Dynamic Membership for Regular Languages

Dynamic Membership for Regular Languages

Antoine Amarilli¹, Louis Jachiet¹, Charles Paperman² July 14, 2021

1T ´el ´ecom Paris

2Universit ´e de Lille

1/18

Problem: dynamic membership for regular languages

• Fix aregular languageL

→ E.g.,L= (ab)^∗

• Read aninput wordwwithn:=|w|

→ E.g.,w=abbbab

• Preprocessit inO(n)

→ E.g., we havew∈/L

• Maintainthe membership ofwtoLundersubstitution updates

→ E.g., replace character at position 3 witha: we now havew∈L

Problem: dynamic membership for regular languages

• Fix aregular languageL

→ E.g.,L= (ab)^∗

• Read aninput wordwwithn:=|w|

→ E.g.,w=abbbab

• Preprocessit inO(n)

→ E.g., we havew∈/ L

• Maintainthe membership ofwtoLundersubstitution updates

→ E.g., replace character at position 3 witha: we now havew∈L

2/18

Problem: dynamic membership for regular languages

• Fix aregular languageL

→ E.g.,L= (ab)^∗

• Read aninput wordwwithn:=|w|

→ E.g.,w=abbbab

• Preprocessit inO(n)

→ E.g., we havew∈/ L

• Maintainthe membership ofwtoLundersubstitution updates

→ E.g., replace character at position 3 witha: we now havew∈L

Design choices

• Model: RAM model

• Cell size inΘ(log(n))

• Unit-cost arithmetics

• Updates: only substitutions(sonnever changes)

• Otherwise, alreadytrickyto maintain the current state of the word

• Memory usage: alwayspolynomial innby definition of the model

• Our upper bounds onlyneedO(n)space

• The lower bounds applywithout this assumption

• Preprocessing:

• The upper bounds onlyneedO(n)preprocessing

• The lower bounds applywithout this assumption

3/18

A general-purpose algorithm inO(logn)

Fix the languageL= (ab)^∗: start 0 1 a b

• Build abalanced binary treeon the input wordw=abaabb

• Label each nodenby thetransition monoidelement: all pairsq q⁰ such that we can go fromqtoq⁰ by reading the factor belown

a b b b a b

• Thetree rootdescribes ifw∈L

• We can update the tree for each substitutioninO(logn)

• Can be improved toO(logn/log logn)with alog-ary tree

A general-purpose algorithm inO(logn)

Fix the languageL= (ab)^∗: start 0 1 a b

• Build abalanced binary treeon the input wordw=abaabb

• Label each nodenby thetransition monoidelement: all pairsq q⁰ such that we can go fromqtoq⁰ by reading the factor belown

a b b b a b

• Thetree rootdescribes ifw∈L

• We can update the tree for each substitutioninO(logn)

• Can be improved toO(logn/log logn)with alog-ary tree

4/18

A general-purpose algorithm inO(logn)

Fix the languageL= (ab)^∗: start 0 1 a b

• Build abalanced binary treeon the input wordw=abaabb

• Label each nodenby thetransition monoidelement: all pairsq q⁰ such that we can go fromqtoq⁰ by reading the factor belown

a b b b a b

• Thetree rootdescribes ifw∈L

• We can update the tree for each substitutioninO(logn)

• Can be improved toO(logn/log logn)with alog-ary tree

A general-purpose algorithm inO(logn)

Fix the languageL= (ab)^∗: start 0 1 a b

• Build abalanced binary treeon the input wordw=abaabb

• Label each nodenby thetransition monoidelement: all pairsq q⁰ such that we can go fromqtoq⁰ by reading the factor belown

a b b b a b

• Thetree rootdescribes ifw∈L

• We can update the tree for each substitutioninO(logn)

• Can be improved toO(logn/log logn)with alog-ary tree

4/18

A general-purpose algorithm inO(logn)

