Unambiguous Tree Languages are Topologically Harder ... - MIMUW
Unambiguous Tree Languages are Topologically Harder Than Deterministic Ones Szczepan Hummel
University of Warsaw
GAMES 2012 & GandALF 2012, Naples, Italy
Context Domain: ●
a
trees – infinite – labelled – binary
b b b
a b
a a a
a
b
a a b
a
Formalism: ●
Parity Tree Automata
Complexity Measure: ●
Topology (Descriptive Set Theory) – Borel+Projective Hierarchies
Why? ● ● ●
deeper understanding separating classes (definability bounds) extending toolbox
Szczepan Hummel
Unambiguous Tree Languages are Topologically Harder Than Deterministic Ones
2/43
Parity Tree Automata ● ● ●
Finite alphabet A Finite set of states Q, initial state q0 q
Transitions:
a q1 ●
q2
Ranks: rank: Q → N
Szczepan Hummel
Unambiguous Tree Languages are Topologically Harder Than Deterministic Ones
3/43
Parity Tree Automata ● ● ●
Finite alphabet A Finite set of states Q, initial state q0 q
Transitions:
a q1 ●
●
Ranks: rank: Q → N Run ρ:
q2 q1 b
q3 b q1 b
Szczepan Hummel
q0 a q2 a
q1 q2 q2 b a a q2 q3 q2 q1 q3 q2 q3 a a a b a b a
Unambiguous Tree Languages are Topologically Harder Than Deterministic Ones
4/43
Parity Tree Automata ● ● ●
Finite alphabet A Finite set of states Q, initial state q0 q
Transitions:
a q1 ●
●
Ranks: rank: Q → N Run ρ:
q2 q1 b
q3 b q1 b
●
q0 a q2 a
q1 q2 q2 b a a q2 q3 q2 q1 q3 q2 q3 a a a b a b a
Acceptance: ● On each branch Parity Condition holds, i.e. limsup rank(ρ) is even
Szczepan Hummel
Unambiguous Tree Languages are Topologically Harder Than Deterministic Ones
5/43
Determinism & Unambiguity Nondeterministic Parity Tree Automaton ● as described above
Szczepan Hummel
Unambiguous Tree Languages are Topologically Harder Than Deterministic Ones
6/43
Determinism & Unambiguity Nondeterministic Parity Tree Automaton ● as described above
(Top-Down) Deterministic Parity Tree Automaton ● For each q and a, there is only one transition of the form
q a q1
Szczepan Hummel
Unambiguous Tree Languages are Topologically Harder Than Deterministic Ones
q2
7/43
Determinism & Unambiguity Nondeterministic Parity Tree Automaton ● as described above
Unambiguous Parity Tree Automaton ● ≤1 accepting run on each input – tradeoff between efficiency and expressivity
(Top-Down) Deterministic Parity Tree Automaton ● For each q and a, there is only one transition of the form
q a q1
Szczepan Hummel
Unambiguous Tree Languages are Topologically Harder Than Deterministic Ones
q2
8/43
Determinism & Unambiguity
Nondeterministic Parity Tree Automaton ● as described above
Unambiguous Parity Tree Automaton ● ≤1 accepting run on each input – tradeoff between efficiency and expressivity
(Top-Down) Deterministic Parity Tree Automaton ● For each q and a, there is only one transition of the form
q a q1
Szczepan Hummel
Unambiguous Tree Languages are Topologically Harder Than Deterministic Ones
q2
9/43
Topology ●
Topological Space: ● TA: trees over (finite) alphabet A ●
Basic open sets: fixed finite tree prefix (initial part of trees) – e.g. set of all trees starting with b b
