Regular cost functions

Thomas Colcombet 23 Mars 2016

Conference dedicated to the scientific legacy of Marcel-Paul Schützenberger

languagesrationalregularThe strength of

languages

Regular languages are effectively equivalently described by - machines: non-deterministic,

alternating automata (…) - expressions: rational expressions,

generalized versions, … - algebra: recognizability by monoid,

deterministic automata, … - logically: definability in monadic

second-order logic (MSO).

rationalregularThe strength of

languages

Regular languages are closed under union, intersection, complement, projection, concatenation, etc…

Regular languages are effectively equivalently described by - machines: non-deterministic,

alternating automata (…) - expressions: rational expressions,

generalized versions, … - algebra: recognizability by monoid,

deterministic automata, … - logically: definability in monadic

second-order logic (MSO).

rationalregularThe strength of

languages

Regular languages are closed under union, intersection, complement, projection, concatenation, etc…

Regular languages have decidable emptiness, inclusion, universality, equivalence.

Regular languages are effectively equivalently described by - machines: non-deterministic,

alternating automata (…) - expressions: rational expressions,

generalized versions, … - algebra: recognizability by monoid,

deterministic automata, … - logically: definability in monadic

second-order logic (MSO).

rationalregularThe strength of

languages

Regular languages are closed under union, intersection, complement, projection, concatenation, etc…

Regular languages have decidable emptiness, inclusion, universality, equivalence.

Regular languages are effectively equivalently described by - machines: non-deterministic,

alternating automata (…) - expressions: rational expressions,

generalized versions, … - algebra: recognizability by monoid,

deterministic automata, … - logically: definability in monadic

second-order logic (MSO).

Can we replicate these strong properties to a more general setting ?

rationalregularThe strength of

languages

Regular languages are closed under union, intersection, complement, projection, concatenation, etc…

Regular languages have decidable emptiness, inclusion, universality, equivalence.

Regular languages are effectively equivalently described by - machines: non-deterministic,

alternating automata (…) - expressions: rational expressions,

generalized versions, … - algebra: recognizability by monoid,

deterministic automata, … - logically: definability in monadic

second-order logic (MSO).

Can we replicate these strong properties to a more general setting ?Models: Infinite words, countable words, trees, infinite trees, graphs….

rationalregularThe strength of

languages

Regular languages are closed under union, intersection, complement, projection, concatenation, etc…

Regular languages have decidable emptiness, inclusion, universality, equivalence.

Regular languages are effectively equivalently described by - machines: non-deterministic,

alternating automata (…) - expressions: rational expressions,

generalized versions, … - algebra: recognizability by monoid,

deterministic automata, … - logically: definability in monadic

second-order logic (MSO).

Can we replicate these strong properties to a more general setting ?Models: Infinite words, countable words, trees, infinite trees, graphs….Range: …

rationalregularThe strength of

Quantitative acceptors

Quantitative acceptors[Schützenberger 61] Automata need not be Boolean, these can be parameterized by a computation domain: a semiring.

Quantitative acceptors[Schützenberger 61] Automata need not be Boolean, these can be parameterized by a computation domain: a semiring.

In a non-deterministic automaton, a word is accepted ifThere exists a sequence of transitions such that

all transitions correspond, and the first state is initial and the last state final.

Quantitative acceptors[Schützenberger 61] Automata need not be Boolean, these can be parameterized by a computation domain: a semiring.

In a non-deterministic automaton, a word is accepted ifThere exists a sequence of transitions such that

all transitions correspond, and the first state is initial and the last state final.

⨁

Quantitative acceptors[Schützenberger 61] Automata need not be Boolean, these can be parameterized by a computation domain: a semiring.

In a non-deterministic automaton, a word is accepted ifThere exists a sequence of transitions such that

all transitions correspond, and the first state is initial and the last state final.

⨁ ⨂

Quantitative acceptors[Schützenberger 61] Automata need not be Boolean, these can be parameterized by a computation domain: a semiring.

In a non-deterministic automaton, a word is accepted ifThere exists a sequence of transitions such that

all transitions correspond, and the first state is initial and the last state final.

⨁ ⨂(S, 0S , 1S ,�,⌦)In which

is a semiring.

Quantitative acceptors[Schützenberger 61] Automata need not be Boolean, these can be parameterized by a computation domain: a semiring.

In a non-deterministic automaton, a word is accepted ifThere exists a sequence of transitions such that

all transitions correspond, and the first state is initial and the last state final.

⨁ ⨂

If one recovers non-deterministic automata.({0, 1}, 0, 1,_,^)

(S, 0S , 1S ,�,⌦)In which

is a semiring.

Quantitative acceptors[Schützenberger 61] Automata need not be Boolean, these can be parameterized by a computation domain: a semiring.

In a non-deterministic automaton, a word is accepted ifThere exists a sequence of transitions such that

all transitions correspond, and the first state is initial and the last state final.

⨁ ⨂

If one recovers non-deterministic automata.({0, 1}, 0, 1,_,^)Fields yield minimizable automata [Schützenberger61].

(S, 0S , 1S ,�,⌦)In which

is a semiring.

Quantitative acceptors[Schützenberger 61] Automata need not be Boolean, these can be parameterized by a computation domain: a semiring.

In a non-deterministic automaton, a word is accepted ifThere exists a sequence of transitions such that

all transitions correspond, and the first state is initial and the last state final.

⨁ ⨂

If one recovers non-deterministic automata.({0, 1}, 0, 1,_,^)Fields yield minimizable automata [Schützenberger61].Probabilistic automata are a particular case.

(S, 0S , 1S ,�,⌦)In which

is a semiring.

Quantitative acceptors[Schützenberger 61] Automata need not be Boolean, these can be parameterized by a computation domain: a semiring.

In a non-deterministic automaton, a word is accepted ifThere exists a sequence of transitions such that

all transitions correspond, and the first state is initial and the last state final.

⨁ ⨂

If one recovers non-deterministic automata.({0, 1}, 0, 1,_,^)Fields yield minimizable automata [Schützenberger61].Probabilistic automata are a particular case.

Min,+ automataMax,+ automataTropical automata: (R [ {�1}, 0,�1,max,+) (R [ {1}, 0,1,min,+)

(S, 0S , 1S ,�,⌦)In which

is a semiring.

Distance automata

Distance automata correspond to the semiring . (N [ {1}, 0,1,min,+)

Distance automata

p q r

a : 1

b : 0 b : 0

a, b : 0 a, b : 0

correspond to the semiring . (N [ {1}, 0,1,min,+)

Distance automata

p q r

a : 1

b : 0 b : 0

a, b : 0 a, b : 0 A distance automaton computes:

the infimum over all accepting runs of the sum of weights.

[[A]] : A⇤ ! N [ {1}u 7!

correspond to the semiring . (N [ {1}, 0,1,min,+)

Distance automata

p q r

a : 1

b : 0 b : 0

a, b : 0 a, b : 0

minblock = The minimum length of a block of consecutive a-letters surrounded by b’s.

A distance automaton computes:

the infimum over all accepting runs of the sum of weights.

[[A]] : A⇤ ! N [ {1}u 7!

correspond to the semiring . (N [ {1}, 0,1,min,+)

Distance automata

p q r

a : 1

b : 0 b : 0

a, b : 0 a, b : 0

minblock = The minimum length of a block of consecutive a-letters surrounded by b’s.

[Hashiguchi 81] The boundedness of distance automata is decidable. (for solving the star-height problem)

A distance automaton computes:

the infimum over all accepting runs of the sum of weights.

[[A]] : A⇤ ! N [ {1}u 7!

correspond to the semiring . (N [ {1}, 0,1,min,+)

Distance automata

p q r

a : 1

b : 0 b : 0

a, b : 0 a, b : 0

minblock = The minimum length of a block of consecutive a-letters surrounded by b’s.

