Automata with timed atoms - mimuw.edu.plsl/SLIDES/Bengaluru.2015.12.pdfInfinity 2015, Bengaluru 1....
Transcript of Automata with timed atoms - mimuw.edu.plsl/SLIDES/Bengaluru.2015.12.pdfInfinity 2015, Bengaluru 1....
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Sławomir Lasota
Automata with timed atoms
joint work with Mikołaj Bojańczyk and Lorenzo Clemente
University of Warsaw
Infinity 2015, Bengaluru1
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Sławomir Lasota
joint work with Mikołaj Bojańczyk and Lorenzo Clemente
University of Warsaw
Infinity 2015, Bengaluru
FO-definable automata
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FO-definable sets offer a right setting for timed models of computation, like timed automata, or timed pushdown automata.
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Plan
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Plan
• Motivation
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Plan
• Motivation
• FO-definable NFA
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Plan
• Motivation
• FO-definable NFA
• FO-definable PDA
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Plan
• Motivation
• FO-definable NFA
• FO-definable PDA
• The core problem: equations over sets of integers
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• reals
• rationals
• integers
}dense time
discrete time
Time domain
any choice of time domain is fine
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• reals
• rationals
• integers
}dense time
discrete time
Time domain
any choice of time domain is fine
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• reals
• rationals
• integers
}dense time
discrete time
Time domain
any choice of time domain is fine
No restriction to non-negative!
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Let input alphabet be reals
• reals
• rationals
• integers
}dense time
discrete time
Time domain
any choice of time domain is fine
No restriction to non-negative!
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Let input alphabet be reals
Monotonic input words :
• reals
• rationals
• integers
}dense time
discrete time
Time domain
any choice of time domain is fine
No restriction to non-negative!
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Timed automatawith uninitialized clocks
[Alur, Dill 1990]
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Timed automatawith uninitialized clocks
[Alur, Dill 1990]
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?
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Timed automatawith uninitialized clocks
t realnumber
[Alur, Dill 1990]
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Timed automatawith uninitialized clocks
c₁ := 0t realnumber
[Alur, Dill 1990]
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Timed automatawith uninitialized clocks
c₁ := 0t
0 < c₁ < 2c₂ := 0
t realnumber
[Alur, Dill 1990]
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Timed automatawith uninitialized clocks
c₁ := 0t
0 < c₁ < 2c₂ := 0
t(2 < c₁ < 3) ∧
(c₂ = 1 ∨ c₂ = 2)
trealnumber
[Alur, Dill 1990]
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Timed automatawith uninitialized clocks
c₁ := 0t
0 < c₁ < 2c₂ := 0
t(2 < c₁ < 3) ∧
(c₂ = 1 ∨ c₂ = 2)
trealnumber
the automaton accepts words t₁ t₂ t₃ ∈ R³ such that
t₁ t₂ t₃
[Alur, Dill 1990]
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Timed automatawith uninitialized clocks
c₁ := 0t
0 < c₁ < 2c₂ := 0
t(2 < c₁ < 3) ∧
(c₂ = 1 ∨ c₂ = 2)
trealnumber
the automaton accepts words t₁ t₂ t₃ ∈ R³ such that
t₁ t₂ t₃
{ 0..2
[Alur, Dill 1990]
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Timed automatawith uninitialized clocks
c₁ := 0t
0 < c₁ < 2c₂ := 0
t(2 < c₁ < 3) ∧
(c₂ = 1 ∨ c₂ = 2)
trealnumber
the automaton accepts words t₁ t₂ t₃ ∈ R³ such that
t₁ t₂ t₃
{ 0..2
{2..3
[Alur, Dill 1990]
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Timed automatawith uninitialized clocks
c₁ := 0t
0 < c₁ < 2c₂ := 0
t(2 < c₁ < 3) ∧
(c₂ = 1 ∨ c₂ = 2)
trealnumber
the automaton accepts words t₁ t₂ t₃ ∈ R³ such that
t₁ t₂ t₃{1 or 2
{ 0..2
{2..3
[Alur, Dill 1990]
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Deterministic timed automata don’t minimize
t₁ t₂ t₃{1 or 2
{ 0..2
{2..3
c₁ := 0t
0 < c₁ < 2c₂ := 0
t(2 < c₁ < 3) ∧
(c₂ = 1 ∨ c₂ = 2)
t
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Deterministic timed automata don’t minimize
⅓0 2⅓
0 2⅓1⅓t₁ t₂ t₃{1 or 2
{ 0..2
{2..3
c₁ := 0t
0 < c₁ < 2c₂ := 0
t(2 < c₁ < 3) ∧
(c₂ = 1 ∨ c₂ = 2)
t
6
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Deterministic timed automata don’t minimize
⅓0 2⅓
0 2⅓1⅓t₁ t₂ t₃{1 or 2
{ 0..2
{2..3
c₁ := 0t
0 < c₁ < 2c₂ := 0
t(2 < c₁ < 3) ∧
(c₂ = 1 ∨ c₂ = 2)
t
(c₁=0, c₂=⅓) ≡ (c₁=0, c₂=1⅓)
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Towards timed pushdown automata
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Towards timed pushdown automata
• timed automata [Alur, Dill 1990]
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Towards timed pushdown automata
• timed automata [Alur, Dill 1990]
• pushdown timed automata [Bouajjani, Echahed, Robbana 1994]
finite stack alphabet
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Towards timed pushdown automata
• timed automata [Alur, Dill 1990]
• pushdown timed automata [Bouajjani, Echahed, Robbana 1994]
• dense-timed pushdown automata [Abdulla, Atig, Stenman 2012]
finite stack alphabet
• clocks can be pushed onto stack• the emptiness problem EXPTIME-complete
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Towards timed pushdown automata
• timed automata [Alur, Dill 1990]
• pushdown timed automata [Bouajjani, Echahed, Robbana 1994]
• dense-timed pushdown automata [Abdulla, Atig, Stenman 2012]
• recursive timed automata [Trivedi, Wojtczak 2010], [Benerecetti, Minopoli, Peron 2010]
finite stack alphabet
• clocks can be pushed onto stack• the emptiness problem EXPTIME-complete
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Dense-timed PDA collapse
Theorem 1: [Clemente, L. 2015]Dense-timed pushdown automata are expressively equivalent to pushdown timed automata.