●
a
Open sets: (countable) unions of such basic sets – e.g. set of all trees containing label b
Szczepan Hummel
Unambiguous Tree Languages are Topologically Harder Than Deterministic Ones
10/43
Topology ●
Topological Space: ● TA: trees over (finite) alphabet A ●
Basic open sets: fixed finite tree prefix (initial part of trees) – e.g. set of all trees starting with b b
●
●
a
Open sets: (countable) unions of such basic sets – e.g. set of all trees containing label b Borel sets: open sets + complements + countable unions
Szczepan Hummel
Unambiguous Tree Languages are Topologically Harder Than Deterministic Ones
11/43
Topology ●
Topological Space: ● TA: trees over (finite) alphabet A ●
Basic open sets: fixed finite tree prefix (initial part of trees) – e.g. set of all trees starting with b b
●
●
●
a
Open sets: (countable) unions of such basic sets – e.g. set of all trees containing label b Borel sets: open sets + complements + countable unions 1 1
Analytic sets (Σ ): projections of Borel sets
Szczepan Hummel
Unambiguous Tree Languages are Topologically Harder Than Deterministic Ones
12/43
Topology ●
Topological Space: ● TA: trees over (finite) alphabet A ●
Basic open sets: fixed finite tree prefix (initial part of trees) – e.g. set of all trees starting with b b
●
●
a
Open sets: (countable) unions of such basic sets – e.g. set of all trees containing label b Borel sets: open sets + complements + countable unions 1 1
●
Analytic sets (Σ ): projections of Borel sets
●
Coanalytic sets (Π1 ): complements of analytic sets
Szczepan Hummel
1
Unambiguous Tree Languages are Topologically Harder Than Deterministic Ones
13/43
Topology ●
Topological Space: ● TA: trees over (finite) alphabet A ●
Basic open sets: fixed finite tree prefix (initial part of trees) – e.g. set of all trees starting with b b
●
●
a
Open sets: (countable) unions of such basic sets – e.g. set of all trees containing label b Borel sets: open sets + complements + countable unions 1 1
●
Analytic sets (Σ ): projections of Borel sets
●
Coanalytic sets (Π1 ): complements of analytic sets
●
1
s ion t c je pro
Projective Hierarchy goes up: 1
Π2 1
Π1 Szczepan Hummel
1
complements s ion t c je pro
Σ2
complements
Σ1
1
Unambiguous Tree Languages are Topologically Harder Than Deterministic Ones
14/43
Results Π12
Σ12
Π11
Σ11
Bor
Szczepan Hummel
Unambiguous Tree Languages are Topologically Harder Than Deterministic Ones
15/43
Results Π12
Δ12
Π11
Σ12
Σ11
Bor
Szczepan Hummel
Unambiguous Tree Languages are Topologically Harder Than Deterministic Ones
16/43
Results Π12
Δ12
Σ12
σ(Σ11 )
Π11
Σ11
Bor
Szczepan Hummel
Unambiguous Tree Languages are Topologically Harder Than Deterministic Ones
17/43
Results Π12
Δ12
Σ12
Reg σ(Σ11 )
Π11
Σ11
Bor
Szczepan Hummel
Unambiguous Tree Languages are Topologically Harder Than Deterministic Ones
18/43
Results Δ12
Π12
Σ12
Reg σ(Σ11 )
1 1
Π
Det
Σ11
Bor
Szczepan Hummel
Unambiguous Tree Languages are Topologically Harder Than Deterministic Ones
19/43
Results Δ12
Π12
Σ12
det
∞