[Hashiguchi 81] The boundedness of distance automata is decidable. (for solving the star-height problem)

A distance automaton computes:

the infimum over all accepting runs of the sum of weights.

[[A]] : A⇤ ! N [ {1}u 7!

[Leung88] [Simon78,94][Kirsten05] [C. & Bojanczyk 06]

[Bojanczyk]

correspond to the semiring . (N [ {1}, 0,1,min,+)

Distance automata

p q r

a : 1

b : 0 b : 0

a, b : 0 a, b : 0

minblock = The minimum length of a block of consecutive a-letters surrounded by b’s.

[Krob 94] The equivalence of distance automata is undecidable.

[Hashiguchi 81] The boundedness of distance automata is decidable. (for solving the star-height problem)

A distance automaton computes:

the infimum over all accepting runs of the sum of weights.

[[A]] : A⇤ ! N [ {1}u 7!

[Leung88] [Simon78,94][Kirsten05] [C. & Bojanczyk 06]

[Bojanczyk]

correspond to the semiring . (N [ {1}, 0,1,min,+)

Cost functions[Krob 94] The equivalence of distance automata is undecidable.

[Hashiguchi 81] The boundedness of distance automata is decidable.

Cost functions[Krob 94] The equivalence of distance automata is undecidable.

[Hashiguchi 81] The boundedness of distance automata is decidable.

How can we hope a theory similar to regular languages ?

Cost functions[Krob 94] The equivalence of distance automata is undecidable.

[Hashiguchi 81] The boundedness of distance automata is decidable.

An answer: let us forget the exact values of functions, but keep sufficient information for boundedness.

How can we hope a theory similar to regular languages ?

Cost functions[Krob 94] The equivalence of distance automata is undecidable.

[Hashiguchi 81] The boundedness of distance automata is decidable.

An answer: let us forget the exact values of functions, but keep sufficient information for boundedness.

f ≼ g (f is dominated by g) if for all sets X (f|X is bounded implies g|X is bounded)

A cost function is an equivalence class for the relation ≈:

How can we hope a theory similar to regular languages ?

Boundedness equivalencef 4 g (f is dominated by g) if for all sets X⊆A*

is bounded implies is bounded f≈g if f≼g and g≼f

cost function = ≈-classf |Xg|X

Boundedness equivalence

Lemma: if and only if

for some non-decreasing map α from naturals to naturals extended with α(∞)=∞.

f 6 ↵ � gf 4 g

f 4 g (f is dominated by g) if for all sets X⊆A* is bounded implies is bounded

f≈g if f≼g and g≼f cost function = ≈-classf |Xg|X

Boundedness equivalence

Lemma: if and only if

for some non-decreasing map α from naturals to naturals extended with α(∞)=∞.

f 6 ↵ � gf 4 g

N[{1

}

A*

f 4 g (f is dominated by g) if for all sets X⊆A* is bounded implies is bounded

f≈g if f≼g and g≼f cost function = ≈-classf |Xg|X

Boundedness equivalence

Lemma: if and only if

for some non-decreasing map α from naturals to naturals extended with α(∞)=∞.

f 6 ↵ � gf 4 g

g

N[{1

}

A*

f 4 g (f is dominated by g) if for all sets X⊆A* is bounded implies is bounded

f≈g if f≼g and g≼f cost function = ≈-classf |Xg|X

Boundedness equivalence

Lemma: if and only if

for some non-decreasing map α from naturals to naturals extended with α(∞)=∞.

f 6 ↵ � gf 4 g

g

f

N[{1

}

A*

f 4 g (f is dominated by g) if for all sets X⊆A* is bounded implies is bounded

f≈g if f≼g and g≼f cost function = ≈-classf |Xg|X

Boundedness equivalence

Lemma: if and only if

for some non-decreasing map α from naturals to naturals extended with α(∞)=∞.

f 6 ↵ � gf 4 g

g

f↵ � g

N[{1

}

A*

f 4 g (f is dominated by g) if for all sets X⊆A* is bounded implies is bounded

f≈g if f≼g and g≼f cost function = ≈-classf |Xg|X

Boundedness equivalence

Lemma: if and only if

for some non-decreasing map α from naturals to naturals extended with α(∞)=∞.

f 6 ↵ � gf 4 g

g

f↵ � g

N[{1

}

A*

f 4 g

f 4 g (f is dominated by g) if for all sets X⊆A* is bounded implies is bounded

f≈g if f≼g and g≼f cost function = ≈-classf |Xg|X

Boundedness equivalence

Lemma: if and only if

for some non-decreasing map α from naturals to naturals extended with α(∞)=∞.

f 6 ↵ � gf 4 g

g

f↵ � g

h

N[{1

}

A*

f 4 g

f 4 g (f is dominated by g) if for all sets X⊆A* is bounded implies is bounded

f≈g if f≼g and g≼f cost function = ≈-classf |Xg|X

Boundedness equivalence

Lemma: if and only if

for some non-decreasing map α from naturals to naturals extended with α(∞)=∞.

f 6 ↵ � gf 4 g

g

f↵ � g

h

N[{1

}

A*

X

f 4 g

f 4 g (f is dominated by g) if for all sets X⊆A* is bounded implies is bounded

f≈g if f≼g and g≼f cost function = ≈-classf |Xg|X

Boundedness equivalence

Lemma: if and only if

for some non-decreasing map α from naturals to naturals extended with α(∞)=∞.

f 6 ↵ � gf 4 g

g

f↵ � g

h

N[{1

}

A*

X

f 4 gh 64 g

f 4 g (f is dominated by g) if for all sets X⊆A* is bounded implies is bounded

f≈g if f≼g and g≼f cost function = ≈-classf |Xg|X

Boundedness equivalence

Lemma: if and only if

for some non-decreasing map α from naturals to naturals extended with α(∞)=∞.

f 6 ↵ � gf 4 g

g

f↵ � g

h

| |b 64 | |aN[{1

}

A*

X

f 4 gh 64 g

f 4 g (f is dominated by g) if for all sets X⊆A* is bounded implies is bounded

f≈g if f≼g and g≼f cost function = ≈-classf |Xg|X

Boundedness equivalence

Lemma: if and only if

for some non-decreasing map α from naturals to naturals extended with α(∞)=∞.

f 6 ↵ � gf 4 g

g

f↵ � g

h

| |b 64 | |aN[{1

}

A*

X

f 4 gh 64 g

| |aa+ 4 | |a

f 4 g (f is dominated by g) if for all sets X⊆A* is bounded implies is bounded

f≈g if f≼g and g≼f cost function = ≈-classf |Xg|X

Boundedness equivalence

Let F (f) = {X ✓ A⇤ : f |X is bounded}

Lemma: if and only if

for some non-decreasing map α from naturals to naturals extended with α(∞)=∞.

f 6 ↵ � gf 4 g

g

f↵ � g

h

| |b 64 | |aN[{1

}

A*

X

f 4 gh 64 g

| |aa+ 4 | |a

f 4 g (f is dominated by g) if for all sets X⊆A* is bounded implies is bounded

f≈g if f≼g and g≼f cost function = ≈-classf |Xg|X

Boundedness equivalence

Let F (f) = {X ✓ A⇤ : f |X is bounded}

Lemma: if and only if

for some non-decreasing map α from naturals to naturals extended with α(∞)=∞.

f 6 ↵ � gf 4 g

g

f↵ � g

h

| |b 64 | |aN[{1

}

A*

X

f 4 gh 64 g

| |aa+ 4 | |a

f 4 g (f is dominated by g) if for all sets X⊆A* is bounded implies is bounded

f≈g if f≼g and g≼f cost function = ≈-classf |Xg|X

- is a filter of countable basis.F (f)

Boundedness equivalence

Let F (f) = {X ✓ A⇤ : f |X is bounded}

Lemma: if and only if

for some non-decreasing map α from naturals to naturals extended with α(∞)=∞.

f 6 ↵ � gf 4 g

g

f↵ � g

h

| |b 64 | |aN[{1

}

A*

X

f 4 gh 64 g

| |aa+ 4 | |a

f 4 g (f is dominated by g) if for all sets X⊆A* is bounded implies is bounded

f≈g if f≼g and g≼f cost function = ≈-classf |Xg|X

- is a filter of countable basis.F (f)- is a bijection between cost functions and filters of countable basis.F

Boundedness equivalence

Let F (f) = {X ✓ A⇤ : f |X is bounded}

Cost functions = filters of countable basis.