Fix the languageL= (ab)^∗: start 0 1 a b

• Build abalanced binary treeon the input wordw=abaabb

• Label each nodenby thetransition monoidelement: all pairsq q⁰ such that we can go fromqtoq⁰ by reading the factor belown

a b b b a b

0 0 ∅ 0 0

∅ 0 0

∅

• Thetree rootdescribes ifw∈L

• We can update the tree for each substitutioninO(logn)

• Can be improved toO(logn/log logn)with alog-ary tree

A general-purpose algorithm inO(logn)

Fix the languageL= (ab)^∗: start 0 1 a b

• Build abalanced binary treeon the input wordw=abaabb

• Label each nodenby thetransition monoidelement: all pairsq q⁰ such that we can go fromqtoq⁰ by reading the factor belown

a b b b a b

0 0 ∅ 0 0

∅ 0 0

∅

• Thetree rootdescribes ifw∈L

• We can update the tree for each substitutioninO(logn)

• Can be improved toO(logn/log logn)with alog-ary tree

4/18

Can we do better thanO(logn)?

For our languageL= (ab)^∗ we can handle updates inO(1):

• Check thatniseven

• Countviolations: a’s ateven positionsandb’s atodd positions

• Maintain this counterin constant time

• We havew∈Liffthere are no violations

Question:what is the complexity of dynamic membership, depending on the fixed regular languageL?

Can we do better thanO(logn)?

For our languageL= (ab)^∗ we can handle updates inO(1):

• Check thatniseven

• Countviolations: a’s ateven positionsandb’s atodd positions

• Maintain this counterin constant time

• We havew∈Liffthere are no violations

Question:what is the complexity of dynamic membership, depending on the fixed regular languageL?

5/18

Can we do better thanO(logn)?

For our languageL= (ab)^∗ we can handle updates inO(1):

• Check thatniseven

• Countviolations: a’s ateven positionsandb’s atodd positions

• Maintain this counterin constant time

• We havew∈Liffthere are no violations

Question:what is the complexity of dynamic membership, depending on the fixed regular languageL?

Dynamic word problem for monoids

To answer the question, we study thedynamic word problem for monoids:

• Problem definition:

• Fix amonoidM(set with associative law and neutral element)

• Input:wordwof elements ofM

• Maintain theproductof the elements under substitution updates

• This is aspecial caseof dynamic membership for regular languages

• e.g., it assumes that there is aneutral element

• This problem was studied by [Skovbjerg Frandsen et al., 1997]:

→ inO(1)forcommutative monoids

→ inO(log logn)forgroup-free monoids

→ inΘ(logn/log logn)for a certain class of monoids

6/18

Dynamic word problem for monoids

To answer the question, we study thedynamic word problem for monoids:

• Problem definition:

• Fix amonoidM(set with associative law and neutral element)

• Input:wordwof elements ofM

• Maintain theproductof the elements under substitution updates

• This is aspecial caseof dynamic membership for regular languages

• e.g., it assumes that there is aneutral element

• This problem was studied by [Skovbjerg Frandsen et al., 1997]:

→ inO(1)forcommutative monoids

→ inO(log logn)forgroup-free monoids

→ inΘ(logn/log logn)for a certain class of monoids

Our results on the dynamic word problem for monoids

ZG: inO(1)

not inO(1)?

• We identify the classZGsatisfyingx^ω+1y=yx^ω+1:

• for any monoidinZG, the problem isinO(1)

• for any monoidnot inZG, we can reduce from a problem that weconjecture is not inO(1)

• We identify the classSGsatisfyingx^ω+1yx^ω =x^ωyx^ω+1

• for any monoidinSG, the problem isinO(log logn)

• for any monoidnot inSG, it isinΩ(logn/log logn) (lower bound of Skovbjerg Frandsen et al.)

• The problem is always inO(logn/log logn)

7/18

Our results on the dynamic word problem for monoids

ZG: inO(1)

SG: inO(log logn) not inO(1)?