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Dense-timed PDA collapse
Theorem 1: [Clemente, L. 2015]Dense-timed pushdown automata are expressively equivalent to pushdown timed automata.
An accidental combination of • stack discipline• monotonicity of time• syntactic restrictions
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9
FO-definable sets offer a right setting for timed models of computation, like timed automata, or timed pushdown automata.
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• do not invent a new definition
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FO-definable sets offer a right setting for timed models of computation, like timed automata, or timed pushdown automata.
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• do not invent a new definition
• re-interpret a classical definition in FO-definable sets, with
finiteness relaxed to orbit-finiteness
9
FO-definable sets offer a right setting for timed models of computation, like timed automata, or timed pushdown automata.
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In search of lost definition
• Motivation
• FO-definable NFA
• FO-definable PDA
• The core problem: equations over sets of integers
10
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In search of lost definition
NFA re-interpreted in FO-definable sets
• Motivation
• FO-definable NFA
• FO-definable PDA
• The core problem: equations over sets of integers
10
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Timed automata are register automata
c₁ := 0t
0 < c₁ < 2c₂ := 0
t(2 < c₁ < 3) ∧
(c₂ = 1 ∨ c₂ = 2)
t
[Bojańczyk, L. 2012]
11
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Timed automata are register automata
c₁ := tt
c₁ := 0t
0 < c₁ < 2c₂ := 0
t(2 < c₁ < 3) ∧
(c₂ = 1 ∨ c₂ = 2)
t
[Bojańczyk, L. 2012]
11
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Timed automata are register automata
c₁ := tt
0 < t-c₁ < 2c₂ := t
t
c₁ := 0t
0 < c₁ < 2c₂ := 0
t(2 < c₁ < 3) ∧
(c₂ = 1 ∨ c₂ = 2)
t
[Bojańczyk, L. 2012]
11
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Timed automata are register automata
c₁ := tt
0 < t-c₁ < 2c₂ := t
t(2 < t-c₁ < 3) ∧
t
c₁ := 0t
0 < c₁ < 2c₂ := 0
t(2 < c₁ < 3) ∧
(c₂ = 1 ∨ c₂ = 2)
t
[Bojańczyk, L. 2012]
11
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Timed automata are register automata
c₁ := tt
0 < t-c₁ < 2c₂ := t
t(2 < t-c₁ < 3) ∧
(t-c₂ = 1 ∨ t-c₂ = 2)
t
c₁ := 0t
0 < c₁ < 2c₂ := 0
t(2 < c₁ < 3) ∧
(c₂ = 1 ∨ c₂ = 2)
t
[Bojańczyk, L. 2012]
11
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Timed automata are register automata
c₁ := tt
0 < t-c₁ < 2c₂ := t
t(2 < t-c₁ < 3) ∧
(t-c₂ = 1 ∨ t-c₂ = 2)
t
c₁ := 0t
0 < c₁ < 2c₂ := 0
t(2 < c₁ < 3) ∧
(c₂ = 1 ∨ c₂ = 2)
t
the guards use the structure (R, <, +1)e.g. 0 < t-c₁ <2 iff c₁ < t < c₁+2
[Bojańczyk, L. 2012]
11
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Timed automata are register automata
c₁ := tt
0 < t-c₁ < 2c₂ := t
t(2 < t-c₁ < 3) ∧
(t-c₂ = 1 ∨ t-c₂ = 2)
t
c₁ := 0t
0 < c₁ < 2c₂ := 0
t(2 < c₁ < 3) ∧
(c₂ = 1 ∨ c₂ = 2)
t
⊥ c₁ 0 < c₂-c₁ < 2 ⊤
the guards use the structure (R, <, +1)e.g. 0 < t-c₁ <2 iff c₁ < t < c₁+2
[Bojańczyk, L. 2012]
11
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Timed automata are register automata
c₁ := tt
0 < t-c₁ < 2c₂ := t
t(2 < t-c₁ < 3) ∧
(t-c₂ = 1 ∨ t-c₂ = 2)
t
c₁ := 0t
0 < c₁ < 2c₂ := 0
t(2 < c₁ < 3) ∧
(c₂ = 1 ∨ c₂ = 2)
t
⊥ c₁ 0 < c₂-c₁ < 2 ⊤
the guards use the structure (R, <, +1)e.g. 0 < t-c₁ <2 iff c₁ < t < c₁+2
[Bojańczyk, L. 2012]
11
the only modifications of a clock: c:= t
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FO(<, +1)-definable sets
�(x1, . . . , xn)FO(<, +1) formula defines a subset of n-tuples of reals, for instance
�(x1, x2) ⌘ 9x3 (x1 < x3 ^ x2 = x3 + 3)
dimension
12
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�(x1, . . . , xn)FO(<, +1) formula defines a subset of n-tuples of reals, for instance
�(x1, x2) ⌘ 9x3 (x1 < x3 ^ x2 = x3 + 3)
dimension
12
FO-definable sets
![Page 49: Automata with timed atoms - mimuw.edu.plsl/SLIDES/Bengaluru.2015.12.pdfInfinity 2015, Bengaluru 1. Sławomir Lasota joint work with Mikołaj Bojańczyk and Lorenzo Clemente University](https://reader033.fdocuments.net/reader033/viewer/2022043021/5f3d2b49d3c88d090b20decf/html5/thumbnails/49.jpg)