Lemma: if and only if

for some non-decreasing map α from naturals to naturals extended with α(∞)=∞.

f 6 ↵ � gf 4 g

g

f↵ � g

h

| |b 64 | |aN[{1

}

A*

X

f 4 gh 64 g

| |aa+ 4 | |a

f 4 g (f is dominated by g) if for all sets X⊆A* is bounded implies is bounded

f≈g if f≼g and g≼f cost function = ≈-classf |Xg|X

- is a filter of countable basis.F (f)- is a bijection between cost functions and filters of countable basis.F

Regular cost functions over finite wordsCost functions are effectively equivalently described by - machines: B-automata, S-automata - expressions: B-rational/S-rational

expressions, - algebra: recognizability by

stabilisation monoids, - logically: definability in cost

monadic second-order logic
(cost MSO).

Regular cost functions over finite wordslike distance automata with several counters and resetsCost functions are effectively equivalently described by

- machines: B-automata, S-automata - expressions: B-rational/S-rational

expressions, - algebra: recognizability by

stabilisation monoids, - logically: definability in cost

monadic second-order logic
(cost MSO).

Regular cost functions over finite wordslike distance automata with several counters and resetsCost functions are effectively equivalently described by

- machines: B-automata, S-automata - expressions: B-rational/S-rational

expressions, - algebra: recognizability by

stabilisation monoids, - logically: definability in cost

monadic second-order logic
(cost MSO).

sup instead of inf

Regular cost functions over finite wordslike distance automata with several counters and resetsCost functions are effectively equivalently described by

- machines: B-automata, S-automata - expressions: B-rational/S-rational

expressions, - algebra: recognizability by

stabilisation monoids, - logically: definability in cost

monadic second-order logic
(cost MSO).

sup instead of inf

L>nnew exponent

Regular cost functions over finite wordslike distance automata with several counters and resetsCost functions are effectively equivalently described by

- machines: B-automata, S-automata - expressions: B-rational/S-rational

expressions, - algebra: recognizability by

stabilisation monoids, - logically: definability in cost

monadic second-order logic
(cost MSO).

sup instead of inf

L>nnew exponent

new exponent L6n

Regular cost functions over finite wordslike distance automata with several counters and resetsCost functions are effectively equivalently described by

- machines: B-automata, S-automata - expressions: B-rational/S-rational

expressions, - algebra: recognizability by

stabilisation monoids, - logically: definability in cost

monadic second-order logic
(cost MSO).

sup instead of inf

L>nnew exponent

new exponent L6n

like ordered monoids with a stabilisation ‘#’ meaning ‘iterating a lot of times’.

Regular cost functions over finite wordslike distance automata with several counters and resetsCost functions are effectively equivalently described by

- machines: B-automata, S-automata - expressions: B-rational/S-rational

expressions, - algebra: recognizability by

stabilisation monoids, - logically: definability in cost

monadic second-order logic
(cost MSO).

sup instead of inf

L>nnew exponent

new exponent L6n

like ordered monoids with a stabilisation ‘#’ meaning ‘iterating a lot of times’.new predicate that appears positively

|X| 6 n

Cost monadic second order logic

Cost monadic second order logicMonadic second order logic (MSO) expresses properties using - the predicates of the structure, and membership, - Boolean connectives, - existential and universal quantifiers over elements, - existential and universal quantifiers over sets of elements.

Cost monadic second order logicMonadic second order logic (MSO) expresses properties using - the predicates of the structure, and membership, - Boolean connectives, - existential and universal quantifiers over elements, - existential and universal quantifiers over sets of elements.

Example: over directed graphs,

Cost monadic second order logicMonadic second order logic (MSO) expresses properties using - the predicates of the structure, and membership, - Boolean connectives, - existential and universal quantifiers over elements, - existential and universal quantifiers over sets of elements.

Example: over directed graphs,= all sets that contain x and are closed under the

edge relation also contain yReach(x, y)

Cost monadic second order logicMonadic second order logic (MSO) expresses properties using - the predicates of the structure, and membership, - Boolean connectives, - existential and universal quantifiers over elements, - existential and universal quantifiers over sets of elements.

Example: over directed graphs,= all sets that contain x and are closed under the

edge relation also contain yReach(x, y)

8Z ((x 2 Z) ^ (8z8z0(z 2 Z ^ Edge(z, z0)) ! z0 2 Z)) ! y 2 Z

Cost monadic second order logicMonadic second order logic (MSO) expresses properties using - the predicates of the structure, and membership, - Boolean connectives, - existential and universal quantifiers over elements, - existential and universal quantifiers over sets of elements.

Example: over directed graphs,= all sets that contain x and are closed under the

edge relation also contain yReach(x, y)

Cost MSO = MSO + appearing positively.|X| 6 n

8Z ((x 2 Z) ^ (8z8z0(z 2 Z ^ Edge(z, z0)) ! z0 2 Z)) ! y 2 Z

Cost monadic second order logicMonadic second order logic (MSO) expresses properties using - the predicates of the structure, and membership, - Boolean connectives, - existential and universal quantifiers over elements, - existential and universal quantifiers over sets of elements.

Example: over directed graphs,= all sets that contain x and are closed under the

edge relation also contain yReach(x, y)

Cost MSO = MSO + appearing positively.|X| 6 n Unique new variable ranging over naturals

8Z ((x 2 Z) ^ (8z8z0(z 2 Z ^ Edge(z, z0)) ! z0 2 Z)) ! y 2 Z

Cost monadic second order logicMonadic second order logic (MSO) expresses properties using - the predicates of the structure, and membership, - Boolean connectives, - existential and universal quantifiers over elements, - existential and universal quantifiers over sets of elements.

Example: over directed graphs,= all sets that contain x and are closed under the

edge relation also contain yReach(x, y)

Cost MSO = MSO + appearing positively.|X| 6 n[[']](A) = inf{n | A |= '(n)}A formula computes:

Unique new variable ranging over naturals

8Z ((x 2 Z) ^ (8z8z0(z 2 Z ^ Edge(z, z0)) ! z0 2 Z)) ! y 2 Z

Cost monadic second order logicMonadic second order logic (MSO) expresses properties using - the predicates of the structure, and membership, - Boolean connectives, - existential and universal quantifiers over elements, - existential and universal quantifiers over sets of elements.

Example: over directed graphs,= all sets that contain x and are closed under the

edge relation also contain yReach(x, y)

Cost MSO = MSO + appearing positively.|X| 6 n[[']](A) = inf{n | A |= '(n)}A formula computes:

Example: over directed graphs,Diameter() = 8x8y9Z Reach(x, y, Z) ^ |Z| 6 n

Unique new variable ranging over naturals

8Z ((x 2 Z) ^ (8z8z0(z 2 Z ^ Edge(z, z0)) ! z0 2 Z)) ! y 2 Z

Cost monadic second order logicMonadic second order logic (MSO) expresses properties using - the predicates of the structure, and membership, - Boolean connectives, - existential and universal quantifiers over elements, - existential and universal quantifiers over sets of elements.

