equality atoms ( A , =) input alphabet: A = A "exactly two different atoms appear" language: number of registers may vary from one location to another Q = A ⁰ ∪ A ⁰ ∪ A ¹ ∪ A ² states: = {init, reject} ∪ A ¹ ∪ A ² transitions: δ : Q × A → Q δ (init, a) = (a) a atom δ ((a), b) = (ab) a ≠ b δ ((a), b) = (a) a = b δ ((ab), c) = reject c ≠ a, b initial state: init accepting states: A ² 7
equality atoms ( A , =) Register automata? Over equality atoms, FO definable NFA slightly generalize register automata (aka finite-memory automata) of [Francez, Kaminsky 1994]: 8
equality atoms ( A , =) Register automata? Over equality atoms, FO definable NFA slightly generalize register automata (aka finite-memory automata) of [Francez, Kaminsky 1994]: • number of registers may vary from one control state to another 8
equality atoms ( A , =) Register automata? Over equality atoms, FO definable NFA slightly generalize register automata (aka finite-memory automata) of [Francez, Kaminsky 1994]: • number of registers may vary from one control state to another • alphabet letters may contain more than one atom 8
equality atoms ( A , =) Register automata? Over equality atoms, FO definable NFA slightly generalize register automata (aka finite-memory automata) of [Francez, Kaminsky 1994]: • number of registers may vary from one control state to another • alphabet letters may contain more than one atom • arbitrary FO constraints on register valuations and transitions 8
equality atoms ( A , =) Register automata? Over equality atoms, FO definable NFA slightly generalize register automata (aka finite-memory automata) of [Francez, Kaminsky 1994]: • number of registers may vary from one control state to another • alphabet letters may contain more than one atom • arbitrary FO constraints on register valuations and transitions • instead of (finite set) × A , disjoint union A ∪ A ∪ ... 8
FO definable Turing machines [Boja ń czyk, Klin, L., Toru ń czyk 2013] [Klin, L., Ochremiak, Toru ń czyk 2014] • tape alphabet A • states Q • transitions δ ⊆ Q × A × Q × A × { ← , → , ↓ } • I, F ⊆ Q 9
FO definable Turing machines [Boja ń czyk, Klin, L., Toru ń czyk 2013] [Klin, L., Ochremiak, Toru ń czyk 2014] } • tape alphabet A • states Q FO definable sets instead of finite ones • transitions δ ⊆ Q × A × Q × A × { ← , → , ↓ } • I, F ⊆ Q 9
FO definable Turing machines [Boja ń czyk, Klin, L., Toru ń czyk 2013] [Klin, L., Ochremiak, Toru ń czyk 2014] } • tape alphabet A • states Q FO definable sets instead of finite ones • transitions δ ⊆ Q × A × Q × A × { ← , → , ↓ } • I, F ⊆ Q Acceptance defined as for classical Turing machines. 9
Finite presentation FO definable NFA, Turing machines, PDA, etc. can be finitely presented. 10
Outline • Re-interpreting models of computation in FO definable sets • FO definable PDA • Well-behaved case: oligomorphic and homogeneous atoms • Reachability in FO definable PDA over oligomorphic atoms • Ill-behaved case: time atoms 11
FO-definable PDA • alphabet A • states Q • stack alphabet S • ρ ⊆ Q × S × (A ∪ { ε }) × Q × S* • I, F ⊆ Q 12
FO-definable PDA } • alphabet A • states Q FO definable sets • stack alphabet S instead of finite ones • ρ ⊆ Q × S × (A ∪ { ε }) × Q × S* • I, F ⊆ Q 12
FO-definable PDA } • alphabet A • states Q FO definable sets • stack alphabet S instead of finite ones ≤ n • ρ ⊆ Q × S × (A ∪ { ε }) × Q × S* • I, F ⊆ Q 12
FO-definable PDA } • alphabet A • states Q FO definable sets • stack alphabet S instead of finite ones ≤ n • ρ ⊆ Q × S × (A ∪ { ε }) × Q × S* • I, F ⊆ Q Acceptance defined as for classical PDA, e.g. configurations = Q × S* 12
total order atoms ( Q , <) input alphabet: A = Q "ordered palindromes" language: states: stack alphabet: transitions: initial state: accepting state: 13
total order atoms ( Q , <) input alphabet: A = Q "ordered palindromes" language: Q = {init, finish, acc} states: stack alphabet: transitions: initial state: init acc accepting state: 13
total order atoms ( Q , <) input alphabet: A = Q "ordered palindromes" language: Q = {init, finish, acc} states: stack alphabet: S = Q ∪ { ⊥ } transitions: initial state: init acc accepting state: 13