�(x1, . . . , xn)FO(<, +1) formula defines a subset of n-tuples of reals, for instance
FO(<, +1) = QF(<, +1) =
�(x1, x2) ⌘ 9x3 (x1 < x3 ^ x2 = x3 + 3)
dimension
12
FO-definable sets
![Page 50: Automata with timed atoms - mimuw.edu.plsl/SLIDES/Bengaluru.2015.12.pdfInfinity 2015, Bengaluru 1. Sławomir Lasota joint work with Mikołaj Bojańczyk and Lorenzo Clemente University](https://reader033.fdocuments.net/reader033/viewer/2022043021/5f3d2b49d3c88d090b20decf/html5/thumbnails/50.jpg)
�(x1, . . . , xn)FO(<, +1) formula defines a subset of n-tuples of reals, for instance
FO(<, +1) = QF(<, +1) = _
finite
^
finite
xi � xj 2 I
| {z }zone
�(x1, x2) ⌘ 9x3 (x1 < x3 ^ x2 = x3 + 3)
dimension
12
FO-definable sets
![Page 51: Automata with timed atoms - mimuw.edu.plsl/SLIDES/Bengaluru.2015.12.pdfInfinity 2015, Bengaluru 1. Sławomir Lasota joint work with Mikołaj Bojańczyk and Lorenzo Clemente University](https://reader033.fdocuments.net/reader033/viewer/2022043021/5f3d2b49d3c88d090b20decf/html5/thumbnails/51.jpg)
�(x1, . . . , xn)FO(<, +1) formula defines a subset of n-tuples of reals, for instance
FO(<, +1) = QF(<, +1) = _
finite
^
finite
xi � xj 2 I
| {z }zone
�(x1, x2) ⌘ 9x3 (x1 < x3 ^ x2 = x3 + 3)
for instance
�(x1, x2) ⌘ x1 + 3 < x2 ⌘ x2 � x1 2 (3,1)
dimension
12
FO-definable sets
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FO-definable NFA
• alphabet A
• states Q
• transitions δ ⊆ Q × A × Q
• I, F ⊆ Q
13
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FO-definable NFA
• alphabet A
• states Q
• transitions δ ⊆ Q × A × Q
• I, F ⊆ Q}definable in FO(<, +1)
13
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FO-definable NFA
• alphabet A
• states Q
• transitions δ ⊆ Q × A × Q
• I, F ⊆ Q
�A(x1, . . . , xn)
�Q(x1, . . . , xm)
��(x1, . . . , xm+n+m)
�I(x1, . . . , xm), �F (x1, . . . , xm)
13
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FO-definable NFA
• alphabet A
• states Q
• transitions δ ⊆ Q × A × Q
• I, F ⊆ Q�A(x1, . . . , xn)
�Q(x1, . . . , xm)
��(x1, . . . , xm+n+m)
�I(x1, . . . , xm), �F (x1, . . . , xm)
}definable in FO(<, +1)
13
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FO-definable NFA
• alphabet A
• states Q
• transitions δ ⊆ Q × A × Q
• I, F ⊆ Q
Runs, acceptance, language recognized, etc. are defined exactly as for classical NFA!
�A(x1, . . . , xn)
�Q(x1, . . . , xm)
��(x1, . . . , xm+n+m)
�I(x1, . . . , xm), �F (x1, . . . , xm)
}definable in FO(<, +1)
13
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FO-definable NFA
• alphabet A
• states Q
• transitions δ ⊆ Q × A × Q
• I, F ⊆ Q
}orbit-finite
Runs, acceptance, language recognized, etc. are defined exactly as for classical NFA!
�A(x1, . . . , xn)
�Q(x1, . . . , xm)
��(x1, . . . , xm+n+m)
�I(x1, . . . , xm), �F (x1, . . . , xm)
}definable in FO(<, +1)
13
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Orbit-finitenessπ
π
π
Automorphisms π of (R, <, +1) act on a definable set thus splitting it into orbits.
14
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Orbit-finitenessπ
π
π
Automorphisms π of (R, <, +1) act on a definable set thus splitting it into orbits.
14
For instance, (-1, ⅓) and (3, 4⅓) and (1⅓, 3) are in the same orbit.
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Orbit-finitenessπ
π
π
Automorphisms π of (R, <, +1) act on a definable set thus splitting it into orbits.
x1 + 3 < x2 ⌘ x2 � x1 2 (3,1) orbit-infinite
Example:
14
For instance, (-1, ⅓) and (3, 4⅓) and (1⅓, 3) are in the same orbit.
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Orbit-finitenessπ
π
π
Automorphisms π of (R, <, +1) act on a definable set thus splitting it into orbits.
x1 + 3 < x2 ⌘ x2 � x1 2 (3,1)
x1 + 3 < x2 x1 + 7 ⌘ x2 � x1 2 (3, 7]
orbit-infiniteorbit-finite
Example:
14
For instance, (-1, ⅓) and (3, 4⅓) and (1⅓, 3) are in the same orbit.
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Orbit-finitenessπ
π
π
Automorphisms π of (R, <, +1) act on a definable set thus splitting it into orbits.
x1 + 3 < x2 ⌘ x2 � x1 2 (3,1)
x1 + 3 < x2 x1 + 7 ⌘ x2 � x1 2 (3, 7]
An FO-definable set is orbit-finite iff
it is defined using bounded intervals only
orbit-infiniteorbit-finite
Example:
14
For instance, (-1, ⅓) and (3, 4⅓) and (1⅓, 3) are in the same orbit.