Example: over directed graphs,= all sets that contain x and are closed under the

edge relation also contain yReach(x, y)

Cost MSO = MSO + appearing positively.|X| 6 n[[']](A) = inf{n | A |= '(n)}A formula computes:

Example: over directed graphs,Diameter() = 8x8y9Z Reach(x, y, Z) ^ |Z| 6 nOver words, , , , distance automata… | |a maxblock minblock

Unique new variable ranging over naturals

8Z ((x 2 Z) ^ (8z8z0(z 2 Z ^ Edge(z, z0)) ! z0 2 Z)) ! y 2 Z

Regular cost functions over finite wordslike distance automata with several counters and resetsCost functions are effectively equivalently described by

- machines: B-automata, S-automata - expressions: B-rational/S-rational

expressions, - algebra: recognizability by

stabilisation monoids, - logically: definability in cost

monadic second-order logic
(cost MSO).

sup instead of inf

L>nnew exponent

new exponent L6n

like ordered monoids with a stabilisation ‘#’ meaning ‘iterating a lot of times’.new predicate that appears positively

|X| 6 n

Regular cost functions are closed under min, max, inf-projection, sup-projection, etc…

Regular cost functions over finite wordslike distance automata with several counters and resetsCost functions are effectively equivalently described by

- machines: B-automata, S-automata - expressions: B-rational/S-rational

expressions, - algebra: recognizability by

stabilisation monoids, - logically: definability in cost

monadic second-order logic
(cost MSO).

sup instead of inf

L>nnew exponent

new exponent L6n

like ordered monoids with a stabilisation ‘#’ meaning ‘iterating a lot of times’.new predicate that appears positively

|X| 6 n

Regular cost functions are closed under min, max, inf-projection, sup-projection, etc…

Regular cost functions over finite wordslike distance automata with several counters and resetsCost functions are effectively equivalently described by

- machines: B-automata, S-automata - expressions: B-rational/S-rational

expressions, - algebra: recognizability by

stabilisation monoids, - logically: definability in cost

monadic second-order logic
(cost MSO).

sup instead of inf

L>nnew exponent

new exponent L6n

like ordered monoids with a stabilisation ‘#’ meaning ‘iterating a lot of times’.new predicate that appears positively

|X| 6 n

Boundedness, divergence, domination, equivalence are decidable.

Regular cost functions are closed under min, max, inf-projection, sup-projection, etc…

Regular cost functions over finite words

These results (in essence) extend regular languages.

like distance automata with several counters and resetsCost functions are effectively equivalently described by

- machines: B-automata, S-automata - expressions: B-rational/S-rational

expressions, - algebra: recognizability by

stabilisation monoids, - logically: definability in cost

monadic second-order logic
(cost MSO).

sup instead of inf

L>nnew exponent

new exponent L6n

like ordered monoids with a stabilisation ‘#’ meaning ‘iterating a lot of times’.new predicate that appears positively

|X| 6 n

Boundedness, divergence, domination, equivalence are decidable.

Languages as cost functionsf 4 g (f is dominated by g) if for all sets X⊆A*

is bounded implies is bounded f≈g if f≼g and g≼f

cost function = ≈-classf |Xg|X

Languages as cost functions

Given a language L, its characteristic map is: �L : A

⇤ ! N [ {1}

u 7!(0 if u 2 L1 otherwise

f 4 g (f is dominated by g) if for all sets X⊆A* is bounded implies is bounded

f≈g if f≼g and g≼f cost function = ≈-classf |Xg|X

Languages as cost functions

Given a language L, its characteristic map is: �L : A

⇤ ! N [ {1}

u 7!(0 if u 2 L1 otherwise

Remark: iff .K ✓ L �L 4 �K

f 4 g (f is dominated by g) if for all sets X⊆A* is bounded implies is bounded

f≈g if f≼g and g≼f cost function = ≈-classf |Xg|X

Languages as cost functions

Given a language L, its characteristic map is: �L : A

⇤ ! N [ {1}

u 7!(0 if u 2 L1 otherwise

Remark: iff .K ✓ L �L 4 �KConsider an MSO formula defining L, then

f 4 g (f is dominated by g) if for all sets X⊆A* is bounded implies is bounded

f≈g if f≼g and g≼f cost function = ≈-classf |Xg|X

Languages as cost functions

Given a language L, its characteristic map is: �L : A

⇤ ! N [ {1}

u 7!(0 if u 2 L1 otherwise

Remark: iff .K ✓ L �L 4 �KConsider an MSO formula defining L, then

[[']](u) = inf{n | u |= '(n)}

f 4 g (f is dominated by g) if for all sets X⊆A* is bounded implies is bounded

f≈g if f≼g and g≼f cost function = ≈-classf |Xg|X

Languages as cost functions

Given a language L, its characteristic map is: �L : A

⇤ ! N [ {1}

u 7!(0 if u 2 L1 otherwise

Remark: iff .K ✓ L �L 4 �KConsider an MSO formula defining L, then

[[']](u) = inf{n | u |= '(n)}=

(0 if u |= '1 otherwise

f 4 g (f is dominated by g) if for all sets X⊆A* is bounded implies is bounded

f≈g if f≼g and g≼f cost function = ≈-classf |Xg|X

Languages as cost functions

Given a language L, its characteristic map is: �L : A

⇤ ! N [ {1}

u 7!(0 if u 2 L1 otherwise

Remark: iff .K ✓ L �L 4 �KConsider an MSO formula defining L, then

[[']](u) = inf{n | u |= '(n)}=

(0 if u |= '1 otherwise

= �L

f 4 g (f is dominated by g) if for all sets X⊆A* is bounded implies is bounded

f≈g if f≼g and g≼f cost function = ≈-classf |Xg|X

Languages as cost functions

Given a language L, its characteristic map is: �L : A

⇤ ! N [ {1}

u 7!(0 if u 2 L1 otherwise

Remark: iff .K ✓ L �L 4 �KConsider an MSO formula defining L, then

[[']](u) = inf{n | u |= '(n)}=

(0 if u |= '1 otherwise

= �L

Similar relation with all classical objects exist.

f 4 g (f is dominated by g) if for all sets X⊆A* is bounded implies is bounded

f≈g if f≼g and g≼f cost function = ≈-classf |Xg|X

Some known results about regular cost functions

Some known results about regular cost functions

Boundedness/domination of cost MSO is decidable

Some known results about regular cost functions

Boundedness/domination of cost MSO is decidable - over finite words [C.09,13]

Some known results about regular cost functions

Boundedness/domination of cost MSO is decidable - over finite words [C.09,13]- over infinite words (of countable length ? !) 
 [C.09], [vanden Boom&Kuperberg12]

Some known results about regular cost functions

Boundedness/domination of cost MSO is decidable - over finite words [C.09,13]- over infinite words (of countable length ? !) 
 [C.09], [vanden Boom&Kuperberg12]- over finite trees [C.&Löding10]

Some known results about regular cost functions

Boundedness/domination of cost MSO is decidable - over finite words [C.09,13]- over infinite words (of countable length ? !) 
 [C.09], [vanden Boom&Kuperberg12]- over finite trees [C.&Löding10]- over graphs of tree-width at most k

Some known results about regular cost functions

Boundedness/domination of cost MSO is decidable - over finite words [C.09,13]- over infinite words (of countable length ? !) 
 [C.09], [vanden Boom&Kuperberg12]- over finite trees [C.&Löding10]- over graphs of tree-width at most k- partially over infinite trees (tamed trees, weak costMSO, …)
 [vanden Boom11], [vanden Boom&Kuperberg12] …

Some known results about regular cost functions

Boundedness/domination of cost MSO is decidable - over finite words [C.09,13]- over infinite words (of countable length ? !) 
 [C.09], [vanden Boom&Kuperberg12]- over finite trees [C.&Löding10]- over graphs of tree-width at most k- partially over infinite trees (tamed trees, weak costMSO, …)
 [vanden Boom11], [vanden Boom&Kuperberg12] …

Regular cost functions = Toolbox of results for deciding boundedness questions.

