Marc Zeitoun Joint work with Thomas Place LaBRI, U ... · Marc Zeitoun Joint work with Thomas Place...

Separating languages with no ambiguity

Marc Zeitoun

Joint work with Thomas Place

LaBRI, U. Bordeaux

ANR Delta@IRIF, March 2018

Concatenation and AlternationHierarchies

The membership problem for classes of languages

Membership problem for a class CI INPUT A (regular) language L.I QUESTION Does L belong to C?

Examples of classes C:I Languages definable in FO.I Languages of GSH k ≥ 1.I Languages of GSH 0 (called star-free, denoted SF).

(ab)+ =

a∅ ∩ ∅b ∩ ∅(aa+ bb)∅ ∩\

c 6=a;b

∅c∅

a Does it belong to C?

Schützenberger ’65For L a regular language, the following are equivalent:I L is star-free.

semantic

I The minimal automaton of L is counter-free.

syntactic

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(ab)+ =

a∅ ∩ ∅b ∩ ∅(aa+ bb)∅ ∩\

c 6=a;b

∅c∅

Schützenberger ’65For L a regular language, the following are equivalent:I L is star-free.

semantic

I The minimal automaton of L is counter-free.

syntactic

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(ab)+ =

a∅ ∩ ∅b ∩ ∅(aa+ bb)∅ ∩\

c 6=a;b

∅c∅

Schützenberger ’65For L a regular language, the following are equivalent:I L is star-free. semanticI The minimal automaton of L is counter-free. syntactic

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(ab)+ = a∅

∩ ∅b ∩ ∅(aa+ bb)∅ ∩\

c 6=a;b

∅c∅

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(ab)+ = a∅ ∩ ∅b

∩ ∅(aa+ bb)∅ ∩\

c 6=a;b

∅c∅

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(ab)+ = a∅ ∩ ∅b ∩ ∅(aa+ bb)∅

c 6=a;b

∅c∅

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(ab)+ = a∅ ∩ ∅b ∩ ∅(aa+ bb)∅ ∩\

c 6=a;b

∅c∅

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(ab)+ =

a∅ ∩ ∅b ∩ ∅(aa+ bb)∅ ∩\

c 6=a;b

∅c∅

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Why is Schützenberger’s theorem interesting?

1. Provides an effective characterization of SF.

2. Link with first-order logic FO.

Schützenberger ’65, McNaughton, Papert ’71For L a regular language, the following are equivalent:I L is star-free. semanticI L is FO-definable. semanticI The minimal automaton of L is counter-free. syntactic

SF FOA∗; ∅ True;False

∪; A∗\ ∨; ¬KaL ∃x a(x) ∧ ’<x

K (x) ∧ ’>xL (x)

3. Constructive proof ⇒ normal forms for star-free expressions, FO sentences.

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∪; A∗\ ∨; ¬KaL ∃x a(x) ∧ ’<x

K (x) ∧ ’>xL (x)

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∪; A∗\ ∨; ¬KaL ∃x a(x) ∧ ’<x

K (x) ∧ ’>xL (x)

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∪; A∗\ ∨; ¬KaL ∃x a(x) ∧ ’<x

K (x) ∧ ’>xL (x)

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Concatenation hierarchies: Motivation

Definition of SF alternate Boolean operations and concatenationI SF = smallest class such that:

I ∅ ∈ SF and A∗ ∈ SF.I SF is closed under Boolean operations over A∗.I SF is closed under marked concatenation.

K;L ∈ SFa ∈ A

=⇒ KaL ∈ SF

GoalI Finer understanding of FO / SF.I Interplay between complement and concatenation.

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Concatenation hierarchies: Motivation

Definition of SF alternate Boolean operations and concatenationI SF = smallest class such that:

I ∅ ∈ SF and A∗ ∈ SF.I SF is closed under Boolean operations over A∗.I SF is closed under marked concatenation.

K;L ∈ SFa ∈ A

=⇒ KaL ∈ SF

GoalI Finer understanding of FO / SF.I Interplay between complement and concatenation.