total order atoms ( Q , <) input alphabet: A = Q "ordered palindromes" language: Q = {init, finish, acc} states: stack alphabet: S = Q ∪ { ⊥ } transitions: δ ⊆ Q × S × (A ∪ { ε }) × Q × (S ⁰ ∪ S ¹ ∪ S ² ) initial state: init acc accepting state: 13
total order atoms ( Q , <) input alphabet: A = Q "ordered palindromes" language: Q = {init, finish, acc} states: stack alphabet: S = Q ∪ { ⊥ } transitions: δ ⊆ Q × S × (A ∪ { ε }) × Q × (S ⁰ ∪ S ¹ ∪ S ² ) init, ⊥ , a init, a ⊥ a atom if in state init , ⊥ is topmost on the stack and atom a is read, stay in state init and push a on the stack initial state: init acc accepting state: 13
total order atoms ( Q , <) input alphabet: A = Q "ordered palindromes" language: Q = {init, finish, acc} states: stack alphabet: S = Q ∪ { ⊥ } transitions: δ ⊆ Q × S × (A ∪ { ε }) × Q × (S ⁰ ∪ S ¹ ∪ S ² ) init, ⊥ , a init, a ⊥ a atom init, b, c init, cb b < c initial state: init acc accepting state: 13
total order atoms ( Q , <) input alphabet: A = Q "ordered palindromes" language: Q = {init, finish, acc} states: stack alphabet: S = Q ∪ { ⊥ } transitions: δ ⊆ Q × S × (A ∪ { ε }) × Q × (S ⁰ ∪ S ¹ ∪ S ² ) init, ⊥ , a init, a ⊥ a atom init, b, c init, cb b < c init, b, ε finish, b b atom init, b, ε finish, ε b atom initial state: init acc accepting state: 13
total order atoms ( Q , <) input alphabet: A = Q "ordered palindromes" language: Q = {init, finish, acc} states: stack alphabet: S = Q ∪ { ⊥ } transitions: δ ⊆ Q × S × (A ∪ { ε }) × Q × (S ⁰ ∪ S ¹ ∪ S ² ) init, ⊥ , a init, a ⊥ a atom init, b, c init, cb b < c init, b, ε finish, b b atom init, b, ε finish, ε b atom finish, b, c finish, ε b = c initial state: init acc accepting state: 13
total order atoms ( Q , <) input alphabet: A = Q "ordered palindromes" language: Q = {init, finish, acc} states: stack alphabet: S = Q ∪ { ⊥ } transitions: δ ⊆ Q × S × (A ∪ { ε }) × Q × (S ⁰ ∪ S ¹ ∪ S ² ) init, ⊥ , a init, a ⊥ a atom init, b, c init, cb b < c init, b, ε finish, b b atom init, b, ε finish, ε b atom finish, b, c finish, ε b = c finish, ⊥ , ε acc , ε initial state: init acc accepting state: 13
equality atoms ( A , =) Pushdown register automata? Over equality atoms, FO definable PDA slightly generalize pushdown register automata of [Murawski, Ramsay, Tzevelekos 2014], exactly like FO definable NFA slightly generalize register automata. 14
FO-definable context-free grammars } • symbols S FO definable sets • terminal symbols A ⊆ S instead of finite ones • an initial symbol • ρ ⊆ (S − A) × S* 15 orbit-finite !set !of !symbols !S
Questions 16 orbit-finite !set !of !symbols !S
Questions • are context-free grammars as expressive as PDA? 16 orbit-finite !set !of !symbols !S
Questions • are context-free grammars as expressive as PDA? • is equivalence of two PDAs decidable? 16 orbit-finite !set !of !symbols !S
Questions • are context-free grammars as expressive as PDA? • is equivalence of two PDAs decidable? • is reachability problem decidable for PDA? 16 orbit-finite !set !of !symbols !S
Questions Under what assumptions on atoms: • are context-free grammars as expressive as PDA? • is equivalence of two PDAs decidable? • is reachability problem decidable for PDA? 16 orbit-finite !set !of !symbols !S
Expressiveness Theorem: [Boja ń czyk, Klin, L. 2014] The following models recognize the same languages: • FO definable context-free grammars • FO definable PDA • FO definable prefix rewriting systems, when A is oligomorphic 17 orbit-finite !set !of !symbols !S
Equivalence-checking Theorem: [Murawski, Ramsay, Tzevelekos 2015] Bisimulation equivalence is undecidable for FO definable PDA over equality atoms. 18 orbit-finite !set !of !symbols !S
Reachability Assumption: From now on assume that FO satisfiability problem in A is decidable. Given : an FO formula over the vocabulary of A Question : is the formula satisfiable in A ? 19 orbit-finite !set !of !symbols !S
Reachability Assumption: From now on assume that FO satisfiability problem in A is decidable. Given : an FO formula over the vocabulary of A Question : is the formula satisfiable in A ? This is necessary but far not enough! Fact: The reachability problem for FO definable NFA over dense-time atoms ( Q , <, +1) is undecidable. 19 orbit-finite !set !of !symbols !S