Some application of regular cost functions

Finite power propertyu \in L^{\leqslant n}

Finite power propertyu \in L^{\leqslant n}

Finite power property: for some ?nL⇤ = L6n

Finite power propertyu \in L^{\leqslant n}

AIF

Finite power property: for some ?nL⇤ = L6n

Finite power propertyu \in L^{\leqslant n}

AIF

" """

Finite power property: for some ?nL⇤ = L6n

Finite power propertyu \in L^{\leqslant n}

AIF

" """

:1:1

:0:0

:0

Finite power property: for some ?nL⇤ = L6n

Finite power propertyu \in L^{\leqslant n}

AIF

" """

:1:1

:0:0

:0

Finite power property: for some ?nL⇤ = L6n

Accepts with weight if and only if

u

u 2 L6n

6 n

Finite power propertyu \in L^{\leqslant n}

AIF

" """

:1:1

:0:0

:0

Finite power property: for some ?nL⇤ = L6n

[[A]](u) = inf{n | u 2 L6n}Hence

Accepts with weight if and only if

u

u 2 L6n

6 n

Finite power propertyu \in L^{\leqslant n}

AIF

" """

:1:1

:0:0

:0

Finite power property: for some ?nL⇤ = L6n

[[A]]Lemma: is bounded over if and only if .L⇤ L⇤ = L6n

[[A]](u) = inf{n | u 2 L6n}Hence

Accepts with weight if and only if

u

u 2 L6n

6 n

Finite power propertyu \in L^{\leqslant n}

AIF

" """

:1:1

:0:0

:0

Finite power property: for some ?nL⇤ = L6n

[[A]]Lemma: is bounded over if and only if .L⇤ L⇤ = L6n

[[A]](u) = inf{n | u 2 L6n}Hence

Accepts with weight if and only if

u

u 2 L6n

6 n

[Simon78] The finite power property is decidable.

Bounding fixpointsu \in L^{\leqslant n}

Bounding fixpointsu \in L^{\leqslant n}

Boundedness: Let be a monadic second-order formula positive in . It has a fixpoint which is the least set s.t. .

'(x, Y )Y {x | A |= '(x, Y )} ✓ YY

Bounding fixpointsu \in L^{\leqslant n}

It is the least fixpoint of . FA : Z 7! {x | A |= '(x, Z)}

Boundedness: Let be a monadic second-order formula positive in . It has a fixpoint which is the least set s.t. .

'(x, Y )Y {x | A |= '(x, Y )} ✓ YY

Bounding fixpointsu \in L^{\leqslant n}

It is the least fixpoint of . FA : Z 7! {x | A |= '(x, Z)}

Does there exist such that for all ?n 2 N AFnA(;) = fixpoint(FA)Problem:

Boundedness: Let be a monadic second-order formula positive in . It has a fixpoint which is the least set s.t. .

'(x, Y )Y {x | A |= '(x, Y )} ✓ YY

Bounding fixpointsu \in L^{\leqslant n}

It is the least fixpoint of . FA : Z 7! {x | A |= '(x, Z)}

tThe map for a node of an infinite tree is computed by an alternating two-way distance parity automaton running over starting from .

x

t x

(t, x) 7! inf{n | x 2 Fnt (;)}

Does there exist such that for all ?n 2 N AFnA(;) = fixpoint(FA)Problem:

Boundedness: Let be a monadic second-order formula positive in . It has a fixpoint which is the least set s.t. .

'(x, Y )Y {x | A |= '(x, Y )} ✓ YY

Bounding fixpointsu \in L^{\leqslant n}

It is the least fixpoint of . FA : Z 7! {x | A |= '(x, Z)}

tThe map for a node of an infinite tree is computed by an alternating two-way distance parity automaton running over starting from .

x

t x

(t, x) 7! inf{n | x 2 Fnt (;)}

Boundedness of such automata is decidable.

Does there exist such that for all ?n 2 N AFnA(;) = fixpoint(FA)Problem:

Boundedness: Let be a monadic second-order formula positive in . It has a fixpoint which is the least set s.t. .

'(x, Y )Y {x | A |= '(x, Y )} ✓ YY

Bounding fixpointsu \in L^{\leqslant n}

It is the least fixpoint of . FA : Z 7! {x | A |= '(x, Z)}

tThe map for a node of an infinite tree is computed by an alternating two-way distance parity automaton running over starting from .

x

t x

(t, x) 7! inf{n | x 2 Fnt (;)}

[Blumensath&Otto&Weyer12] The boundedness of monadic second-order logic fixpoints is decidable over infinite trees.

Boundedness of such automata is decidable.

Does there exist such that for all ?n 2 N AFnA(;) = fixpoint(FA)Problem:

Boundedness: Let be a monadic second-order formula positive in . It has a fixpoint which is the least set s.t. .

'(x, Y )Y {x | A |= '(x, Y )} ✓ YY

Membership to (non-deterministic) classes

Membership to (non-deterministic) classes

A very inspiring family of questions over regular languages is the ability to decide the membership of regular languages to subclasses.

Membership to (non-deterministic) classes

A very inspiring family of questions over regular languages is the ability to decide the membership of regular languages to subclasses.

[Schützenberger65] A regular language is definable by a star-free expressions if and only if its syntactic monoid is aperiodic. This is decidable.

Membership to (non-deterministic) classes

A very inspiring family of questions over regular languages is the ability to decide the membership of regular languages to subclasses.

[Schützenberger65] A regular language is definable by a star-free expressions if and only if its syntactic monoid is aperiodic. This is decidable.

Many other classes known, over finite words,⌃1 ⌃2 ⌃3 ⌃4 B(⌃1) B(⌃2)FO2 etc…

The star-height problem

The star-height problemDef: A regular language is of star height if it can be described by a rational expression using at most nesting of Kleene stars, but not less.

kk

The star-height problemDef: A regular language is of star height if it can be described by a rational expression using at most nesting of Kleene stars, but not less.

kk

[Eggan 63] The star-height forms a strict hierarchy over regular languages.

The star-height problemDef: A regular language is of star height if it can be described by a rational expression using at most nesting of Kleene stars, but not less.

kk

[Eggan 63] The star-height forms a strict hierarchy over regular languages.

(*) This is the restricted star height, in which intersection and negations are not allowed. Nobody knows if the hierarchy is strict in the unrestricted case.

(*)

The star-height problemDef: A regular language is of star height if it can be described by a rational expression using at most nesting of Kleene stars, but not less.

kk

[Eggan 63] The star-height forms a strict hierarchy over regular languages.

(*) This is the restricted star height, in which intersection and negations are not allowed. Nobody knows if the hierarchy is strict in the unrestricted case.

But, is the star-height of a language computable ?

(*)

The star-height problemDef: A regular language is of star height if it can be described by a rational expression using at most nesting of Kleene stars, but not less.

kk

[Eggan 63] The star-height forms a strict hierarchy over regular languages.

(*) This is the restricted star height, in which intersection and negations are not allowed. Nobody knows if the hierarchy is strict in the unrestricted case.

But, is the star-height of a language computable ?

[Hashiguchi 81–88] Yes!

(*)

The star-height problemDef: A regular language is of star height if it can be described by a rational expression using at most nesting of Kleene stars, but not less.

kk

[Eggan 63] The star-height forms a strict hierarchy over regular languages.

(*) This is the restricted star height, in which intersection and negations are not allowed. Nobody knows if the hierarchy is strict in the unrestricted case.

But, is the star-height of a language computable ?