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c₁ := tt
0 < t-c₁ < 2c₂ := t
t(2 < t-c₁ < 3) ∧
(t-c₂ = 1 ∨ t-c₂ = 2)
t
⊥
Register automata are FO-definable NFA
15
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states:
c₁ := tt
0 < t-c₁ < 2c₂ := t
t(2 < t-c₁ < 3) ∧
(t-c₂ = 1 ∨ t-c₂ = 2)
t
⊥
Register automata are FO-definable NFA
Q = {⊥} ∪ {c₁∊R} ∪ { (c₁, c₂)∊R×R : 0 < c₂-c₁ < 2 } ∪ {⊤}
15
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states:
c₁ := tt
0 < t-c₁ < 2c₂ := t
t(2 < t-c₁ < 3) ∧
(t-c₂ = 1 ∨ t-c₂ = 2)
t
⊥
Register automata are FO-definable NFA
Q = {⊥} ∪ {c₁∊R} ∪ { (c₁, c₂)∊R×R : 0 < c₂-c₁ < 2 } ∪ {⊤}𝜙Q(c0, c₁, c₂) ≡ c0 = c₁ = c₂ ∨ c0+1 = c₁ = c₂ ∨ c0+2 = c₁ < c₂ < c₁+2 ∨ c0+3 = c₁ = c₂
15
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states:
c₁ := tt
0 < t-c₁ < 2c₂ := t
t(2 < t-c₁ < 3) ∧
(t-c₂ = 1 ∨ t-c₂ = 2)
t
transitions:
⊥
Register automata are FO-definable NFA
Q = {⊥} ∪ {c₁∊R} ∪ { (c₁, c₂)∊R×R : 0 < c₂-c₁ < 2 } ∪ {⊤}
𝛿 = { (⊥, t, c₁’) : c₁’ = t } ∪
𝜙Q(c0, c₁, c₂) ≡ c0 = c₁ = c₂ ∨ c0+1 = c₁ = c₂ ∨ c0+2 = c₁ < c₂ < c₁+2 ∨ c0+3 = c₁ = c₂
15
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states:
c₁ := tt
0 < t-c₁ < 2c₂ := t
t(2 < t-c₁ < 3) ∧
(t-c₂ = 1 ∨ t-c₂ = 2)
t
transitions:
⊥
Register automata are FO-definable NFA
Q = {⊥} ∪ {c₁∊R} ∪ { (c₁, c₂)∊R×R : 0 < c₂-c₁ < 2 } ∪ {⊤}
𝛿 = { (⊥, t, c₁’) : c₁’ = t } ∪ { (c₁, t, (c₁’, c₂’)) : 0 < t-c₁ < 2 ∧ c₁ = c₁’ ∧ c₂’ = t } ∪
𝜙Q(c0, c₁, c₂) ≡ c0 = c₁ = c₂ ∨ c0+1 = c₁ = c₂ ∨ c0+2 = c₁ < c₂ < c₁+2 ∨ c0+3 = c₁ = c₂
15
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states:
c₁ := tt
0 < t-c₁ < 2c₂ := t
t(2 < t-c₁ < 3) ∧
(t-c₂ = 1 ∨ t-c₂ = 2)
t
transitions:
⊥
Register automata are FO-definable NFA
Q = {⊥} ∪ {c₁∊R} ∪ { (c₁, c₂)∊R×R : 0 < c₂-c₁ < 2 } ∪ {⊤}
𝛿 = { (⊥, t, c₁’) : c₁’ = t } ∪ { (c₁, t, (c₁’, c₂’)) : 0 < t-c₁ < 2 ∧ c₁ = c₁’ ∧ c₂’ = t } ∪ { ((c₁, c₂), t, ⊤) : (2 < t-c₁ < 3) ∧ (t-c₂ = 1 ∨ t-c₂ = 2) }
𝜙Q(c0, c₁, c₂) ≡ c0 = c₁ = c₂ ∨ c0+1 = c₁ = c₂ ∨ c0+2 = c₁ < c₂ < c₁+2 ∨ c0+3 = c₁ = c₂
15
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states:
c₁ := tt
0 < t-c₁ < 2c₂ := t
t(2 < t-c₁ < 3) ∧
(t-c₂ = 1 ∨ t-c₂ = 2)
t
transitions:
⊥
Register automata are FO-definable NFA
Q = {⊥} ∪ {c₁∊R} ∪ { (c₁, c₂)∊R×R : 0 < c₂-c₁ < 2 } ∪ {⊤}
𝛿 = { (⊥, t, c₁’) : c₁’ = t } ∪ { (c₁, t, (c₁’, c₂’)) : 0 < t-c₁ < 2 ∧ c₁ = c₁’ ∧ c₂’ = t } ∪ { ((c₁, c₂), t, ⊤) : (2 < t-c₁ < 3) ∧ (t-c₂ = 1 ∨ t-c₂ = 2) }
𝜙Q(c0, c₁, c₂) ≡ c0 = c₁ = c₂ ∨ c0+1 = c₁ = c₂ ∨ c0+2 = c₁ < c₂ < c₁+2 ∨ c0+3 = c₁ = c₂
𝜙𝛿(c0, c₁, c₂, t, c0’, c₁’, c₂’) ≡ ...
15
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Timed automata vs. FO-definable NFA
FO-definable NFA are like updatable timed automata [Bouyer, Duford, Fleury 2000], but:
16
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Timed automata vs. FO-definable NFA
• in every location, clock valuations are restricted by an orbit-finite constraint (invariant)
FO-definable NFA are like updatable timed automata [Bouyer, Duford, Fleury 2000], but:
16
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Timed automata vs. FO-definable NFA
• in every location, clock valuations are restricted by an orbit-finite constraint (invariant)
• number of clocks may vary from one location to another
FO-definable NFA are like updatable timed automata [Bouyer, Duford, Fleury 2000], but:
16
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Timed automata vs. FO-definable NFA
• in every location, clock valuations are restricted by an orbit-finite constraint (invariant)
• number of clocks may vary from one location to another • the input needs not be monotonic (but can be enforced to be)
FO-definable NFA are like updatable timed automata [Bouyer, Duford, Fleury 2000], but:
16
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Timed automata vs. FO-definable NFA
• in every location, clock valuations are restricted by an orbit-finite constraint (invariant)
• number of clocks may vary from one location to another • the input needs not be monotonic (but can be enforced to be) • alphabet letters may be a tuples of timestamps
FO-definable NFA are like updatable timed automata [Bouyer, Duford, Fleury 2000], but:
16
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FO-definable NFA
timed automata with uninitialized clocks
Timed automata vs. FO-definable NFA
17
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FO-definable NFA
timed automata with uninitialized clocks
Timed automata vs. FO-definable NFA
deterministic
deterministic
17
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FO-definable NFA
timed automata with uninitialized clocks
Timed automata vs. FO-definable NFA
{integer
deterministic
deterministic
17
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FO-definable NFA
timed automata with uninitialized clocks
Timed automata vs. FO-definable NFA
{integer
{< 2
deterministic
deterministic
17
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FO-definable NFA
timed automata with uninitialized clocks
Timed automata vs. FO-definable NFA
{integer
{< 2 {< 2
deterministic
deterministic
17
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FO-definable NFA
timed automata with uninitialized clocks
Timed automata vs. FO-definable NFA
{integer
{< 2 {< 2...