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Classical hierarchies inside SFTwo classes built on top of CI Boolean closure Bool(C).I Polynomial closure Pol(C) = closure under marked concatenation + union

How many needed alternations between Boolean operations and concatenations?

Straubing-Thérien hierarchy ’81I ST [0] = {∅; A∗}.I ST

ˆn + 1

˜= Pol(ST [n]).

I ST [n + 1] = Bool(STˆn + 1

Brzozowski-Cohen hierarchy ’71I DD [0] = {∅; {"}; A+; A∗}.I DD

ˆn + 1

˜= Pol(DD [n]).

I DD [n + 1] = Bool(DDˆn + 1

0 12 1 3

2 2 52

PolBool

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ˆn + 1

˜= Pol(ST [n]).

ˆn + 1

˜= Pol(DD [n]).

0 12 1 3

2 2 52

PolBool

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Straubing-Thérien hierarchy ’81I ST [0] = {∅; A∗}.

I STˆn + 1

˜= Pol(ST [n]).

ˆn + 1

˜= Pol(DD [n]).

0 12 1 3

2 2 52

PolBool

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ˆn + 1

˜= Pol(ST [n]).

ˆn + 1

˜= Pol(DD [n]).

0 12 1 3

2 2 52

PolBool

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ˆn + 1

˜= Pol(ST [n]).

ˆn + 1

˜= Pol(DD [n]).

0 12 1 3

2 2 52

PolBool

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ˆn + 1

˜= Pol(ST [n]).

ˆn + 1

˜= Pol(DD [n]).

0 12 1 3

2 2 52

PolBool

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ˆn + 1

˜= Pol(ST [n]).

ˆn + 1

˜= Pol(DD [n]).

0 12 1 3

2 2 52

PolBool

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Brzozowski-Cohen and Straubing-Thérien hierarchies

Natural questionsI Are the hierarchies strict?I Logical description of each level?I What is known about membership?

What is known1. Both hierarchies are strict (Brzozowski-Knast 1978 + interleaving),

(a · · · (a(ab)∗b)∗ · · · b)∗

2. Natural logical description wihin FO.3. Membership for BC reduces to membership for ST.4. Membership solved for only few levels.

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Brzozowski-Cohen and Straubing-Thérien hierarchies

What is known1. Both hierarchies are strict (Brzozowski-Knast 1978 + interleaving),

(a · · · (a(ab)∗b)∗ · · · b)∗

2. Natural logical description wihin FO.3. Membership for BC reduces to membership for ST.4. Membership solved for only few levels.

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Logical counterpart: quantifier alternation hierarchies

Intuition: marked concatenation corresponds to existential quantification.I Σi = ∃∗∀∗∃∗∀∗∃∗ · · ·| {z }

at most i blocks ∃∗ or ∀∗’, (’ quantifier free).

I BΣi = Finite Boolean combinations of Σi .

Quantifier Alternation Hierarchies

Σ1 BΣ1

Σ2 BΣ2

Σ3 BΣ3

FO( ( ( ( ( ( (

Two versionsI Order signature: < and a().I Enriched signature: <, a(), +1, min(), max() and ".

Logical Correspondence Theorem (Thomas ’82, Perrin-Pin ’86)I Brzozowski-Cohen hierarchy = enriched quantifier alternation hierarchy.I Straubing-Thérien hierarchy = order quantifier alternation hierarchy.

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Σ1 BΣ1

Σ2 BΣ2

Σ3 BΣ3

FO( ( ( ( ( ( (

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Σ1 BΣ1

Σ2 BΣ2

Σ3 BΣ3

FO( ( ( ( ( ( (

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Σ1 BΣ1

Σ2 BΣ2

Σ3 BΣ3

FO( ( ( ( ( ( (

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The membership problem for BC and ST hierarchies

Σ1 BΣ1 Σ2 BΣ2 Σ3 BΣ3 Σ4 FO( ( ( ( ( ( (

Schützenberger’65McNaughton-Papert’71

Simon’75 Place,Z.’14

Arfi’87Pin, Weil’95

Place’15

Enrichment Theorem for membership (Straubing ’85 – Pin, Weil ’97)

Membership for a level in the enriched hierarchy (ie, BC)reduces to

Membership for the same level in the order hierarchy (ie, ST).