[Hashiguchi 81–88] Yes!

Decidability of the boundedness

of distance automata

(*)

The star-height problemDef: A regular language is of star height if it can be described by a rational expression using at most nesting of Kleene stars, but not less.

kk

[Eggan 63] The star-height forms a strict hierarchy over regular languages.

(*) This is the restricted star height, in which intersection and negations are not allowed. Nobody knows if the hierarchy is strict in the unrestricted case.

But, is the star-height of a language computable ?

[Hashiguchi 81–88] Yes! (but very complicated)

Decidability of the boundedness

of distance automata

(*)

The star-height problemDef: A regular language is of star height if it can be described by a rational expression using at most nesting of Kleene stars, but not less.

kk

[Eggan 63] The star-height forms a strict hierarchy over regular languages.

(*) This is the restricted star height, in which intersection and negations are not allowed. Nobody knows if the hierarchy is strict in the unrestricted case.

But, is the star-height of a language computable ?

[Hashiguchi 81–88] Yes!

[Kirsten 05] Yes, with a self contained and simpler proof.

(but very complicated)

Decidability of the boundedness

of distance automata

(*)

The star-height problemDef: A regular language is of star height if it can be described by a rational expression using at most nesting of Kleene stars, but not less.

kk

[Eggan 63] The star-height forms a strict hierarchy over regular languages.

(*) This is the restricted star height, in which intersection and negations are not allowed. Nobody knows if the hierarchy is strict in the unrestricted case.

But, is the star-height of a language computable ?

[Hashiguchi 81–88] Yes!

[Kirsten 05] Yes, with a self contained and simpler proof.

(but very complicated)

Decidability of the boundedness

of distance automata

(*)

boundedness of hB-automata

The star-height problemDef: A regular language is of star height if it can be described by a rational expression using at most nesting of Kleene stars, but not less.

kk

[Eggan 63] The star-height forms a strict hierarchy over regular languages.

(*) This is the restricted star height, in which intersection and negations are not allowed. Nobody knows if the hierarchy is strict in the unrestricted case.

But, is the star-height of a language computable ?

[Hashiguchi 81–88] Yes!

[Kirsten 05] Yes, with a self contained and simpler proof.

[C.&Löding 08] The star-height is decidable over trees.

(but very complicated)

Decidability of the boundedness

of distance automata

(*)

boundedness of hB-automata

Star-height and boundedness

Star-height and boundednessFix a regular language L, and an integer k. Define:

fL,k : A⇤ ! N [ {1}u 7! inf{size(E) | there exists an expression E,

sh(E) 6 k, u 2 LE ✓ L}

Star-height and boundedness

Lemma: is bounded over L if and only if L has star-height at most k.fL,k

Fix a regular language L, and an integer k. Define:

fL,k : A⇤ ! N [ {1}u 7! inf{size(E) | there exists an expression E,

sh(E) 6 k, u 2 LE ✓ L}

Star-height and boundedness

Lemma: is bounded over L if and only if L has star-height at most k.fL,k

Fix a regular language L, and an integer k. Define:

fL,k : A⇤ ! N [ {1}u 7! inf{size(E) | there exists an expression E,

sh(E) 6 k, u 2 LE ✓ L}

Proof:Assume L is of star-height k.Let E be the corresponding expression.It is a witness that is bounded over L(by the size of E).

fL,k

Star-height and boundedness

Lemma: is bounded over L if and only if L has star-height at most k.fL,k

Fix a regular language L, and an integer k. Define:

fL,k : A⇤ ! N [ {1}u 7! inf{size(E) | there exists an expression E,

sh(E) 6 k, u 2 LE ✓ L}

Proof:Assume L is of star-height k.Let E be the corresponding expression.It is a witness that is bounded over L(by the size of E).

fL,k

Assume bounded over L by n. Define:

fL,k

EL,n =X

size(E) 6 nsh(E) 6 kLE ✓ L

E

Star-height and boundedness

Lemma: is bounded over L if and only if L has star-height at most k.fL,k

Fix a regular language L, and an integer k. Define:

fL,k : A⇤ ! N [ {1}u 7! inf{size(E) | there exists an expression E,

sh(E) 6 k, u 2 LE ✓ L}

Proof:Assume L is of star-height k.Let E be the corresponding expression.It is a witness that is bounded over L(by the size of E).

fL,k

Assume bounded over L by n. Define:

fL,k

EL,n =X

size(E) 6 nsh(E) 6 kLE ✓ L

E

L ✓ L(LE,n)We have .

Star-height and boundedness

Star-height and boundednessFix a regular language L, and an integer k. Define:

fL,k : A⇤ ! N [ {1}u 7! inf{size(E) | there exists an expression E,

sh(E) 6 k, u 2 LE ✓ L}

Star-height and boundedness

Lemma: is bounded over L if and only if L has star-height at most k.fL,k

Fix a regular language L, and an integer k. Define:

fL,k : A⇤ ! N [ {1}u 7! inf{size(E) | there exists an expression E,

sh(E) 6 k, u 2 LE ✓ L}

Star-height and boundedness

Lemma: is bounded over L if and only if L has star-height at most k.fL,k

Fix a regular language L, and an integer k. Define:

[Kirsten05] There exists a hB-automaton computing .fL,k

fL,k : A⇤ ! N [ {1}u 7! inf{size(E) | there exists an expression E,

sh(E) 6 k, u 2 LE ✓ L}

Star-height and boundedness

Lemma: is bounded over L if and only if L has star-height at most k.fL,k

Fix a regular language L, and an integer k. Define:

[Kirsten05] There exists a hB-automaton computing .fL,k

[Kirsten05] The star-height problem is decidable.

fL,k : A⇤ ! N [ {1}u 7! inf{size(E) | there exists an expression E,

sh(E) 6 k, u 2 LE ✓ L}

Star-height and boundedness

Lemma: is bounded over L if and only if L has star-height at most k.fL,k

Fix a regular language L, and an integer k. Define:

[Kirsten05] There exists a hB-automaton computing .fL,k

[Kirsten05] The star-height problem is decidable.

This approach is generic. It works for trees, and for the Mostowski index of languages of infinite trees (open).

fL,k : A⇤ ! N [ {1}u 7! inf{size(E) | there exists an expression E,

sh(E) 6 k, u 2 LE ✓ L}

Star-height and boundedness

Lemma: is bounded over L if and only if L has star-height at most k.fL,k

Fix a regular language L, and an integer k. Define:

[Kirsten05] There exists a hB-automaton computing .fL,k

[Kirsten05] The star-height problem is decidable.

This approach is generic. It works for trees, and for the Mostowski index of languages of infinite trees (open).

It works also for separation.

fL,k : A⇤ ! N [ {1}u 7! inf{size(E) | there exists an expression E,

sh(E) 6 k, u 2 LE ✓ L}

Church synthesis with bounded error

Church synthesis with bounded error

[Church problem] Two players (controller and environment) alternatively choose a letter, producing in the end an infinite word. Controller wins if he can guarantee this word to be in a given ω-regular language. - decide the winner, - construct a finite automaton implementing the strategy.

Church synthesis with bounded error

[Church problem] Two players (controller and environment) alternatively choose a letter, producing in the end an infinite word. Controller wins if he can guarantee this word to be in a given ω-regular language. - decide the winner, - construct a finite automaton implementing the strategy.

[Büchi&Landweber69] This is doable.

Church synthesis with bounded error

[Rabinowich&Velner11] is it possible to decide if there exists n such that controller can guarantee being in L up to a change of n of his letters.

[Church problem] Two players (controller and environment) alternatively choose a letter, producing in the end an infinite word. Controller wins if he can guarantee this word to be in a given ω-regular language. - decide the winner, - construct a finite automaton implementing the strategy.

[Büchi&Landweber69] This is doable.