All: inΘ(logn/log logn)

• We identify the classZGsatisfyingx^ω+1y=yx^ω+1:

• for any monoidinZG, the problem isinO(1)

• for any monoidnot inZG, we can reduce from a problem that weconjecture is not inO(1)

• We identify the classSGsatisfyingx^ω+1yx^ω =x^ωyx^ω+1

• for any monoidinSG, the problem isinO(log logn)

• for any monoidnot inSG, it isinΩ(logn/log logn) (lower bound of Skovbjerg Frandsen et al.)

• The problem is always inO(logn/log logn)

Our results on the dynamic word problem for monoids

ZG: inO(1)

SG: inO(log logn) not inO(1)?

All: inΘ(logn/log logn)

• We identify the classZGsatisfyingx^ω+1y=yx^ω+1:

• for any monoidinZG, the problem isinO(1)

• for any monoidnot inZG, we can reduce from a problem that weconjecture is not inO(1)

• We identify the classSGsatisfyingx^ω+1yx^ω =x^ωyx^ω+1

• for any monoidinSG, the problem isinO(log logn)

• for any monoidnot inSG, it isinΩ(logn/log logn) (lower bound of Skovbjerg Frandsen et al.)

• The problem is always inO(logn/log logn)

7/18

Results on the dynamic membership problem for regular languages

QLZG: inO(1)

QSG: inO(log logn) not inO(1)?

All: inΘ(logn/log logn)

Our results extend to regular language classes called QLZGandQSG

→ We define them in the sequel

Results on monoids

O(1)upper bound for monoids

Theorem

The dynamic word problem forcommutative monoidsis inO(1) Algorithm:

• Countthe numbernmof occurrences of each elementmofMinw

• Maintainthe countsnm under updates

• Evaluatethe product asQ

m∈Mm^nminO(1)

Lemma (Closure under monoid variety operations)

Thesubmonoids,direct products,quotientsof tractable monoids are also tractable

O(1)upper bound for monoids

Theorem

The dynamic word problem forcommutative monoidsis inO(1) Algorithm:

• Countthe numbernmof occurrences of each elementmofMinw

• Maintainthe countsnm under updates

• Evaluatethe product asQ

m∈Mm^nminO(1) Lemma (Closure under monoid variety operations)

Thesubmonoids,direct products,quotientsof tractable monoids are also tractable

9/18

O(1)upper bound for monoids (cont’d)

Theorem

The monoidsS¹where we add an identity to anilpotent semigroupSare inO(1) Idea of the proof:considere^∗ae^∗be^∗

• Preprocessing: prepare adoubly-linked listLof the positions containinga’s andb’s

• Maintainthe (unsorted) list whena’s andb’s are added/removed

• Evaluation:

• If there are not exactly two positions inL, answerno

• Otherwise, check that thesmallest positionof these two is anaand thelargest is ab

This technique applies to monoids where we intuitively need to tracka constant number of non-neutral elements

O(1)upper bound for monoids (cont’d)

Theorem

The monoidsS¹where we add an identity to anilpotent semigroupSare inO(1) Idea of the proof:considere^∗ae^∗be^∗

• Preprocessing: prepare adoubly-linked listLof the positions containinga’s andb’s

• Maintainthe (unsorted) list whena’s andb’s are added/removed

• Evaluation:

• If there are not exactly two positions inL, answerno

• Otherwise, check that thesmallest positionof these two is anaand thelargest is ab

This technique applies to monoids where we intuitively need to tracka constant number of non-neutral elements

10/18

O(1)upper bound for monoids (cont’d)

Theorem

The monoidsS¹where we add an identity to anilpotent semigroupSare inO(1) Idea of the proof:considere^∗ae^∗be^∗

• Preprocessing: prepare adoubly-linked listLof the positions containinga’s andb’s

• Maintainthe (unsorted) list whena’s andb’s are added/removed

• Evaluation:

• If there are not exactly two positions inL, answerno

• Otherwise, check that thesmallest positionof these two is anaand thelargest is ab

This technique applies to monoids where we intuitively need to tracka constant number of non-neutral elements