∀ path∀ a
Reg σ(Σ11 )
1 1
Π
Det
Σ11
Bor
Szczepan Hummel
Unambiguous Tree Languages are Topologically Harder Than Deterministic Ones
20/43
Results Δ12
Π12
Σ12
∞
∃! path∃ a
Reg
unambiguous det
∞
∀ path∀ a
σ(Σ11 )
1 1
Π
Det
Σ11
Bor
Szczepan Hummel
Unambiguous Tree Languages are Topologically Harder Than Deterministic Ones
21/43
Results Δ12
Π12
Σ12
∞
∃! path∃ a G
Reg
unambiguous det
∞
∀ path∀ a
σ(Σ11 )
1 1
Π
Det
Σ11
Bor
Szczepan Hummel
Unambiguous Tree Languages are Topologically Harder Than Deterministic Ones
22/43
Results Δ12
Π12
Σ12
∞
∃! path∃ a G
Reg
σ(G)
σ(Σ11 )
unambiguous det
∞
∀ path∀ a
Unamb 1 1
Π
Det
Σ11
Bor
Szczepan Hummel
Unambiguous Tree Languages are Topologically Harder Than Deterministic Ones
23/43
Results Δ12
Π12
Σ12
∞
∃! path∃ a G
Reg
σ(G)
σ(Σ11 )
unambiguous det
∞
∀ path∀ a
Unamb 1 1
Π
Det
Σ11
Bor
Szczepan Hummel
Unambiguous Tree Languages are Topologically Harder Than Deterministic Ones
24/43
Analytic-Complete Set First attempt: ∃ path∃∞ a – a well known analytic-complete set ●
recognise unambiguously
Szczepan Hummel
Unambiguous Tree Languages are Topologically Harder Than Deterministic Ones
25/43
Analytic-Complete Set First attempt: ∃ path∃∞ a – a well known analytic-complete set ● ●
recognise unambiguously which branch to choose?
Szczepan Hummel
Unambiguous Tree Languages are Topologically Harder Than Deterministic Ones
26/43
Analytic-Complete Set First attempt: ∃ path∃∞ a – a well known analytic-complete set ● ● ●
recognise unambiguously which branch to choose? left-most?
Szczepan Hummel
Unambiguous Tree Languages are Topologically Harder Than Deterministic Ones
27/43
Analytic-Complete Set First attempt: ∃ path∃∞ a – a well known analytic-complete set ● ● ●
recognise unambiguously which branch to choose? left-most? – may not exist b b b
a
b
a a a
Szczepan Hummel
Unambiguous Tree Languages are Topologically Harder Than Deterministic Ones
28/43
Analytic-Complete Set First attempt: ∃ path∃∞ a – a well known analytic-complete set ● ● ●
recognise unambiguously which branch to choose? right-most?
Szczepan Hummel
Unambiguous Tree Languages are Topologically Harder Than Deterministic Ones
29/43
Analytic-Complete Set First attempt: ∃ path∃∞ a – a well known analytic-complete set ● ● ●
recognise unambiguously which branch to choose? right-most? – may not exist
b b b
a
b
a a a
Szczepan Hummel
Unambiguous Tree Languages are Topologically Harder Than Deterministic Ones
30/43
Analytic-Complete Set First attempt: ∃ path∃∞ a – a well known analytic-complete set ● ●
●
recognise unambiguously which branch to choose?
ambiguous! (not-unambiguous) –
Szczepan Hummel
impossible to a define choice function in MSO [Gurevich, Shelah '96] [Carayol, Löding, Niwiński, Walukiewicz '10]
Unambiguous Tree Languages are Topologically Harder Than Deterministic Ones
31/43
Analytic-Complete Unambiguous Set G G = there exists a branch – –
labelled only with a's turning left ∞ many times
good branch
good tree
Proposition 1: If there is a good branch there is the left-most good branch.
