Summary
We have shown a (conditional)trichotomyon the dynamic word problem for monoids and semigroups, and on dynamic membership forregular languages
• Can one show asuperconstant lower boundonprefix-U1?
→ Help welcome!but new techniques probably needed
• What aboutintermediate casesbetweenO(1)andO(log logn)
• Yes withrandomization: one languageinΘ(log logn)and oneinO(√
log logn)
• Question:can the intermediate classes becharacterized?
• Meta-dichotomy: what is the complexity of finding which case occurs?
→ ProbablyPSPACE-complete(depends on the representation)
• What about a dichotomy for theprefix problemorinfix problem?
→ We have such a result butinelegant characterization
• What about languages that arenon-regular?
Thanks for your attention!
18/18
Summary
We have shown a (conditional)trichotomyon the dynamic word problem for monoids and semigroups, and on dynamic membership forregular languages
• Can one show asuperconstant lower boundonprefix-U1?
→ Help welcome!but new techniques probably needed
• What aboutintermediate casesbetweenO(1)andO(log logn)
• Yes withrandomization: one languageinΘ(log logn)and oneinO(√
log logn)
• Question:can the intermediate classes becharacterized?
• Meta-dichotomy: what is the complexity of finding which case occurs?
→ ProbablyPSPACE-complete(depends on the representation)
• What about a dichotomy for theprefix problemorinfix problem?
→ We have such a result butinelegant characterization
• What about languages that arenon-regular?
Thanks for your attention!
Summary
We have shown a (conditional)trichotomyon the dynamic word problem for monoids and semigroups, and on dynamic membership forregular languages
• Can one show asuperconstant lower boundonprefix-U1?
→ Help welcome!but new techniques probably needed
• What aboutintermediate casesbetweenO(1)andO(log logn)
• Yes withrandomization: one languageinΘ(log logn)and oneinO(√
log logn)
• Question:can the intermediate classes becharacterized?
• Meta-dichotomy: what is the complexity of finding which case occurs?
→ ProbablyPSPACE-complete(depends on the representation)
• What about a dichotomy for theprefix problemorinfix problem?
→ We have such a result butinelegant characterization
• What about languages that arenon-regular?
Thanks for your attention!
18/18
Summary
We have shown a (conditional)trichotomyon the dynamic word problem for monoids and semigroups, and on dynamic membership forregular languages
• Can one show asuperconstant lower boundonprefix-U1?
→ Help welcome!but new techniques probably needed
• What aboutintermediate casesbetweenO(1)andO(log logn)
• Yes withrandomization: one languageinΘ(log logn)and oneinO(√
log logn)
• Question:can the intermediate classes becharacterized?
• Meta-dichotomy: what is the complexity of finding which case occurs?
→ ProbablyPSPACE-complete(depends on the representation)
• What about a dichotomy for theprefix problemorinfix problem?
→ We have such a result butinelegant characterization
• What about languages that arenon-regular?
Thanks for your attention!
Summary
We have shown a (conditional)trichotomyon the dynamic word problem for monoids and semigroups, and on dynamic membership forregular languages
• Can one show asuperconstant lower boundonprefix-U1?
→ Help welcome!but new techniques probably needed
• What aboutintermediate casesbetweenO(1)andO(log logn)
• Yes withrandomization: one languageinΘ(log logn)and oneinO(√
log logn)
• Question:can the intermediate classes becharacterized?
• Meta-dichotomy: what is the complexity of finding which case occurs?
→ ProbablyPSPACE-complete(depends on the representation)
• What about a dichotomy for theprefix problemorinfix problem?
→ We have such a result butinelegant characterization
• What about languages that arenon-regular?
Thanks for your attention!
18/18
Summary
We have shown a (conditional)trichotomyon the dynamic word problem for monoids and semigroups, and on dynamic membership forregular languages
• Can one show asuperconstant lower boundonprefix-U1?
→ Help welcome!but new techniques probably needed
• What aboutintermediate casesbetweenO(1)andO(log logn)
• Yes withrandomization: one languageinΘ(log logn)and oneinO(√
log logn)
• Question:can the intermediate classes becharacterized?
• Meta-dichotomy: what is the complexity of finding which case occurs?
→ ProbablyPSPACE-complete(depends on the representation)
• What about a dichotomy for theprefix problemorinfix problem?
→ We have such a result butinelegant characterization
• What about languages that arenon-regular?
Thanks for your attention!
Summary
We have shown a (conditional)trichotomyon the dynamic word problem for monoids and semigroups, and on dynamic membership forregular languages
• Can one show asuperconstant lower boundonprefix-U1?
→ Help welcome!but new techniques probably needed
• What aboutintermediate casesbetweenO(1)andO(log logn)
• Yes withrandomization: one languageinΘ(log logn)and oneinO(√
log logn)
• Question:can the intermediate classes becharacterized?
• Meta-dichotomy: what is the complexity of finding which case occurs?
→ ProbablyPSPACE-complete(depends on the representation)
• What about a dichotomy for theprefix problemorinfix problem?
→ We have such a result butinelegant characterization
• What about languages that arenon-regular?
Thanks for your attention! 18/18
References i
Fredman, M. and Saks, M. (1989).
The cell probe complexity of dynamic data structures.
InSTOC, pages 345–354.
Patrascu, M. (2008).
Lower bound techniques for data structures.
PhD thesis, Massachusetts Institute of Technology.
Skovbjerg Frandsen, G., Miltersen, P. B., and Skyum, S. (1997).
Dynamic word problems.
JACM, 44(2):257–271.
O(log logn)upper bound for monoids (proof sketch)
Example :Σ∗(ae∗a)Σ∗onΣ ={a,b,e}
• Idea: maintain the count of factorsae∗a
• Problem:to do this, we need to “jump over” thee’s
→ Van Emde Boas treedata structure:
• maintaina subset of{1, . . . ,n}underinsertions/deletions
• jumpto theprev/next elementinO(log logn)
Full proof:induction onJ-classesandRees-Sushkevich theorem
ExtendingSGto semigroups
We can show that, for semigroups:
Lemma
A semigroup satisfies the equation ofSGiff it is inLSG
Hence, as the algorithm forSGworks forsemigroupsas well as monoids:
Theorem
For any semigroupS:
• IfSisinSG, then the dynamic word problem isin O(log logn)
• Otherwise, the dynamic word problem isinΘ(logn/log logn)
Case ofZG
We haveZG6=LZG, but we can still show:
Theorem
For any semigroupS:
• IfSisinLZG, then the dynamic word problem isin O(1)
• Otherwise, it has a reductionfrom Prefix-U1
Proof sketch:only need to show theupper bound:
• We show theO(1)upper bound on thesemidirect productZG∗D ofZGwithdefinite semigroups
• We show an independentlocality result: LZG=ZG∗D
→ Technical proof relying onfinite categoriesandStraubing’s delay theorem
Case ofZG
We haveZG6=LZG, but we can still show:
Theorem
For any semigroupS:
• IfSisinLZG, then the dynamic word problem isin O(1)
• Otherwise, it has a reductionfrom Prefix-U1
Proof sketch:only need to show theupper bound:
• We show theO(1)upper bound on thesemidirect productZG∗D ofZGwithdefinite semigroups
• We show an independentlocality result: LZG=ZG∗D
→ Technical proof relying onfinite categoriesandStraubing’s delay theorem
Difference between the stable semigroup and syntactic semigroup
• Dynamic membership for(aa)∗ba∗ isinO(1): count theb’s atevenandodd positions
• The dynamic word problem for its syntactic semigroup has areduction from Z2