O(1)upper bound for monoids (end)

CallZGthe variety of monoids satisfyingx^ω+1y=yx^ω+1 for allx,y

→ Elements of the formx^ω+1are those belonging to asubgroupof the monoid

→ This includes in particular allidempotents(xx=x)

→ Thex^ω+1 arecentral: they commute with all other elements

Lemma

ZGis exactly the monoids obtainable fromcommutative monoidsandmonoids of the form S¹ for a nilpotent semigroupSvia themonoid variety operators

Theorem

The dynamic word problem for monoids inZGisin O(1)

11/18

O(1)upper bound for monoids (end)

CallZGthe variety of monoids satisfyingx^ω+1y=yx^ω+1 for allx,y

→ Elements of the formx^ω+1are those belonging to asubgroupof the monoid

→ This includes in particular allidempotents(xx=x)

→ Thex^ω+1 arecentral: they commute with all other elements Lemma

ZGis exactly the monoids obtainable fromcommutative monoidsandmonoids of the form S¹ for a nilpotent semigroupSvia themonoid variety operators

Theorem

The dynamic word problem for monoids inZGisin O(1)

O(log logn)upper bound for monoids

CallSGthe variety of monoids satisfyingx^ω+1yx^ω=x^ωyx^ω+1for allx,y

→ Intuition:we canswapthe elements of any given subgroup of the monoid Examples:

• AllZGmonoids(where elementsx^ω+1 commute with everything)

• Allgroup-free monoids(where subgroups are trivial)

• ProductsofZGmonoids and group-free monoids

Theorem

The dynamic word problem for monoids inSGis inO(log logn)

Tools:induction onJ-classes, Rees-Sushkevich theorem, Van Emde Boas trees

12/18

O(log logn)upper bound for monoids

CallSGthe variety of monoids satisfyingx^ω+1yx^ω=x^ωyx^ω+1for allx,y

→ Intuition:we canswapthe elements of any given subgroup of the monoid Examples:

• AllZGmonoids(where elementsx^ω+1 commute with everything)

• Allgroup-free monoids(where subgroups are trivial)

• ProductsofZGmonoids and group-free monoids Theorem

The dynamic word problem for monoids inSGis inO(log logn)

Tools:induction onJ-classes, Rees-Sushkevich theorem, Van Emde Boas trees

Lower bounds

All lower bounds reduce from theprefix problemfor some languageL:

• Maintain a word undersubstitution updates

• Answer queries asking if agiven prefixof the current word is inL

Specifically:

• Prefix-Zd: forΣ ={0, . . . ,d−1}, does the input prefixsum to 0 modulod?

→ Knownlower boundofΩ(logn/log logn)

• Prefix-U1:forΣ ={0,1}, does the queried prefixcontain a 0?

→ Weconjecturethat this cannot be done inO(1) Theorem (Lower bounds on a monoidM)

• IfMisnot inSG, then for somed∈NthePrefix-Zdproblem reduces to the dynamic word problem forM

• IfMisinSG\ZG, thenPrefix-U1reduces to the dynamic word problem forM

13/18

Lower bounds

All lower bounds reduce from theprefix problemfor some languageL:

• Maintain a word undersubstitution updates

• Answer queries asking if agiven prefixof the current word is inL Specifically:

• Prefix-Zd: forΣ ={0, . . . ,d−1}, does the input prefixsum to 0 modulod?

→ Knownlower boundofΩ(logn/log logn)

• Prefix-U1:forΣ ={0,1}, does the queried prefixcontain a 0?

→ Weconjecturethat this cannot be done inO(1)

Theorem (Lower bounds on a monoidM)

• IfMisnot inSG, then for somed∈NthePrefix-Zdproblem reduces to the dynamic word problem forM

• IfMisinSG\ZG, thenPrefix-U1reduces to the dynamic word problem forM

Lower bounds

All lower bounds reduce from theprefix problemfor some languageL:

• Maintain a word undersubstitution updates

• Answer queries asking if agiven prefixof the current word is inL Specifically:

• Prefix-Zd: forΣ ={0, . . . ,d−1}, does the input prefixsum to 0 modulod?