Szczepan Hummel
Unambiguous Tree Languages are Topologically Harder Than Deterministic Ones
32/43
Analytic-Complete Unambiguous Set G G = there exists a branch – –
labelled only with a's turning left ∞ many times
good branch
good tree
Proposition 1: If there is a good branch there is the left-most good branch. Construction: a
good
Szczepan Hummel
Unambiguous Tree Languages are Topologically Harder Than Deterministic Ones
33/43
Analytic-Complete Unambiguous Set G G = there exists a branch – –
labelled only with a's turning left ∞ many times
good branch
good tree
Proposition 1: If there is a good branch there is the left-most good branch. Construction: a a
bad
Szczepan Hummel
a
good
Unambiguous Tree Languages are Topologically Harder Than Deterministic Ones
34/43
Analytic-Complete Unambiguous Set G G = there exists a branch – –
labelled only with a's turning left ∞ many times
good branch
good tree
Proposition 1: If there is a good branch there is the left-most good branch. Construction: a a b
a
bad
Szczepan Hummel
a
good
Unambiguous Tree Languages are Topologically Harder Than Deterministic Ones
35/43
Analytic-Complete Unambiguous Set G G = there exists a branch – –
labelled only with a's turning left ∞ many times
good branch
good tree
Proposition 1: If there is a good branch there is the left-most good branch. Construction: a a b
a a
a
a
good
Szczepan Hummel
Unambiguous Tree Languages are Topologically Harder Than Deterministic Ones
36/43
Analytic-Complete Unambiguous Set G G = there exists a branch – –
labelled only with a's turning left ∞ many times
good branch
good tree
Proposition 1: If there is a good branch there is the left-most good branch. Construction: a a b
a a
a
a a
Szczepan Hummel
b
Unambiguous Tree Languages are Topologically Harder Than Deterministic Ones
37/43
Analytic-Complete Unambiguous Set G G = there exists a branch – –
labelled only with a's turning left ∞ many times
good branch
good tree
Proposition 1: If there is a good branch there is the left-most good branch. → unambiguous: ● show a good branch ● show that none branch to the left is good Note: G is deterministic
Szczepan Hummel
Unambiguous Tree Languages are Topologically Harder Than Deterministic Ones
38/43
Analytic-Complete Unambiguous Set G G = there exists a branch – –
labelled only with a's turning left ∞ many times
good branch
good tree
Proposition 1: If there is a good branch there is the left-most good branch. → unambiguous: ● show a good branch ● show that none branch to the left is good Note: G is deterministic ● Analytic-complete ● Continously reduce set IF of ill-founded trees to G Szczepan Hummel
Unambiguous Tree Languages are Topologically Harder Than Deterministic Ones
39/43
Even Higher σ(G) – combines G and G ● trees over {a,b,,}
Szczepan Hummel
Unambiguous Tree Languages are Topologically Harder Than Deterministic Ones
40/43
Even Higher σ(G) – combines G and G ● trees over {a,b,,} 1 ● topologically harder than any set in σ(Σ ) 1 ● class of sets reducible to σ(G) is closed under: –
countable union
⋁
Uxi: complement X: t
Szczepan Hummel
f
t
⋁
f
⌙
–
⋁
⋁
f0(t) f1(t)
f2(t)
f3(t) f(t)
Unambiguous Tree Languages are Topologically Harder Than Deterministic Ones
41/43
Even Higher σ(G) – combines G and G ● trees over {a,b,,} 1 ● topologically harder than any set in σ(Σ ) 1 ● class of sets reducible to σ(G) is closed under: –
countable union
⋁
Uxi:
●
complement X: t
f
t
⋁
f
⌙
–
⋁
⋁
f0(t) f1(t)
f2(t)
f3(t) f(t)
unambigous ● choose first possible turning right on branch 1 ● use unambiguous aut. for G for Σ1 atoms 1 ● use deterministic aut. for G for Π1 atoms of the formula
Szczepan Hummel
Unambiguous Tree Languages are Topologically Harder Than Deterministic Ones
42/43