deterministic
deterministic
17
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FO-definable NFA
timed automata with uninitialized clocks
closed underminimization
Timed automata vs. FO-definable NFA
{integer
{< 2 {< 2...
deterministic
deterministic
17
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FO-definable NFA
timed automata with uninitialized clocks
closed underminimization
minimal automata for languages of deterministic timed automata
with uninitialized clocks
Timed automata vs. FO-definable NFA
{integer
{< 2 {< 2...
deterministic
deterministic
17
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FO-definable DFA do minimize
deterministic FO-definable NFA
deterministic timed automata with uninitialized clocks
minimal automata for languages of deterministic timed automata
with uninitialized clocks
18
[Bojańczyk, L. 2012]
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FO-definable DFA do minimize
deterministic FO-definable NFA
deterministic timed automata with uninitialized clocks
minimal automata for languages of deterministic timed automata
with uninitialized clocks
c₁ := tt
0 < t-c₁ < 2c₂ := t
t(2 < t-c₁ < 3) ∧
(t-c₂ = 1 ∨ t-c₂ = 2)
t
c1 c2 t{1 or 2
{ 0..2
{
2..3
0 < c₂-c₁ < 2⅓0 2⅓
0 2⅓1⅓
18
[Bojańczyk, L. 2012]
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FO-definable DFA do minimize
deterministic FO-definable NFA
deterministic timed automata with uninitialized clocks
minimal automata for languages of deterministic timed automata
with uninitialized clocks
c₁ := tt
0 < t-c₁ < 2c₂ := t
t(2 < t-c₁ < 3) ∧
(t-c₂ = 1 ∨ t-c₂ = 2)
t
c1 c2 t{1 or 2
{ 0..2
{
2..3
0 < c₂-c₁ < 2
c₁ := tt
if 0 < t-c₁ <= 1c₂ := t
if 1 < t-c₁ < 2c₂ := t-1
t(2 < t-c₁ < 3) ∧
(t-c₂ = 1 ∨ t-c₂ = 2)
t
0 < c₂-c₁ <= 1
⅓0 2⅓
0 2⅓1⅓
18
[Bojańczyk, L. 2012]
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Presburger NFA
Minimization holds also if FO(<, +1) is replaced by FO(<, +)
19
[Bojańczyk, L. 2012]
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• Motivation
• FO-definable NFA
• FO-definable PDA
• The core problem: equations over sets of integers
20
In search of lost definition
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PDA re-interpreted in FO-definable sets
• Motivation
• FO-definable NFA
• FO-definable PDA
• The core problem: equations over sets of integers
20
In search of lost definition
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FO-definable PDA
• alphabet A
• states Q
• stack alphabet S
• push ⊆ Q × A × Q × S
• pop ⊆ Q × S × A × Q
• I, F ⊆ Q
}definable in FO(<, +1)
}orbit-finite
21
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FO-definable PDA
• alphabet A
• states Q
• stack alphabet S
• push ⊆ Q × A × Q × S
• pop ⊆ Q × S × A × Q
• I, F ⊆ Q
}orbit-finite�A(x1
, . . . , xn)
�Q(x1
, . . . , xm)
�S(x1
, . . . , xk)
�
push
(x1
, . . . , xm+n+m+k)
�
pop
(x1
, . . . , xm+k+n+m)
�I(x1
, . . . , xm), �F (x1
, . . . , xm)
21
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FO-definable PDA
• alphabet A
• states Q
• stack alphabet S
• push ⊆ Q × A × Q × S
• pop ⊆ Q × S × A × Q
• I, F ⊆ Q
}orbit-finite
Acceptance defined as for classical PDA.
�A(x1
, . . . , xn)
�Q(x1
, . . . , xm)
�S(x1
, . . . , xk)
�
push
(x1
, . . . , xm+n+m+k)
�
pop
(x1
, . . . , xm+k+n+m)
�I(x1
, . . . , xm), �F (x1
, . . . , xm)
}definable in FO(<, +1)
21
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language: "ordered palindromes of even length over reals"input alphabet: A = reals ⨄ {ε}
states:stack alphabet:
transitions:
accepting state: initial state:
Example
22
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language:
Q = reals ⨄ {init, finish, acc}
"ordered palindromes of even length over reals"input alphabet: A = reals ⨄ {ε}
states:stack alphabet:
transitions:
accepting state: initial state: init
acc
Example
22
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language:
Q = reals ⨄ {init, finish, acc}
"ordered palindromes of even length over reals"input alphabet: A = reals ⨄ {ε}
states:stack alphabet:
transitions:
accepting state: initial state: init
acc
S = reals ⨄ {⊥}
Example
22
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language:
Q = reals ⨄ {init, finish, acc}
"ordered palindromes of even length over reals"input alphabet: A = reals ⨄ {ε}
push ⊆ Q × A × Q × S
states:stack alphabet:
transitions:
accepting state: initial state: init
acc
S = reals ⨄ {⊥}
in state init, without reading input, change state to an arbitrary real t, and push ⊥ on stack
Example
(finish, t, t, finish)(finish, ⊥, ε, acc)
pop ⊆ Q × S × A × Q
(init, ε, t, ⊥)(t, u, u, u) t < u(t, u, finish, u) t < u
22
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language:
Q = reals ⨄ {init, finish, acc}
"ordered palindromes of even length over reals"input alphabet: A = reals ⨄ {ε}
push ⊆ Q × A × Q × S
states:stack alphabet:
transitions:
accepting state: initial state: init
acc
S = reals ⨄ {⊥}
Example
(finish, t, t, finish)(finish, ⊥, ε, acc)
pop ⊆ Q × S × A × Q
(init, ε, t, ⊥)(t, u, u, u) t < u(t, u, finish, u) t < u
in state finish, pop a real t from stack, read the same t from input, and stay in the same state
22
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FO-definable prefix rewriting
• alphabet A
• states Q
• stack alphabet S
• ρ ⊆ Q × S* × A × Q × S*
• I, F ⊆ Q
}definable in FO(<, +1)}orbit-finite
23
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FO-definable prefix rewriting
• alphabet A
• states Q
• stack alphabet S
• ρ ⊆ Q × S* × A × Q × S*
• I, F ⊆ Q
}definable in FO(<, +1)}orbit-finite
≤n ≤m
23
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FO-definable prefix rewriting
• alphabet A
• states Q
• stack alphabet S
• ρ ⊆ Q × S* × A × Q × S*
• I, F ⊆ Q
}definable in FO(<, +1)}orbit-finite
Acceptance defined as for classical prefix rewriting.