Church synthesis with bounded error

[Rabinowich&Velner11] is it possible to decide if there exists n such that controller can guarantee being in L up to a change of n of his letters.

[Church problem] Two players (controller and environment) alternatively choose a letter, producing in the end an infinite word. Controller wins if he can guarantee this word to be in a given ω-regular language. - decide the winner, - construct a finite automaton implementing the strategy.

[Büchi&Landweber69] This is doable.

[Regular cost functions] immediately yes.

Regular cost functions as languages

Motivation

MotivationWorking with regular cost functions usually involves the terms large and small used (with care) as Boolean predicates:

'f ≼ g if, whenever f(u) is large, g(u) is large’.

MotivationWorking with regular cost functions usually involves the terms large and small used (with care) as Boolean predicates:

'f ≼ g if, whenever f(u) is large, g(u) is large’.

Example: Given a tree, thenmax(maxdegree,height) ≈ size

MotivationWorking with regular cost functions usually involves the terms large and small used (with care) as Boolean predicates:

'f ≼ g if, whenever f(u) is large, g(u) is large’.

Example: Given a tree, thenmax(maxdegree,height) ≈ size

Proof:If maxdegree is large, then there is a large number of nodes. If height is large, then there is a large number of nodes.

MotivationWorking with regular cost functions usually involves the terms large and small used (with care) as Boolean predicates:

'f ≼ g if, whenever f(u) is large, g(u) is large’.

Example: Given a tree, thenmax(maxdegree,height) ≈ size

Proof:If maxdegree is large, then there is a large number of nodes. If height is large, then there is a large number of nodes.Conversely, assume both maxdegree and height would be small, then there would be a small number of nodes.

MotivationWorking with regular cost functions usually involves the terms large and small used (with care) as Boolean predicates:

'f ≼ g if, whenever f(u) is large, g(u) is large’.

Example: Given a tree, thenmax(maxdegree,height) ≈ size

Proof:If maxdegree is large, then there is a large number of nodes. If height is large, then there is a large number of nodes.Conversely, assume both maxdegree and height would be small, then there would be a small number of nodes.

[Toruńczyk 11] One can identify regular cost functions with a language of profinite words, in a natural way.

Non-standard analysis (internal set theory)

Non-standard analysis (internal set theory)[Robinson 66] …

Non-standard analysis (internal set theory)

[Nelson 77] Internal set theory (IST) a conservative extension of ZFC.

[Robinson 66] …

Non-standard analysis (internal set theory)

[Nelson 77] Internal set theory (IST) a conservative extension of ZFC.

[Robinson 66] … This is an extension of set theory (ZFC) in which one can prove exactly the same statements of ZFC.

Non-standard analysis (internal set theory)

[Nelson 77] Internal set theory (IST) a conservative extension of ZFC.

All the language of ZFC is available.

[Robinson 66] … This is an extension of set theory (ZFC) in which one can prove exactly the same statements of ZFC.

Non-standard analysis (internal set theory)

[Nelson 77] Internal set theory (IST) a conservative extension of ZFC.

All the language of ZFC is available.

[Robinson 66] … This is an extension of set theory (ZFC) in which one can prove exactly the same statements of ZFC.

There is a new unary symbol to be read ‘ is standard’.St(x) x

Non-standard analysis (internal set theory)

[Nelson 77] Internal set theory (IST) a conservative extension of ZFC.

All the language of ZFC is available.

All the proof arguments usable in ZFC are available.

[Robinson 66] … This is an extension of set theory (ZFC) in which one can prove exactly the same statements of ZFC.

There is a new unary symbol to be read ‘ is standard’.St(x) x

Non-standard analysis (internal set theory)

[Nelson 77] Internal set theory (IST) a conservative extension of ZFC.

All the language of ZFC is available.

All the proof arguments usable in ZFC are available.

Three new axioms (schema) are available:

[Robinson 66] … This is an extension of set theory (ZFC) in which one can prove exactly the same statements of ZFC.

There is a new unary symbol to be read ‘ is standard’.St(x) x

Non-standard analysis (internal set theory)

[Nelson 77] Internal set theory (IST) a conservative extension of ZFC.

All the language of ZFC is available.

All the proof arguments usable in ZFC are available.

Three new axioms (schema) are available:

[Robinson 66] … This is an extension of set theory (ZFC) in which one can prove exactly the same statements of ZFC.

- transfer: 8Stx'8x' iff

' has to be an internal formula, i.e., no symbol .St(x)

9x' 9Stx'iff( )

There is a new unary symbol to be read ‘ is standard’.St(x) x

Non-standard analysis (internal set theory)

[Nelson 77] Internal set theory (IST) a conservative extension of ZFC.

All the language of ZFC is available.

All the proof arguments usable in ZFC are available.

Three new axioms (schema) are available:

[Robinson 66] … This is an extension of set theory (ZFC) in which one can prove exactly the same statements of ZFC.

- transfer: 8Stx'8x' iff

' has to be an internal formula, i.e., no symbol .St(x)

9x' 9Stx'iff( )

There is a new unary symbol to be read ‘ is standard’.St(x) x

- standardization: given a standard set and a formula , there exists a standard set such that for all standard elements ,

X '

Yx

x 2 Xx 2 Yiffx |= 'and

Non-standard analysis (internal set theory)

[Nelson 77] Internal set theory (IST) a conservative extension of ZFC.

All the language of ZFC is available.

All the proof arguments usable in ZFC are available.

Three new axioms (schema) are available:

- idealization: (simplified) there exists a non-standard integer.

[Robinson 66] … This is an extension of set theory (ZFC) in which one can prove exactly the same statements of ZFC.

- transfer: 8Stx'8x' iff

' has to be an internal formula, i.e., no symbol .St(x)

9x' 9Stx'iff( )

There is a new unary symbol to be read ‘ is standard’.St(x) x

- standardization: given a standard set and a formula , there exists a standard set such that for all standard elements ,

X '

Yx

x 2 Xx 2 Yiffx |= 'and

More on IST

More on IST- transfer: 8Stx'8x' iff

' has be an internal formula, i.e., no symbol .St(x)

9x' 9Stx'iff( )

More on IST- transfer: 8Stx'8x' iff

' has be an internal formula, i.e., no symbol .St(x)

9x' 9Stx'iff( ) All objects definable in standard mathematics are standard.

More on IST- transfer: 8Stx'8x' iff

' has be an internal formula, i.e., no symbol .St(x)

9x' 9Stx'iff( ) All objects definable in standard mathematics are standard.

This is the case for 0, 1,⇡,N,R, A⇤ . . .

More on IST- transfer: 8Stx'8x' iff

' has be an internal formula, i.e., no symbol .St(x)

9x' 9Stx'iff( ) All objects definable in standard mathematics are standard.

Things definable by internal formulae with standard parameters are standard. → Successor, predecessor, square of standard integers are integers.

This is the case for 0, 1,⇡,N,R, A⇤ . . .

More on IST- transfer: 8Stx'8x' iff

' has be an internal formula, i.e., no symbol .St(x)

9x' 9Stx'iff( ) All objects definable in standard mathematics are standard.

Things definable by internal formulae with standard parameters are standard. → Successor, predecessor, square of standard integers are integers.

Lemma: If m

More on IST- transfer: 8Stx'8x' iff

' has be an internal formula, i.e., no symbol .St(x)

9x' 9Stx'iff( ) All objects definable in standard mathematics are standard.

Things definable by internal formulae with standard parameters are standard. → Successor, predecessor, square of standard integers are integers.

0 1 2Lemma: If m

More on IST- transfer: 8Stx'8x' iff

' has be an internal formula, i.e., no symbol .St(x)

9x' 9Stx'iff( ) All objects definable in standard mathematics are standard.