References Main result (set G) = [NW'03] [Bi'10] [NW'03] D. Niwiński, I. Walukiewicz. A Gap Property of Deterministic Tree Languages. TCS 2003 [Bi'10] M. Bilkowski. Unambiguous Complements of Deterministic Languages. Unpublished 2010 [CLNW'10] A. Carayol, C. Löding, D. Niwiński, I. Walukiewicz. Choice Functions and Well Orderings Over the Infinite Binary Tree. CEJM 2010 [Co'12] T. Colcombet. Forms of Determinism for Automata. STACS 2012 [FS'109] O. Finkel, P. Simonnet. On Recognizable Tree Languages Beyond the Borel Hierarchy. FI 2009 Szczepan Hummel
Unambiguous Tree Languages are Topologically Harder Than Deterministic Ones
43/43
University of Warsaw
GAMES 2012 & GandALF 2012, Naples, Italy
Context Domain: ●
a
trees – infinite – labelled – binary
b b b
a b
a a a
a
b
a a b
a
Formalism: ●
Parity Tree Automata
Complexity Measure: ●
Topology (Descriptive Set Theory) – Borel+Projective Hierarchies
Why? ● ● ●
deeper understanding separating classes (definability bounds) extending toolbox
Szczepan Hummel
Unambiguous Tree Languages are Topologically Harder Than Deterministic Ones
2/43
Parity Tree Automata ● ● ●
Finite alphabet A Finite set of states Q, initial state q0 q
Transitions:
a q1 ●
q2
Ranks: rank: Q → N
Szczepan Hummel
Unambiguous Tree Languages are Topologically Harder Than Deterministic Ones
3/43
Parity Tree Automata ● ● ●
Finite alphabet A Finite set of states Q, initial state q0 q
Transitions:
a q1 ●
●
Ranks: rank: Q → N Run ρ:
q2 q1 b
q3 b q1 b
Szczepan Hummel
q0 a q2 a
q1 q2 q2 b a a q2 q3 q2 q1 q3 q2 q3 a a a b a b a
Unambiguous Tree Languages are Topologically Harder Than Deterministic Ones
4/43
Parity Tree Automata ● ● ●
Finite alphabet A Finite set of states Q, initial state q0 q
Transitions:
a q1 ●
●
Ranks: rank: Q → N Run ρ:
q2 q1 b
q3 b q1 b
●
q0 a q2 a
q1 q2 q2 b a a q2 q3 q2 q1 q3 q2 q3 a a a b a b a
Acceptance: ● On each branch Parity Condition holds, i.e. limsup rank(ρ) is even
Szczepan Hummel
Unambiguous Tree Languages are Topologically Harder Than Deterministic Ones
5/43
Determinism & Unambiguity Nondeterministic Parity Tree Automaton ● as described above
Szczepan Hummel
Unambiguous Tree Languages are Topologically Harder Than Deterministic Ones
6/43
Determinism & Unambiguity Nondeterministic Parity Tree Automaton ● as described above
(Top-Down) Deterministic Parity Tree Automaton ● For each q and a, there is only one transition of the form
q a q1
Szczepan Hummel
Unambiguous Tree Languages are Topologically Harder Than Deterministic Ones
q2
7/43
Determinism & Unambiguity Nondeterministic Parity Tree Automaton ● as described above
Unambiguous Parity Tree Automaton ● ≤1 accepting run on each input – tradeoff between efficiency and expressivity
(Top-Down) Deterministic Parity Tree Automaton ● For each q and a, there is only one transition of the form
q a q1
Szczepan Hummel
Unambiguous Tree Languages are Topologically Harder Than Deterministic Ones
q2
8/43
Determinism & Unambiguity
Nondeterministic Parity Tree Automaton ● as described above
Unambiguous Parity Tree Automaton ● ≤1 accepting run on each input – tradeoff between efficiency and expressivity
(Top-Down) Deterministic Parity Tree Automaton ● For each q and a, there is only one transition of the form
q a q1
Szczepan Hummel
Unambiguous Tree Languages are Topologically Harder Than Deterministic Ones
q2
9/43
Topology ●
Topological Space: ● TA: trees over (finite) alphabet A ●
Basic open sets: fixed finite tree prefix (initial part of trees) – e.g. set of all trees starting with b b
●
a
Open sets: (countable) unions of such basic sets – e.g. set of all trees containing label b
Szczepan Hummel
Unambiguous Tree Languages are Topologically Harder Than Deterministic Ones
10/43
Topology ●
Topological Space: ● TA: trees over (finite) alphabet A ●
Basic open sets: fixed finite tree prefix (initial part of trees) – e.g. set of all trees starting with b b
●
●
a