≤n ≤m
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orbit-finitesetofsymbolsS
FO-definable context-free grammars
• nonterminal symbols S
• terminal symbols A
• an initial nonterminal symbol
• ρ ⊆ S×(S⨄A)*}definable in FO(<, +1)
} orbit-finite
24
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orbit-finitesetofsymbolsS
FO-definable context-free grammars
• nonterminal symbols S
• terminal symbols A
• an initial nonterminal symbol
• ρ ⊆ S×(S⨄A)*}definable in FO(<, +1)
} orbit-finite
Generated language defined as for classical PDA.
24
≤n
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prefix rewriting
CFG
Expressiveness of FO-definable models
dense-timed PDA with uninitialized clocks
[Abdulla, Atig, Stenman 2012]
PDA with timeless stack
(finite stack alphabet)
PDA
25
[Clemente, L. 2015]
![Page 103: Automata with timed atoms - mimuw.edu.plsl/SLIDES/Bengaluru.2015.12.pdfInfinity 2015, Bengaluru 1. Sławomir Lasota joint work with Mikołaj Bojańczyk and Lorenzo Clemente University](https://reader033.fdocuments.net/reader033/viewer/2022043021/5f3d2b49d3c88d090b20decf/html5/thumbnails/103.jpg)
prefix rewriting
CFG
Expressiveness of FO-definable models
palindromes
dense-timed PDA with uninitialized clocks
[Abdulla, Atig, Stenman 2012]
PDA with timeless stack
(finite stack alphabet)
PDA
25
[Clemente, L. 2015]
![Page 104: Automata with timed atoms - mimuw.edu.plsl/SLIDES/Bengaluru.2015.12.pdfInfinity 2015, Bengaluru 1. Sławomir Lasota joint work with Mikołaj Bojańczyk and Lorenzo Clemente University](https://reader033.fdocuments.net/reader033/viewer/2022043021/5f3d2b49d3c88d090b20decf/html5/thumbnails/104.jpg)
prefix rewriting
CFG
Expressiveness of FO-definable models
palindromes
dense-timed PDA with uninitialized clocks
[Abdulla, Atig, Stenman 2012]
PDA with timeless stack
(finite stack alphabet)
PDA
constrained PDA
25
[Clemente, L. 2015]
![Page 105: Automata with timed atoms - mimuw.edu.plsl/SLIDES/Bengaluru.2015.12.pdfInfinity 2015, Bengaluru 1. Sławomir Lasota joint work with Mikołaj Bojańczyk and Lorenzo Clemente University](https://reader033.fdocuments.net/reader033/viewer/2022043021/5f3d2b49d3c88d090b20decf/html5/thumbnails/105.jpg)
prefix rewriting
CFG
Expressiveness of FO-definable models
palindromes
dense-timed PDA with uninitialized clocks
[Abdulla, Atig, Stenman 2012]
PDA with timeless stack
(finite stack alphabet)
PDA
constrained PDA
palindromes over {a,b}×reals with the same number of a’s and b’s
25
[Clemente, L. 2015]
![Page 106: Automata with timed atoms - mimuw.edu.plsl/SLIDES/Bengaluru.2015.12.pdfInfinity 2015, Bengaluru 1. Sławomir Lasota joint work with Mikołaj Bojańczyk and Lorenzo Clemente University](https://reader033.fdocuments.net/reader033/viewer/2022043021/5f3d2b49d3c88d090b20decf/html5/thumbnails/106.jpg)
Constrained FO-definable PDA?
• alphabet A
• states Q
• stack alphabet S
• push ⊆ Q × A × Q × S
• pop ⊆ Q × S × A × Q
• I, F ⊆ Q
}definable in FO(<, +1)
}orbit-finite
26
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Constrained FO-definable PDA?
• alphabet A
• states Q
• stack alphabet S
• push ⊆ Q × A × Q × S
• pop ⊆ Q × S × A × Q
• I, F ⊆ Q
}orbit-finite}orbit-finite?
26
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Constrained FO-definable PDA?
• alphabet A
• states Q
• stack alphabet S
• push ⊆ Q × A × Q × S
• pop ⊆ Q × S × A × Q
• I, F ⊆ Q
}orbit-finite}orbit-finite?
Too strong restriction! Span of transitions is bounded.
26
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Constrained FO-definable PDA?
• alphabet A
• states Q
• stack alphabet S
• push ⊆ Q × A × Q × S
• pop ⊆ Q × S × A × Q
• I, F ⊆ Q
}orbit-finite}orbit-finite?
Too strong restriction! Span of transitions is bounded. For instance, such PDA do not recognize palindromes over reals.
26
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Constrained FO-definable PDA
• alphabet A
• states Q
• stack alphabet S
• push ⊆ Q × A × Q × S
• pop ⊆ Q × S × A × Q
• I, F ⊆ Q
}definable in FO(<, +1)
}orbit-finite
27
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Constrained FO-definable PDA
• alphabet A
• states Q
• stack alphabet S
• push ⊆ Q × A × Q × S
• pop ⊆ Q × S × A × Q
• I, F ⊆ Q
}definable in FO(<, +1)
}orbit-finite
}
orbit-finite
}
orbit-finite
27
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Constrained FO-definable PDA
• alphabet A
• states Q
• stack alphabet S
• push ⊆ Q × A × Q × S
• pop ⊆ Q × S × A × Q
• I, F ⊆ Q
}definable in FO(<, +1)
}orbit-finite
}
orbit-finite
}
orbit-finite
Theorem 2: [Clemente, L. 2015]The non-emptiness problem is in NEXPTIME.For finite stack alphabet, EXPTIME-complete.