Things definable by internal formulae with standard parameters are standard. → Successor, predecessor, square of standard integers are integers.

0 1 2

| {z }Standard numbers

Lemma: If m

More on IST- transfer: 8Stx'8x' iff

' has be an internal formula, i.e., no symbol .St(x)

9x' 9Stx'iff( ) All objects definable in standard mathematics are standard.

Things definable by internal formulae with standard parameters are standard. → Successor, predecessor, square of standard integers are integers.

0 1 2

| {z }Standard numbers

| {z }Non-standard numbers

Lemma: If m

More on IST- transfer: 8Stx'8x' iff

' has be an internal formula, i.e., no symbol .St(x)

9x' 9Stx'iff( ) All objects definable in standard mathematics are standard.

Things definable by internal formulae with standard parameters are standard. → Successor, predecessor, square of standard integers are integers.

0 1 2

| {z }Standard numbers

| {z }Non-standard numbers

Lemma: If m

More on IST- transfer: 8Stx'8x' iff

' has be an internal formula, i.e., no symbol .St(x)

9x' 9Stx'iff( ) All objects definable in standard mathematics are standard.

Things definable by internal formulae with standard parameters are standard. → Successor, predecessor, square of standard integers are integers.

0 1 2

| {z }Standard numbers

| {z }Non-standard numbers

Lemma: If m

More on IST- transfer: 8Stx'8x' iff

' has be an internal formula, i.e., no symbol .St(x)

9x' 9Stx'iff( ) All objects definable in standard mathematics are standard.

Things definable by internal formulae with standard parameters are standard. → Successor, predecessor, square of standard integers are integers.

0 1 2

| {z }Standard numbers

| {z }Non-standard numbers

Lemma: If m

More on IST- transfer: 8Stx'8x' iff

' has be an internal formula, i.e., no symbol .St(x)

9x' 9Stx'iff( ) All objects definable in standard mathematics are standard.

Things definable by internal formulae with standard parameters are standard. → Successor, predecessor, square of standard integers are integers.

0 1 2

| {z }Standard numbers

| {z }Non-standard numbers

Lemma: If m

Regular cost functions in IST0 1 2

| {z } | {z }Standard numbers Non-standard numbers

Regular cost functions in IST0 1 2

| {z } | {z }Standard numbers Non-standard numbers

Small

Regular cost functions in IST0 1 2

| {z } | {z }Standard numbers Non-standard numbers

Small Large

Regular cost functions in IST0 1 2

| {z } | {z }Standard numbers Non-standard numbers

Small Large

Given a standard function , define: f : A⇤ ! N [ {1}

| {z }is smallf(u)

Lf = {u 2 A⇤ | St(f(u)) ^ f(u) 6= 1}external

Regular cost functions in IST0 1 2

| {z } | {z }Standard numbers Non-standard numbers

Small Large

Given a standard function , define: f : A⇤ ! N [ {1}

| {z }is smallf(u)

Lf = {u 2 A⇤ | St(f(u)) ^ f(u) 6= 1}external

Then if and only if . f 4 g Lg ✓ Lf

Regular cost functions in IST0 1 2

| {z } | {z }Standard numbers Non-standard numbers

Small Large

Given a standard function , define: f : A⇤ ! N [ {1}

| {z }is smallf(u)

Lf = {u 2 A⇤ | St(f(u)) ^ f(u) 6= 1}external

Then if and only if . f 4 g Lg ✓ Lf

Under this view, the theory of regular cost functions can be recast into a theory of regular external languages.

Regular external languages

B-rational expression rational expression + LSt

(aStb)⇤aSt

Regular external languages

B-rational expression rational expression + LSt

(aStb)⇤aSt

S-rational expression rational expression + L¬St

(a⇤b)⇤a¬St(ba⇤)⇤

Regular external languages

B-rational expression rational expression + LSt

(aStb)⇤aSt

S-rational expression rational expression + L¬St

(a⇤b)⇤a¬St(ba⇤)⇤

Regular external languages

complement

B-rational expression rational expression + LSt

(aStb)⇤aSt

S-rational expression rational expression + L¬St

(a⇤b)⇤a¬St(ba⇤)⇤

B-automata ND automata + counters

with standardness constraints

Regular external languages

complement

B-rational expression rational expression + LSt

(aStb)⇤aSt

S-rational expression rational expression + L¬St

(a⇤b)⇤a¬St(ba⇤)⇤

B-automata ND automata + counters

with standardness constraints

S-automata ND automata + counters

with non-standardness constraints

Regular external languages

complement

B-rational expression rational expression + LSt

(aStb)⇤aSt

S-rational expression rational expression + L¬St

(a⇤b)⇤a¬St(ba⇤)⇤

B-automata ND automata + counters

with standardness constraints

S-automata ND automata + counters

with non-standardness constraints

cost MSO MSO + (positively )St(|X|)

‘there is a maximal set of consecutive a’s of standard size’

Regular external languages

complement

B-rational expression rational expression + LSt

(aStb)⇤aSt

S-rational expression rational expression + L¬St

(a⇤b)⇤a¬St(ba⇤)⇤

B-automata ND automata + counters

with standardness constraints

S-automata ND automata + counters

with non-standardness constraints

cost MSO MSO + (positively )St(|X|)

‘there is a maximal set of consecutive a’s of standard size’

cost MSO’ MSO + (positively )¬St(|X|)‘all maximal sets of consecutive

a’s have non-standard size’

Regular external languages

complement

B-rational expression rational expression + LSt

(aStb)⇤aSt

S-rational expression rational expression + L¬St

(a⇤b)⇤a¬St(ba⇤)⇤

B-automata ND automata + counters

with standardness constraints

S-automata ND automata + counters

with non-standardness constraints

stabilisation monoids ordered monoid + stabilisation + upward closed accepting set

cost MSO MSO + (positively )St(|X|)

‘there is a maximal set of consecutive a’s of standard size’

cost MSO’ MSO + (positively )¬St(|X|)‘all maximal sets of consecutive

a’s have non-standard size’

Regular external languages

complement

B-rational expression rational expression + LSt

(aStb)⇤aSt

S-rational expression rational expression + L¬St

(a⇤b)⇤a¬St(ba⇤)⇤

B-automata ND automata + counters

with standardness constraints

S-automata ND automata + counters

with non-standardness constraints

stabilisation monoids ordered monoid + stabilisation + upward closed accepting set

stabilisation monoids ordered monoid + stabilisation

+ downward closed accepting set

cost MSO MSO + (positively )St(|X|)

‘there is a maximal set of consecutive a’s of standard size’

cost MSO’ MSO + (positively )¬St(|X|)‘all maximal sets of consecutive

a’s have non-standard size’

Regular external languages

complement

B-rational expression rational expression + LSt

(aStb)⇤aSt

S-rational expression rational expression + L¬St

(a⇤b)⇤a¬St(ba⇤)⇤

B-automata ND automata + counters

with standardness constraints

S-automata ND automata + counters

with non-standardness constraints

stabilisation monoids ordered monoid + stabilisation + upward closed accepting set

stabilisation monoids ordered monoid + stabilisation

+ downward closed accepting set

cost MSO MSO + (positively )St(|X|)

‘there is a maximal set of consecutive a’s of standard size’

cost MSO’ MSO + (positively )¬St(|X|)‘all maximal sets of consecutive

a’s have non-standard size’

Regular external languages

complement

Emptiness of Boolean combinations is decidable.

ConclusionRegular cost functions are functions computed by variants of tropical automata modulo a ‘boundedness equivalence relation’.

It can be used to decide several problems involving the existence of bounds, be it explicitly or implicitly.

Thanks to non-standard analysis, the results can be seen as a language theory over less usual words, in the same spirit as infinite words/words of countable length.

The most important results that hold for regular languages are still valid.