Open sets: (countable) unions of such basic sets – e.g. set of all trees containing label b Borel sets: open sets + complements + countable unions
Szczepan Hummel
Unambiguous Tree Languages are Topologically Harder Than Deterministic Ones
11/43
Topology ●
Topological Space: ● TA: trees over (finite) alphabet A ●
Basic open sets: fixed finite tree prefix (initial part of trees) – e.g. set of all trees starting with b b
●
●
●
a
Open sets: (countable) unions of such basic sets – e.g. set of all trees containing label b Borel sets: open sets + complements + countable unions 1 1
Analytic sets (Σ ): projections of Borel sets
Szczepan Hummel
Unambiguous Tree Languages are Topologically Harder Than Deterministic Ones
12/43
Topology ●
Topological Space: ● TA: trees over (finite) alphabet A ●
Basic open sets: fixed finite tree prefix (initial part of trees) – e.g. set of all trees starting with b b
●
●
a
Open sets: (countable) unions of such basic sets – e.g. set of all trees containing label b Borel sets: open sets + complements + countable unions 1 1
●
Analytic sets (Σ ): projections of Borel sets
●
Coanalytic sets (Π1 ): complements of analytic sets
Szczepan Hummel
1
Unambiguous Tree Languages are Topologically Harder Than Deterministic Ones
13/43
Topology ●
Topological Space: ● TA: trees over (finite) alphabet A ●
Basic open sets: fixed finite tree prefix (initial part of trees) – e.g. set of all trees starting with b b
●
●
a
Open sets: (countable) unions of such basic sets – e.g. set of all trees containing label b Borel sets: open sets + complements + countable unions 1 1
●
Analytic sets (Σ ): projections of Borel sets
●
Coanalytic sets (Π1 ): complements of analytic sets
●
1
s ion t c je pro
Projective Hierarchy goes up: 1
Π2 1
Π1 Szczepan Hummel
1
complements s ion t c je pro
Σ2
complements
Σ1
1
Unambiguous Tree Languages are Topologically Harder Than Deterministic Ones
14/43
Results Π12
Σ12
Π11
Σ11
Bor
Szczepan Hummel
Unambiguous Tree Languages are Topologically Harder Than Deterministic Ones
15/43
Results Π12
Δ12
Π11
Σ12
Σ11
Bor
Szczepan Hummel
Unambiguous Tree Languages are Topologically Harder Than Deterministic Ones
16/43
Results Π12
Δ12
Σ12
σ(Σ11 )
Π11
Σ11
Bor
Szczepan Hummel
Unambiguous Tree Languages are Topologically Harder Than Deterministic Ones
17/43
Results Π12
Δ12
Σ12
Reg σ(Σ11 )
Π11
Σ11
Bor
Szczepan Hummel
Unambiguous Tree Languages are Topologically Harder Than Deterministic Ones
18/43
Results Δ12
Π12
Σ12
Reg σ(Σ11 )
1 1
Π
Det
Σ11
Bor
Szczepan Hummel
Unambiguous Tree Languages are Topologically Harder Than Deterministic Ones
19/43
Results Δ12
Π12
Σ12
det
∞
∀ path∀ a
Reg σ(Σ11 )
1 1
Π
Det
Σ11
Bor
Szczepan Hummel
Unambiguous Tree Languages are Topologically Harder Than Deterministic Ones
20/43
Results Δ12
Π12
Σ12
∞
∃! path∃ a
Reg
unambiguous det
∞
∀ path∀ a
σ(Σ11 )
1 1
Π
Det
Σ11
Bor
Szczepan Hummel
Unambiguous Tree Languages are Topologically Harder Than Deterministic Ones
21/43
Results Δ12
Π12
Σ12
∞
∃! path∃ a G
Reg
unambiguous det
∞
∀ path∀ a
σ(Σ11 )
1 1
Π
Det
Σ11
Bor
Szczepan Hummel
Unambiguous Tree Languages are Topologically Harder Than Deterministic Ones
22/43
Results Δ12
Π12
Σ12
∞
∃! path∃ a G
Reg
σ(G)
σ(Σ11 )
unambiguous det
∞
∀ path∀ a
Unamb 1 1
Π
Det
Σ11
Bor
Szczepan Hummel
Unambiguous Tree Languages are Topologically Harder Than Deterministic Ones
23/43
Results Δ12
Π12
Σ12
∞
∃! path∃ a G
Reg
σ(G)
σ(Σ11 )
unambiguous det
∞
∀ path∀ a
Unamb 1 1
Π
Det
Σ11
Bor
Szczepan Hummel
Unambiguous Tree Languages are Topologically Harder Than Deterministic Ones
24/43
Analytic-Complete Set First attempt: ∃ path∃∞ a – a well known analytic-complete set ●
recognise unambiguously
Szczepan Hummel
Unambiguous Tree Languages are Topologically Harder Than Deterministic Ones
25/43
Analytic-Complete Set First attempt: ∃ path∃∞ a – a well known analytic-complete set ● ●
recognise unambiguously which branch to choose?