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Σ1 BΣ1 Σ2 BΣ2 Σ3 BΣ3 Σ4 FO( ( ( ( ( ( (

Simon’75

Place,Z.’14

Place’15

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Σ1 BΣ1 Σ2 BΣ2 Σ3 BΣ3 Σ4 FO( ( ( ( ( ( (

Simon’75

Place,Z.’14

Place’15

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Σ1 BΣ1 Σ2 BΣ2 Σ3 BΣ3 Σ4 FO( ( ( ( ( ( (

Place’15

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Σ1 BΣ1 Σ2 BΣ2 Σ3 BΣ3 Σ4 FO( ( ( ( ( ( (

Place’15

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Σ1 BΣ1 Σ2 BΣ2 Σ3 BΣ3 Σ4 FO( ( ( ( ( ( (

Place’15

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Generalizations in two directions

1. Proofs are ad hoc for DD and ST: obtain generic theorems.For given C, what about Pol(C), Bool(Pol(C)),. . .

2. Recent results via generalizations of membership: separation and covering.

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Generic concatenation hierarchies

Generic pattern parametrized by the basisI C[0] (basis) Boolean algebra in REG closed under left/right quotients

C[n + 12 ]: close C[n] under K;L 7→ KaL and ∪.

C[n + 1]: close C[n + 12 ] under Boolean operations.

0 12 1 3

2 2 52

PolBool

Examples

I Straubing-Thérien: C[0] = {∅; A∗}.I Brzozowski-Cohen: C[0] = {∅; {"}; A∗; A+}.I Pin-Margolis: C[0] = group languages.

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C[n + 12 ]: close C[n] under K; L 7→ KaL and ∪.

0 12 1 3

2 2 52

PolBool

Examples

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C[n + 12 ]: close C[n] under K; L 7→ KaL and ∪.

0 12 1 3

2 2 52

PolBool

Examples

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Generic Hierarchies

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Basic properties of generic hierarchies

Basic Structure (Place, Z. ’17)

For any hierarchy:I All levels are closed under quotient and in REG.I Full levels are closed under Bool.I Half levels are closed under union, intersection (and concatenation).

Quotienting: closed under u−1L = {x | ux ∈ L} and Lu−1:

Strictness Theorem (Place, Z. ’17)

Any hierarchy whose basis is finite is strict.

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Basic properties of generic hierarchies

Basic Structure (Place, Z. ’17)

For any hierarchy:I All levels are closed under quotient and in REG.I Full levels are closed under Bool.I Half levels are closed under union, intersection (and concatenation).

Quotienting: closed under u−1L = {x | ux ∈ L} and Lu−1:

Strictness Theorem (Place, Z. ’17)

Any hierarchy whose basis is finite is strict.

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Generic logical correspondence

Logical Correspondence Theorem (Place, Z. ’17)

For any basis C, there is a natural set S of first order predicates, st.

Concatenation hierarchy of basis C=

Quantifier alternation hierarchy over signature S

Generalizes the correspondences discovered for BC and ST hierarchies.

IntuitionFor each L ∈ C, add 4 predicates in addition to < and a(); b(); : : :

I w |= IL(x; y) when x < y and w ]x; y [ ∈ L (Infix).I w |= PL(y) when w [1; y [ ∈ L (Prefix).I w |= SL(x) when w ]x; n] ∈ L (Suffix).I w |= WL when w ∈ L (Whole word).

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I w |= IL(x; y) when x < y and w ]x; y [ ∈ L (Infix).

I w |= PL(y) when w [1; y [ ∈ L (Prefix).I w |= SL(x) when w ]x; n] ∈ L (Suffix).I w |= WL when w ∈ L (Whole word).

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I w |= IL(x; y) when x < y and w ]x; y [ ∈ L (Infix).I w |= PL(y) when w [1; y [ ∈ L (Prefix).I w |= SL(x) when w ]x; n] ∈ L (Suffix).

I w |= WL when w ∈ L (Whole word).

