu6fpqy6ece">fatcat:gzkyubh2ireizdsuu6fpqy6ece</a> </span>
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Complexity of Left-Ideal, Suffix-Closed and Suffix-Free Regular Languages [chapter]

Janusz A. Brzozowski, Corwin Sinnamon
<span title="">2017</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;
In particular, we study the quotient/state complexity of boolean operations, product (concatenation), star, and reversal on these languages, as well as the size of their syntactic semigroups, and the quotient  ...  We examine complexity properties of these three special classes of suffix-convex regular languages.  ...  Similarly, if L ′ m and L n are regular languages of complexities m and n respectively, • is a binary operation, then the complexity of • is the maximal value of κ(L ′ m •L n ), expressed as a function  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1007/978-3-319-53733-7_12">doi:10.1007/978-3-319-53733-7_12</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/vuc2j7cefzgubgx3vlkhrhzfum">fatcat:vuc2j7cefzgubgx3vlkhrhzfum</a> </span>
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Complexity of Left-Ideal, Suffix-Closed and Suffix-Free Regular Languages [article]

Janusz Brzozowski, Corwin Sinnamom
<span title="2016-10-03">2016</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
In particular, we study the quotient/state complexity of boolean operations, product (concatenation), star, and reversal on these languages, as well as the size of their syntactic semigroups, and the quotient  ...  We examine complexity properties of these three special classes of suffix-convex regular languages.  ...  Similarly, if L ′ m and L n are regular languages of complexities m and n respectively, • is a binary operation, then the complexity of • is the maximal value of κ(L ′ m •L n ), expressed as a function  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/1610.00728v1">arXiv:1610.00728v1</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/2ndc53ji5jfglbjzzpi6e4hzbu">fatcat:2ndc53ji5jfglbjzzpi6e4hzbu</a> </span>
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Complexity of Right-Ideal, Prefix-Closed, and Prefix-Free Regular Languages

Janusz A. Brzozowski, Corwin Sinnamon
<span title="">2017</span> <i title="University of Szeged"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/ldneqkirkzflfmmjbnihcafne4" style="color: black;">Acta Cybernetica</a> </i> &nbsp;
We first present a most complex regular language stream similar to that of [5], but one that is better suited for prefix-convex languages.  ...  In particular, we study the quotient/state complexity of boolean operations, product (concatenation), star, and reversal, the size of the syntactic semigroup, and the quotient complexity of atoms.  ...  If L ′ m and L n are regular languages of quotient complexities m and n respectively, and • is a binary operation, then the quotient/state complexity of • is the maximal value of κ(L ′ m •L n ), expressed  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.14232/actacyb.23.1.2017.3">doi:10.14232/actacyb.23.1.2017.3</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/lcomhhtiavhv5budwgy4zsg6qi">fatcat:lcomhhtiavhv5budwgy4zsg6qi</a> </span>
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Complexity of Prefix-Convex Regular Languages [article]

Janusz Brzozowski, Corwin Sinnamon
<span title="2016-06-24">2016</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
Prefix-convex languages include right-ideal, prefix-closed, and prefix-free languages. We study complexity properties of prefix-convex regular languages.  ...  In particular, we find the quotient/state complexity of boolean operations, product (concatenation), star, and reversal, the size of the syntactic semigroup, and the quotient complexity of atoms.  ...  If L ′ m and L n are regular languages of quotient complexities m and n respectively, and • is a binary operation, then the quotient/state complexity of • is the maximal value of κ(L ′ m • L n ), expressed  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/1605.06697v3">arXiv:1605.06697v3</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/mfgvafwe7nctbkfgnrkw7jesda">fatcat:mfgvafwe7nctbkfgnrkw7jesda</a> </span>
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A New Technique for Reachability of States in Concatenation Automata [article]

Sylvie Davies
<span title="2017-10-17">2017</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
Such demonstrations are a necessary step in establishing the state complexity of the concatenation of two languages, and thus in establishing the state complexity of concatenation as an operation.  ...  We present a new technique for demonstrating the reachability of states in deterministic finite automata representing the concatenation of two languages.  ...  This work was supported by the Natural Sciences and Engineering Research Council of Canada under grant No. OGP0000871.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/1710.05061v2">arXiv:1710.05061v2</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/cakpdb3t2raybpunxyk3mv6o44">fatcat:cakpdb3t2raybpunxyk3mv6o44</a> </span>
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A New Technique for Reachability of States in Concatenation Automata [chapter]

Sylvie Davies
<span title="">2018</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;
Such demonstrations are a necessary step in establishing the state complexity of the concatenation of two languages, and thus in establishing the state complexity of concatenation as an operation.  ...  We present a new technique for demonstrating the reachability of states in deterministic finite automata representing the concatenation of two languages.  ...  This work was supported by the Natural Sciences and Engineering Research Council of Canada under grant No. OGP0000871.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1007/978-3-319-94631-3_7">doi:10.1007/978-3-319-94631-3_7</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/cm2ib6aar5djjdylm32l2c6que">fatcat:cm2ib6aar5djjdylm32l2c6que</a> </span>
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Complexity of Suffix-Free Regular Languages [article]

