-
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.