→ Knownlower boundofΩ(logn/log logn)

• Prefix-U1:forΣ ={0,1}, does the queried prefixcontain a 0?

→ Weconjecturethat this cannot be done inO(1) Theorem (Lower bounds on a monoidM)

• IfMisnot inSG, then for somed∈NthePrefix-Zdproblem reduces to the dynamic word problem forM

• IfMisinSG\ZG, thenPrefix-U1reduces to the dynamic word problem forM

13/18

Results on languages

(via semigroups)

From monoids to semigroups

• Semigroup: like a monoid but possibly without a neutral element

• Dynamic word problem for semigroups: defined like for monoids What is the difference?

• The languageΣ^∗(ae^∗a)Σ^∗onΣ ={a,b,e}has aneutral lettere that we intuitively need to “jump over”

• The languageΣ^∗aaΣ^∗ onΣ ={a,b}withoute

can bemaintained inO(1)by counting the factorsaa

14/18

Submonoids in semigroups

• Asubmonoidof a semigroupSis a subset ofSthat has aneutral element

→ IfShas a submonoidMthen the dynamic word problem forMreducestoS

→ Lower boundsonMthus apply toS

• Hence, we define:

• LSG: all submonoids areinSG

→ We showLSG=SGand extend our bounds tosemigroups inSG

• LZG: all submonoids areinZG

→ We haveLZG6=ZGand show bounds forsemigroups inLZG

Submonoids in semigroups

• Asubmonoidof a semigroupSis a subset ofSthat has aneutral element

→ IfShas a submonoidMthen the dynamic word problem forMreducestoS

→ Lower boundsonMthus apply toS

• Hence, we define:

• LSG: all submonoids areinSG

→ We showLSG=SGand extend our bounds tosemigroups inSG

• LZG: all submonoids areinZG

→ We haveLZG6=ZGand show bounds forsemigroups inLZG

15/18

Submonoids in semigroups

• Asubmonoidof a semigroupSis a subset ofSthat has aneutral element

→ IfShas a submonoidMthen the dynamic word problem forMreducestoS

→ Lower boundsonMthus apply toS

• Hence, we define:

• LSG: all submonoids areinSG

→ We showLSG=SGand extend our bounds tosemigroups inSG

• LZG: all submonoids areinZG

→ We haveLZG6=ZGand show bounds forsemigroups inLZG

From semigroups to languages

We now move back todynamic membership for regular languages

• Dynamic membership for a regular languageLis like the dynamic word problem for itssyntactic semigroup

→ This is like the transition monoid but without theneutral element

• Difference: not all elements of the syntactic semigroup can be achieved as one letter

→ We use instead thestable semigroup, which intuitively groups letters together intoblocksof a constant size

16/18

From semigroups to languages (cont’d)

CallQLZGandQSGthe languages whosestable semigroupis inLZGandSG Theorem

Our results onsemigroupsinSGandLZGextend toregular languagesinQSG andQLZG

For any regular languageL:

• IfLisinQLZGthen dynamic membership isin O(1)

• IfLisinQSG\QLZGthen dynamic membership isin O(log logn)and has a reductionfrom prefix-U1

• IfLisnot inQSGthen dynamic membership isinΘ(logn/log logn)

From semigroups to languages (cont’d)

CallQLZGandQSGthe languages whosestable semigroupis inLZGandSG Theorem

Our results onsemigroupsinSGandLZGextend toregular languagesinQSG andQLZG

For any regular languageL:

• IfLisinQLZGthen dynamic membership isin O(1)

• IfLisinQSG\QLZGthen dynamic membership isin O(log logn)and has a reductionfrom prefix-U1

• IfLisnot inQSGthen dynamic membership isinΘ(logn/log logn)

17/18

Conclusion and future work

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.