Szczepan Hummel
Unambiguous Tree Languages are Topologically Harder Than Deterministic Ones
26/43
Analytic-Complete Set First attempt: ∃ path∃∞ a – a well known analytic-complete set ● ● ●
recognise unambiguously which branch to choose? left-most?
Szczepan Hummel
Unambiguous Tree Languages are Topologically Harder Than Deterministic Ones
27/43
Analytic-Complete Set First attempt: ∃ path∃∞ a – a well known analytic-complete set ● ● ●
recognise unambiguously which branch to choose? left-most? – may not exist b b b
a
b
a a a
Szczepan Hummel
Unambiguous Tree Languages are Topologically Harder Than Deterministic Ones
28/43
Analytic-Complete Set First attempt: ∃ path∃∞ a – a well known analytic-complete set ● ● ●
recognise unambiguously which branch to choose? right-most?
Szczepan Hummel
Unambiguous Tree Languages are Topologically Harder Than Deterministic Ones
29/43
Analytic-Complete Set First attempt: ∃ path∃∞ a – a well known analytic-complete set ● ● ●
recognise unambiguously which branch to choose? right-most? – may not exist
b b b
a
b
a a a
Szczepan Hummel
Unambiguous Tree Languages are Topologically Harder Than Deterministic Ones
30/43
Analytic-Complete Set First attempt: ∃ path∃∞ a – a well known analytic-complete set ● ●
●
recognise unambiguously which branch to choose?
ambiguous! (not-unambiguous) –
Szczepan Hummel
impossible to a define choice function in MSO [Gurevich, Shelah '96] [Carayol, Löding, Niwiński, Walukiewicz '10]
Unambiguous Tree Languages are Topologically Harder Than Deterministic Ones
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Analytic-Complete Unambiguous Set G G = there exists a branch – –
labelled only with a's turning left ∞ many times
good branch
good tree
Proposition 1: If there is a good branch there is the left-most good branch.