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I w |= IL(x; y) when x < y and w ]x; y [ ∈ L (Infix).I w |= PL(y) when w [1; y [ ∈ L (Prefix).I w |= SL(x) when w ]x; n] ∈ L (Suffix).I w |= WL when w ∈ L (Whole word).

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Generic membership theorem

Generic membership Theorem (Place, Z. ’17, Place ’15)

For any finite basis C, levels 12 ; 1;

52 have decidable membership.

Remember: decidability known up to level 72 for ST and BC hierarchies.

The alphabet trick

Languages in STˆ32

˜(Pin and Straubing ’85)

Languages of level STˆ32

˜are unions of languages of the form B∗

0a1B∗1 · · · anB∗

STˆ32

˜= level 1

2 with basis AT = {B∗ | B ⊆ A}.

ST[q] is also level (q − 1) in another hierarchy with finite basis.ST[q] = AT[q − 1].

Corollary (by Alphabet trick)

and BC

hierarchy, levels 12 ; 1;

32 ; 2;

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The alphabet trick

Languages in STˆ32

0a1B∗1 · · · anB∗

STˆ32

˜= level 1

and BC

32 ; 2;

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The alphabet trick

Languages in STˆ32

0a1B∗1 · · · anB∗

STˆ32

˜= level 1

and BC

32 ; 2;

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The alphabet trick

Languages in STˆ32

0a1B∗1 · · · anB∗

STˆ32

˜= level 1

and BC

32 ; 2;

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The alphabet trick

Languages in STˆ32

0a1B∗1 · · · anB∗

STˆ32

˜= level 1

and BC

32 ; 2;

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The alphabet trick

Languages in STˆ32

0a1B∗1 · · · anB∗

STˆ32

˜= level 1

In ST and BC hierarchy, levels 12 ; 1;

32 ; 2;

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I Generic construction process for concatenation hierarchies.I Generic logical correspondence.I Generic strictness theorem.I Generic membership theorem.

Recent results required solving harder problems than membership.

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I Generic construction process for concatenation hierarchies.I Generic logical correspondence.I Generic strictness theorem.I Generic membership theorem.

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Beyond Membership:Separation Problems

Beyond membership: Separation

Motivation:

I Classe C with decidable membership.I Class Op(C) built on top of C with undecidable membership.

Nice idea, Henckell and Rhodes ’88

Prove more on C to recover membership decidability for Op(C).

Nice statement, Almeida ’96Almeida’96: a problem introduced by Henckell can be formulated as separation.

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Motivation:

I Classe C with decidable membership.I Class Op(C) built on top of C with undecidable membership.

Nice idea, Henckell and Rhodes ’88

Prove more on C to recover membership decidability for Op(C).

Nice statement, Almeida ’96Almeida’96: a problem introduced by Henckell can be formulated as separation.

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Decide the following problem:

Take 2 regular languages L1; L2

a b b b

b a b b

Can L1 be separated from L2

with a language from C?

in CC-separable from complement⇔in C

Membership can be formally reduced to separation

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a b b b

b a b b

in CC-separable from complement⇔in C

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a b b b

b a b b

C-separable from complement⇔in C

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a b b b

b a b b

L2 = A∗ \ L1 L1

C-separable from complement⇔in C

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Separation for classical hierarchiesSe

Σ1 BΣ1 Σ2 BΣ2 Σ3 BΣ3 Σ4 FO( ( ( ( ( ( (

Henckell’88Henckell, Rhodes, Steinberg’10

Place,Z.’14

Simon’75

Almeida,Z.’97Czerwinski,Martens,Masopust’13

Place,Van Rooijen,Z.’13

Place,Z.’14Arfi’87

Pin, Weil’95

Place,Z.’14

Place,Z.’17

Place’15

Some membership algorithms come from separation algorithms for simpler levels

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Separation for classical hierarchiesSe

Σ1 BΣ1 Σ2 BΣ2 Σ3 BΣ3 Σ4 FO( ( ( ( ( ( (

Henckell’88Henckell, Rhodes, Steinberg’10

Place,Z.’14

Simon’75

Almeida,Z.’97Czerwinski,Martens,Masopust’13

Place,Van Rooijen,Z.’13

Place,Z.’14Arfi’87

Pin, Weil’95

Place,Z.’14

Place,Z.’17

Place’15

Some membership algorithms come from separation algorithms for simpler levels

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Separation results for generic hierarchies

All what we know for ST and BC follow from:

Generic Separation Theorem (Place, Z. ’17, Place ’15)

Levels 12 ; 1 and 3

2 have decidable separation for any hierarchy whose basis is finite.