27
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Constrained FO-definable PDA
• alphabet A
• states Q
• stack alphabet S
• push ⊆ Q × A × Q × S
• pop ⊆ Q × S × A × Q
• I, F ⊆ Q
}definable in FO(<, +1)
}orbit-finite
}
orbit-finite
}
orbit-finite
Theorem 2: [Clemente, L. 2015]The non-emptiness problem is in NEXPTIME.For finite stack alphabet, EXPTIME-complete.
Fact: The model subsumes dense-timed PDA with uninitialized clocks. 27
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Complexity of non-emptiness
prefix rewriting
CFG
dense-timed PDA with uninitialized clocks
[Abdulla, Atig, Stenman 2012]
PDA with finite stack alphabet
PDA
constrained PDA
28
[Clemente, L. 2015]
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Complexity of non-emptiness
prefix rewriting
CFG
dense-timed PDA with uninitialized clocks
[Abdulla, Atig, Stenman 2012]
PDA with finite stack alphabet
PDA
constrained PDA in NEXPTIME
28
[Clemente, L. 2015]
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Complexity of non-emptiness
prefix rewriting
CFG
dense-timed PDA with uninitialized clocks
[Abdulla, Atig, Stenman 2012]
PDA with finite stack alphabet
PDA
constrained PDA
EXPTIME-c.
in NEXPTIME
28
[Clemente, L. 2015]
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undecidable
Complexity of non-emptiness
prefix rewriting
CFG
dense-timed PDA with uninitialized clocks
[Abdulla, Atig, Stenman 2012]
PDA with finite stack alphabet
PDA
constrained PDA
EXPTIME-c.
in NEXPTIME
28
[Clemente, L. 2015]
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undecidable
Complexity of non-emptiness
prefix rewriting
CFG
dense-timed PDA with uninitialized clocks
[Abdulla, Atig, Stenman 2012]
PDA with finite stack alphabet
PDA
constrained PDA
EXPTIME-c.
in NEXPTIME
EXPT
IME-c
.
28
[Clemente, L. 2015]
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undecidable
Complexity of non-emptiness
prefix rewriting
CFG
dense-timed PDA with uninitialized clocks
[Abdulla, Atig, Stenman 2012]
PDA with finite stack alphabet
PDA
constrained PDA
EXPTIME-c.
in NEXPTIME
EXPT
IME-c
.
?
28
[Clemente, L. 2015]
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Plan
• Motivation
• FO-definable NFA
• FO-definable PDA
• The core problem: equations over sets of integers
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orbit-finitesetofsymbolsS
The core problemSystems of equations over sets of integers
where right-hand sides use:{x1 = t1
x2 = t2
. . .
xn = tn
30
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orbit-finitesetofsymbolsS
The core problemSystems of equations over sets of integers
• constants {-1}, {0}, {1}where right-hand sides use:{x1 = t1
x2 = t2
. . .
xn = tn
30
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orbit-finitesetofsymbolsS
The core problemSystems of equations over sets of integers
• constants {-1}, {0}, {1}• set union ∪
where right-hand sides use:{x1 = t1
x2 = t2
. . .
xn = tn
30
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orbit-finitesetofsymbolsS
The core problemSystems of equations over sets of integers
• constants {-1}, {0}, {1}• set union ∪• point-wise addition +
where right-hand sides use:{x1 = t1
x2 = t2
. . .
xn = tn
30
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orbit-finitesetofsymbolsS
The core problemSystems of equations over sets of integers
• constants {-1}, {0}, {1}• set union ∪• point-wise addition + • limited intersection ∩
where right-hand sides use:{x1 = t1
x2 = t2
. . .
xn = tn
30
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orbit-finitesetofsymbolsS
The core problemSystems of equations over sets of integers
• constants {-1}, {0}, {1}• set union ∪• point-wise addition + • limited intersection ∩
where right-hand sides use:{x1 = t1
x2 = t2
. . .
xn = tn
for instance:
x1 = {0} [ x2 + {1} [ x2 + {�1}x2 = x1 + {1} [ x1 + {�1}{
30
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orbit-finitesetofsymbolsS
The core problemSystems of equations over sets of integers
• constants {-1}, {0}, {1}• set union ∪• point-wise addition + • limited intersection ∩
where right-hand sides use:{x1 = t1
x2 = t2
. . .
xn = tn
for instance:
x1 = {0} [ x2 + {1} [ x2 + {�1}x2 = x1 + {1} [ x1 + {�1}{
What is the least solution with respect to inclusion?
30
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orbit-finitesetofsymbolsS
The core problem - no intersectionsGiven a systems of equations
decide, whether its least solution assigns a non-empty set to ?x1
{x1 = t1
x2 = t2
. . .
xn = tn
• constants {-1}, {0}, {1}• set union ∪• point-wise addition + • limited intersection ∩
31
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orbit-finitesetofsymbolsS
The core problem - no intersectionsGiven a systems of equations
decide, whether its least solution assigns a non-empty set to ?x1
{x1 = t1
x2 = t2
. . .
xn = tn
• constants {-1}, {0}, {1}• set union ∪• point-wise addition + • limited intersection ∩
How to solve the problem in absence of intersections?