Szczepan Hummel
Unambiguous Tree Languages are Topologically Harder Than Deterministic Ones
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Analytic-Complete Unambiguous Set G G = there exists a branch – –
labelled only with a's turning left ∞ many times
good branch
good tree
Proposition 1: If there is a good branch there is the left-most good branch. Construction: a
good
Szczepan Hummel
Unambiguous Tree Languages are Topologically Harder Than Deterministic Ones
33/43
Analytic-Complete Unambiguous Set G G = there exists a branch – –
labelled only with a's turning left ∞ many times
good branch
good tree
Proposition 1: If there is a good branch there is the left-most good branch. Construction: a a
bad
Szczepan Hummel
a
good
Unambiguous Tree Languages are Topologically Harder Than Deterministic Ones
34/43
Analytic-Complete Unambiguous Set G G = there exists a branch – –
labelled only with a's turning left ∞ many times
good branch
good tree
Proposition 1: If there is a good branch there is the left-most good branch. Construction: a a b
a
bad
Szczepan Hummel
a
good
Unambiguous Tree Languages are Topologically Harder Than Deterministic Ones
35/43
Analytic-Complete Unambiguous Set G G = there exists a branch – –
labelled only with a's turning left ∞ many times
good branch
good tree
Proposition 1: If there is a good branch there is the left-most good branch. Construction: a a b
a a
a
a
good
Szczepan Hummel
Unambiguous Tree Languages are Topologically Harder Than Deterministic Ones
36/43
Analytic-Complete Unambiguous Set G G = there exists a branch – –
labelled only with a's turning left ∞ many times
good branch
good tree
Proposition 1: If there is a good branch there is the left-most good branch. Construction: a a b
a a
a
a a
Szczepan Hummel
b
Unambiguous Tree Languages are Topologically Harder Than Deterministic Ones
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Analytic-Complete Unambiguous Set G G = there exists a branch – –
labelled only with a's turning left ∞ many times
good branch
good tree
Proposition 1: If there is a good branch there is the left-most good branch. → unambiguous: ● show a good branch ● show that none branch to the left is good Note: G is deterministic
Szczepan Hummel
Unambiguous Tree Languages are Topologically Harder Than Deterministic Ones
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Analytic-Complete Unambiguous Set G G = there exists a branch – –
labelled only with a's turning left ∞ many times
good branch
good tree
Proposition 1: If there is a good branch there is the left-most good branch. → unambiguous: ● show a good branch ● show that none branch to the left is good Note: G is deterministic ● Analytic-complete ● Continously reduce set IF of ill-founded trees to G Szczepan Hummel
Unambiguous Tree Languages are Topologically Harder Than Deterministic Ones
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Even Higher σ(G) – combines G and G ● trees over {a,b,,}
Szczepan Hummel
Unambiguous Tree Languages are Topologically Harder Than Deterministic Ones
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Even Higher σ(G) – combines G and G ● trees over {a,b,,} 1 ● topologically harder than any set in σ(Σ ) 1 ● class of sets reducible to σ(G) is closed under: –
countable union
⋁
Uxi: complement X: t
Szczepan Hummel
f
t
⋁
f
⌙
–
⋁
⋁
f0(t) f1(t)
f2(t)
f3(t) f(t)
Unambiguous Tree Languages are Topologically Harder Than Deterministic Ones
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Even Higher σ(G) – combines G and G ● trees over {a,b,,} 1 ● topologically harder than any set in σ(Σ ) 1 ● class of sets reducible to σ(G) is closed under: –
countable union
⋁
Uxi:
●
complement X: t
f
t
⋁
f
⌙
–
⋁
⋁
f0(t) f1(t)
f2(t)
f3(t) f(t)
unambigous ● choose first possible turning right on branch 1 ● use unambiguous aut. for G for Σ1 atoms 1 ● use deterministic aut. for G for Π1 atoms of the formula
Szczepan Hummel
Unambiguous Tree Languages are Topologically Harder Than Deterministic Ones
42/43
References Main result (set G) = [NW'03] [Bi'10] [NW'03] D. Niwiński, I. Walukiewicz. A Gap Property of Deterministic Tree Languages. TCS 2003 [Bi'10] M. Bilkowski. Unambiguous Complements of Deterministic Languages. Unpublished 2010 [CLNW'10] A. Carayol, C. Löding, D. Niwiński, I. Walukiewicz. Choice Functions and Well Orderings Over the Infinite Binary Tree. CEJM 2010 [Co'12] T. Colcombet. Forms of Determinism for Automata. STACS 2012 [FS'109] O. Finkel, P. Simonnet. On Recognizable Tree Languages Beyond the Borel Hierarchy. FI 2009 Szczepan Hummel
Unambiguous Tree Languages are Topologically Harder Than Deterministic Ones
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