Corollary (by Alphabet trick + Enrichment)

Levels 12 ; 1,

32 , 2 and 5

2 have decidable separation in ST and BC hierarchies.

Jump Theorem for quotienting lattices (Place, Z. ’15)

For any basis, membership for level n + 12 reduces to separation for level n − 1

Enrichment Theorem (Place, Z. ’15)

Separation for a level in the enriched hierarchy (ie, BC)reduces to

Separation for the same level in the order hierarchy (ie, ST).

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Levels 12 ; 1 and 3

Levels 12 ; 1,

32 , 2 and 5

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Levels 12 ; 1 and 3

Levels 12 ; 1,

32 , 2 and 5

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Levels 12 ; 1 and 3

Levels 12 ; 1,

32 , 2 and 5

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The Jump Theorem

The Jump Theorem on Automata

A regular language is in Cˆn+ 1

˜iff its minimal automaton has no pattern:

not final final

where Lp;q is not Cˆn− 1

˜-separable from Lp;p ∩ Lq;q

Lp;q = {w | p w−−−−→ q}

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The Jump Theorem

not final final

Lp;q = {w | p w−−−−→ q}

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The Jump Theorem

not final final

Lp;q = {w | p w−−−−→ q}

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The Jump Theorem

not final final

Lp;q = {w | p w−−−−→ q}

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Current knowledge is captured by few generic results:

1. Separation theoremC finite ⇒ separation decidable for Pol(C), BPol(C) and Pol(BPol(C)).In particular, cannot yet deal with 2 levels of complement.

2. Jump theoremC-separation decidable ⇒ Pol(C)-membership decidable.

3. Enrichment theorem (see also Varun’s talk).

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Unambiguous Polynomial Closure

History and motivations

I Schützenberger, 1976: UL = Unambiguous Polynomial closure(AT).

AT = {B∗ | B ⊆ A}:

I Robust: FO2(<), ∆2(<), UTL, TL(rankers), 2way turtle DFAs, DA. . .

I Pin, 1980: Generic investigation of unambiguous polynomial closure

I Margolis Pin Thérien, 1988 UPol(C) membership reduces to C-membershipAlgebraic proofs (relational morphisms, categories, bilateral kernels, etc.)

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AT = {B∗ | B ⊆ A}:

I Robust: FO2(<), ∆2(<),

UTL, TL(rankers), 2way turtle DFAs, DA. . .

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AT = {B∗ | B ⊆ A}:

I Robust: FO2(<), ∆2(<), UTL, TL(rankers),

2way turtle DFAs, DA. . .

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AT = {B∗ | B ⊆ A}:

I Robust: FO2(<), ∆2(<), UTL, TL(rankers), 2way turtle DFAs,

DA. . .

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AT = {B∗ | B ⊆ A}:

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AT = {B∗ | B ⊆ A}:

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AT = {B∗ | B ⊆ A}:

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AT = {B∗ | B ⊆ A}:

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Unambiguous polynomial closure

I KL is unambiguous if each w ∈ KL has a single factorization in K × L.Generalizes to n languages (and so to).

I A∗aA∗? (A \ {a})∗aA∗?

I UPol(C) = closure of C underI unambiguous concatenation,I disjoint union.

I ExamplesI UPol({∅; A∗})?I UPol(AT) = UL from Schützenberger.

I Not even clear whether UPol(C) is closed under union.

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I A∗aA∗?

(A \ {a})∗aA∗?

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I A∗aA∗? (A \ {a})∗aA∗?

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I A∗aA∗? (A \ {a})∗aA∗?

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I A∗aA∗? (A \ {a})∗aA∗?