Janusz Brzozowski, Marek Szykuła
<span title="2016-12-12">2016</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
The quotient complexity of an operation on regular languages is the maximal quotient complexity of the result of the operation expressed as a function of the quotient complexities of the operands.  ...  It is known that there exist such most complex streams in the class of regular languages, in the class of prefix-free languages, and also in the classes of right, left, and two-sided ideals.  ...  Similarly, to establish the state/quotient complexity of a regularity preserving binary operationon regular languages we have to find two streams (K m , m h) and (L n , n k) of languages meeting this  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/1504.05159v4">arXiv:1504.05159v4</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/jivxp23gwrb3xe6mrlnh3qqnoq">fatcat:jivxp23gwrb3xe6mrlnh3qqnoq</a> </span>
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Most Complex Regular Ideal Languages [article]

Janusz Brzozowski, Sylvie Davies, Bo Yang Victor Liu
<span title="2016-10-13">2016</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
left quotients), the quotient complexities of the atoms, and the quotient complexities of reversal, star, product (concatenation), and all binary boolean operations.  ...  We show that there exist sequences (L_n | n > k ) of right, left, and two-sided regular ideals, where L_n has quotient complexity (state complexity) n, such that L_n is most complex in its class under  ...  As this paper was written well before [5] , all our results are in terms of restricted complexity. A study of unrestricted complexity of binary operations on ideals can be found in [9] .  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/1511.00157v3">arXiv:1511.00157v3</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/5ssf2cwvf5fnhpv3feucvq2sau">fatcat:5ssf2cwvf5fnhpv3feucvq2sau</a> </span>
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Complexity of Suffix-Free Regular Languages [chapter]

Janusz Brzozowski, Marek Szykuła
<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;
We study various complexity properties of suffix-free regular languages.  ...  A sequence (L k , L k+1 , . . . ) of regular languages in some class, where n is the quotient complexity of L n , is most complex if its languages L n meet the complexity upper bounds for all basic measures  ...  Acknowledgments This work was supported by the Natural Sciences and Engineering Research Council of Canada (NSERC) grant No.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1007/978-3-319-22177-9_12">doi:10.1007/978-3-319-22177-9_12</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/sducrq3fi5aclpyzddrtwmwqfy">fatcat:sducrq3fi5aclpyzddrtwmwqfy</a> </span>
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A General Approach to State Complexity of Operations: Formalization and Limitations [chapter]

Sylvie Davies
<span title="">2018</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;
Our main result is that the worst-case state complexity of a uniform operation can be determined by considering a finite set of witnesses, and this set depends only on the arity of the operation and the  ...  When this correlation holds, the state complexity of a unary operation can be maximized using languages in which there is one letter corresponding to each possible transformation; for operations of higher  ...  I thank Jason Bell, Janusz Brzozowski, and the referees of the DLT 2018 version of this paper for their careful proofreading and helpful comments.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1007/978-3-319-98654-8_21">doi:10.1007/978-3-319-98654-8_21</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/2tr43qbs3jhaxeflgi3ccwqri4">fatcat:2tr43qbs3jhaxeflgi3ccwqri4</a> </span>
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A General Approach to State Complexity of Operations: Formalization and Limitations [article]

Sylvie Davies
<span title="2018-09-05">2018</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
Our main result is that the worst-case state complexity of a uniform operation can be determined by considering a finite set of witnesses, and this set depends only on the arity of the operation and the  ...  When this correlation holds, the state complexity of a unary operation can be maximized using languages in which there is one letter corresponding to each possible transformation; for operations of higher  ...  I thank Jason Bell, Janusz Brzozowski, and the referees of the DLT 2018 version of this paper for careful proofreading and helpful comments.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/1806.08476v2">arXiv:1806.08476v2</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/72wmyfzv3rgnvgdyocy3eestva">fatcat:72wmyfzv3rgnvgdyocy3eestva</a> </span>
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Complexity of Proper Suffix-Convex Regular Languages [article]

Corwin Sinnamon
<span title="2018-05-12">2018</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
We find tight upper bounds for reversal, star, product, and boolean operations of proper suffix-convex languages, and we conjecture on the size of the largest syntactic semigroup.  ...  Suffix-convex languages include left ideals, suffix-closed languages, and suffix-free languages, which were studied previously.  ...  I am extremely grateful for his guidance and mentorship.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/1805.03375v2">arXiv:1805.03375v2</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/eobdtd4nvvf45l26fqnsukrw5e">fatcat:eobdtd4nvvf45l26fqnsukrw5e</a> </span>
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Complexity of Proper Prefix-Convex Regular Languages [chapter]

Janusz A. Brzozowski, Corwin Sinnamon
<span title="">2017</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;
We find tight upper bounds for reversal, star, product, and boolean operations of proper suffix-convex languages, and we conjecture on the size of the largest syntactic semigroup.  ...  Suffix-convex languages include left ideals, suffix-closed languages, and suffix-free languages, which were studied previously.  ...  I am extremely grateful for his guidance and mentorship.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1007/978-3-319-60134-2_5">doi:10.1007/978-3-319-60134-2_5</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/6bkcrle7sjhnfofjd7r4vq2cpe">fatcat:6bkcrle7sjhnfofjd7r4vq2cpe</a> </span>
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