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