I ExamplesI UPol({∅; A∗})?

I UPol(AT) = UL from Schützenberger.

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I A∗aA∗? (A \ {a})∗aA∗?

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I A∗aA∗? (A \ {a})∗aA∗?

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Alternating polynomial closure

I KaL is left deterministic if KaA∗ ∩K = ∅.Note. Does not depend on L.

I Right deterministic: symmetric

I Unambiguous concatenation which is not left neither right deterministic?(A \ {a})∗aA∗b(A \ {b})∗.

I ADet(C) = closure of C under left+right deterministic marked concatenation

Easy remark

ADet(C) ⊆ UPol(C)

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Easy remark

ADet(C) ⊆ UPol(C)

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I Unambiguous concatenation which is not left neither right deterministic?

(A \ {a})∗aA∗b(A \ {b})∗.

Easy remark

ADet(C) ⊆ UPol(C)

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Easy remark

ADet(C) ⊆ UPol(C)

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Easy remark

ADet(C) ⊆ UPol(C)

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Quasi separation wrt. a surjective morphism

A∗ M

¸−1(s)

¸−1(t)

¸−1(s)

¸−1(s); ¸−1(t) are C-separable by some ¸−1(P )⇔

(s; t) is not a quasi C-pair

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A∗ M

¸−1(s)

¸−1(t)

¸−1(s)

¸−1(s); ¸−1(t) are C-separable⇔

(s; t) is not a C-pair

(s; t) is not a quasi C-pair

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A∗ M

¸−1(s)

¸−1(t)

¸−1(s)

(s; t) is not a quasi C-pair25 / 29

Quasi C-pairs

(s; t) quasi C-pair = ¸−1(s); ¸−1(t) not C-separable by language recognized by ¸

I C-membership decidable =⇒ quasi C-pairs computable.

I C-pair ⇒ quasi C-pair.

I The “C-pair” relation is not transitive.Ex. for C =AT: L1 = {ab}, L2 = {ba; cd}, L3 = {dc}.

I The “quasi C-pair” relation is a congruence.

I Quasi C-pair relation = transitive closure of C-pair relation.

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Quasi C-pairs

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Quasi C-pairs

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Quasi C-pairs

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Quasi C-pairs

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Membership resultsGeneric Separation Theorem (Place, Z. ’18)

Let C be a quotienting Boolean algebra of regular languages.Let L ⊆ A∗ be regular and ¸ : A∗ → M be its syntactic morphism. TFAE:

1. L ∈ UPol(C).2. L ∈ ADet(C).3. L ∈ Pol(C) ∩ co-Pol(C).

4. For all C-pairs (s; t) ∈ M2, we have s!+1 = s!ts!.

5. For all quasi C-pairs (s; t) ∈ M2, we have s!+1 = s!ts!.

2 =⇒ 1 =⇒ 4 ⇐⇒ 3 =⇒ 5 =⇒ 2

Corollary1. If C is a quotienting Boolean algebra, so is UPol(C).2. UPol(C) membership reduces to C membership.

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2 =⇒ 1 =⇒ 4 ⇐⇒ 3 =⇒ 5 =⇒ 2

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2 =⇒ 1 =⇒ 4 ⇐⇒ 3 =⇒ 5 =⇒ 2

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Separation result

Generic Separation Theorem (Place, Z. ’18)

If C is a finite quotienting Boolean algebra, separation for UPol(C) is decidable.

1. This result is stronger than the one for membership!2. Solved by considering a more general problem: covering.

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Separation result

Generic Separation Theorem (Place, Z. ’18)

If C is a finite quotienting Boolean algebra, separation for UPol(C) is decidable.

1. This result is stronger than the one for membership!2. Solved by considering a more general problem: covering.

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Summary of results

I C finite+. . . : separation OK for Pol(C), BPol(C), Pol(BPol(C)), UPol(C).I Via transfer results, imply all known algorithms.

Thanks!

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Summary of results

I C finite+. . . : separation OK for Pol(C), BPol(C), Pol(BPol(C)), UPol(C).I Via transfer results, imply all known algorithms.

Thanks!

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