31
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orbit-finitesetofsymbolsS
The core problem - no intersectionsGiven a systems of equations
decide, whether its least solution assigns a non-empty set to ?x1
{x1 = t1
x2 = t2
. . .
xn = tn
• constants {-1}, {0}, {1}• set union ∪• point-wise addition + • limited intersection ∩
How to solve the problem in absence of intersections?
x1 = {0} [ x2 + {1} [ x2 + {�1}x2 = x1 + {1} [ x1 + {�1}{
31
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orbit-finitesetofsymbolsS
The core problem - no intersectionsGiven a systems of equations
decide, whether its least solution assigns a non-empty set to ?x1
{x1 = t1
x2 = t2
. . .
xn = tn
• constants {-1}, {0}, {1}• set union ∪• point-wise addition + • limited intersection ∩
How to solve the problem in absence of intersections?
x1 = {0} [ x2 + {1} [ x2 + {�1}x2 = x1 + {1} [ x1 + {�1}{
Decidable in P
31
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orbit-finitesetofsymbolsS
The core problem - intersectionsGiven a systems of equations
decide, whether its least solution assigns a non-empty set to ?x1
{x1 = t1
x2 = t2
. . .
xn = tn
• constants {-1}, {0}, {1}• set union ∪• point-wise addition + • limited intersection ∩
32
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orbit-finitesetofsymbolsS
The core problem - intersectionsGiven a systems of equations
decide, whether its least solution assigns a non-empty set to ?x1
{x1 = t1
x2 = t2
. . .
xn = tn
The problem is undecidable for unlimited intersections. [Jeż, Okhotin 2010]
• constants {-1}, {0}, {1}• set union ∪• point-wise addition + • limited intersection ∩
32
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orbit-finitesetofsymbolsS
The core problem - limited intersectionGiven a systems of equations
decide, whether its least solution assigns a non-empty set to ?x1
{x1 = t1
x2 = t2
. . .
xn = tn
• constants {-1}, {0}, {1}• set union ∪• point-wise addition + • limited intersection ∩
33
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orbit-finitesetofsymbolsS
The core problem - limited intersectionGiven a systems of equations
decide, whether its least solution assigns a non-empty set to ?x1
{x1 = t1
x2 = t2
. . .
xn = tn
What about limited intersections: _ ∩ I, for I a finite interval?
• constants {-1}, {0}, {1}• set union ∪• point-wise addition + • limited intersection ∩
33
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orbit-finitesetofsymbolsS
The core problem - limited intersectionGiven a systems of equations
decide, whether its least solution assigns a non-empty set to ?x1
{x1 = t1
x2 = t2
. . .
xn = tn
What about limited intersections: _ ∩ I, for I a finite interval?
{x1 = {0} [ x2 + {1} [ x2 + {�1}x2 = (x1 + {1} [ x1 + {�1}) \ {1}
• constants {-1}, {0}, {1}• set union ∪• point-wise addition + • limited intersection ∩
33
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orbit-finitesetofsymbolsS
The core problem - limited intersectionGiven a systems of equations
decide, whether its least solution assigns a non-empty set to ?x1
{x1 = t1
x2 = t2
. . .
xn = tn
What about limited intersections: _ ∩ I, for I a finite interval?
{x1 = {0} [ x2 + {1} [ x2 + {�1}x2 = x1 + {1} [ x1 + {�1}
• constants {-1}, {0}, {1}• set union ∪• point-wise addition + • limited intersection ∩
membership problem
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orbit-finitesetofsymbolsS
The core problem - limited intersectionGiven a systems of equations
decide, whether its least solution assigns a non-empty set to ?x1
{x1 = t1
x2 = t2
. . .
xn = tn
What about limited intersections: _ ∩ I, for I a finite interval?
{x1 = {0} [ x2 + {1} [ x2 + {�1}x2 = {1}
• constants {-1}, {0}, {1}• set union ∪• point-wise addition + • limited intersection ∩
33
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orbit-finitesetofsymbolsS
Given a systems of equations
decide, whether its least solution assigns a non-empty set to ?x1
{x1 = t1
x2 = t2
. . .
xn = tn
What about limited intersections: _ ∩ I, for I a finite interval?
• constants {-1}, {0}, {1}• set union ∪• point-wise addition + • limited intersection ∩
The core problem - limited intersection
34
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orbit-finitesetofsymbolsS
Given a systems of equations
decide, whether its least solution assigns a non-empty set to ?x1
{x1 = t1
x2 = t2
. . .
xn = tn
What about limited intersections: _ ∩ I, for I a finite interval?
• NP-complete
• constants {-1}, {0}, {1}• set union ∪• point-wise addition + • limited intersection ∩
The core problem - limited intersection
34
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orbit-finitesetofsymbolsS
Given a systems of equations
decide, whether its least solution assigns a non-empty set to ?x1
{x1 = t1
x2 = t2
. . .
xn = tn
What about limited intersections: _ ∩ I, for I a finite interval?
• NP-complete• non-emptiness of constrained FO-definable PDA reduces to
the core problem (with exponential blow-up)
• constants {-1}, {0}, {1}• set union ∪• point-wise addition + • limited intersection ∩
The core problem - limited intersection
34
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orbit-finitesetofsymbolsS
Given a systems of equations
decide, whether its least solution assigns a non-empty set to ?x1
{x1 = t1
x2 = t2
. . .
xn = tn
• constants {-1}, {0}, {1}• set union ∪• point-wise addition + • limited intersection ∩
The core problem - limited intersection
35
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orbit-finitesetofsymbolsS
Given a systems of equations
decide, whether its least solution assigns a non-empty set to ?x1
{x1 = t1
x2 = t2
. . .
xn = tn
What about _ ∩ I, for I an arbitrary interval?
• constants {-1}, {0}, {1}• set union ∪• point-wise addition + • limited intersection ∩
The core problem - limited intersection
35
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orbit-finitesetofsymbolsS
Given a systems of equations
decide, whether its least solution assigns a non-empty set to ?x1
{x1 = t1
x2 = t2
. . .
xn = tn
What about _ ∩ I, for I an arbitrary interval?
• decidability status open!
• constants {-1}, {0}, {1}• set union ∪• point-wise addition + • limited intersection ∩
The core problem - limited intersection
35
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orbit-finitesetofsymbolsS
Given a systems of equations
decide, whether its least solution assigns a non-empty set to ?x1
{x1 = t1
x2 = t2
. . .
xn = tn
What about _ ∩ I, for I an arbitrary interval?
• decidability status open!• non-emptiness of FO-definable PDA reduces to the core problem
(with exponential blow-up)
• constants {-1}, {0}, {1}• set union ∪• point-wise addition + • limited intersection ∩
The core problem - limited intersection
35
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