Outline • Re-interpreting models of computation in FO definable sets • FO definable PDA • Well-behaved case: oligomorphic and homogeneous atoms • Reachability in FO definable PDA over oligomorphic atoms • Ill-behaved case: time atoms 20
Atom automorphisms atoms atom automorphisms equality atoms ( A , =) all bijections total order atoms ( Q , <) monotonic bijections monotonic bijections dense-time atoms ( Q , <, +1) preserving integer differences discrete-time atoms ( Z , <, +1) translations equivalence atoms ( A , R, =) equivalence-preserving bijections random graph ( V , E, =) random graph automorphisms ... ... 21
Orbits Atom automorphisms π act on thus splitting it into orbits. n A π π π 22
Orbits Atom automorphisms π act on thus splitting it into orbits. n A π π π Examples: x ₁ = x ₂ ≠ x ₃ x ₁ < x ₂ < x ₃ x ₁ < x ₂ = x ₃ < x ₁ +1 22
Orbits Atom automorphisms π act on thus splitting it into orbits. n A π π π Examples: Non-examples: x ₁ = x ₂ ≠ x ₃ x ₁ = x ₂ ≠ x ₃ ∨ x ₁ ≠ x ₂ = x ₃ x ₁ < x ₂ ≤ x ₃ x ₁ < x ₂ < x ₃ x ₁ < x ₂ = x ₃ < x ₁ +1 x ₁ < x ₂ ≤ x ₃ ≤ x ₁ +1+1 22
Oligomorphic structures 23
Oligomorphic structures A relational structure A is oligomorphic if 23
Oligomorphic structures A relational structure A is oligomorphic if n for every n, is orbit-finite, i.e. splits into finitely many orbits. A 23
Oligomorphic structures A relational structure A is oligomorphic if n for every n, is orbit-finite, i.e. splits into finitely many orbits. A As a consequence, FO definable sets are orbit-finite. 23
Oligomorphic structures A relational structure A is oligomorphic if n for every n, is orbit-finite, i.e. splits into finitely many orbits. A As a consequence, FO definable sets are orbit-finite. Example: ( Q , <) 23
Oligomorphic structures A relational structure A is oligomorphic if n for every n, is orbit-finite, i.e. splits into finitely many orbits. A As a consequence, FO definable sets are orbit-finite. Example: ( Q , <) Q ² has 3 orbits: 23
Oligomorphic structures A relational structure A is oligomorphic if n for every n, is orbit-finite, i.e. splits into finitely many orbits. A As a consequence, FO definable sets are orbit-finite. Example: ( Q , <) Q ² has 3 orbits: • { (x, y) : x < y } • { (x, y) : x = y } • { (x, y) : x > y } 23
Oligomorphic structures A relational structure A is oligomorphic if n for every n, is orbit-finite, i.e. splits into finitely many orbits. A As a consequence, FO definable sets are orbit-finite. Example: ( Q , <) Q ² has 3 orbits: • { (x, y) : x < y } • { (x, y) : x = y } • { (x, y) : x > y } Q ³ has 13 orbits 23
Homogeneous structures 24
Homogeneous structures A relational structure A is homogeneous if 24
Homogeneous structures A relational structure A is homogeneous if every isomorphism of finite induced substructures of A extends to an automorphism of the whole structure. 24
Homogeneous structures A relational structure A is homogeneous if every isomorphism of finite induced substructures of A extends to an automorphism of the whole structure. Example: ( Q , ≤ ) 24
Homogeneous structures A relational structure A is homogeneous if every isomorphism of finite induced substructures of A extends to an automorphism of the whole structure. Example: ( Q , ≤ ) 24
Homogeneous structures A relational structure A is homogeneous if every isomorphism of finite induced substructures of A extends to an automorphism of the whole structure. Example: ( Q , ≤ ) 24
Homogeneous structures A relational structure A is homogeneous if every isomorphism of finite induced substructures of A extends to an automorphism of the whole structure. Example: ( Q , ≤ ) 24
Homogeneous structures A relational structure A is homogeneous if every isomorphism of finite induced substructures of A extends to an automorphism of the whole structure. Example: ( Q , ≤ ) 24
Homogeneous structures A relational structure A is homogeneous if every isomorphism of finite induced substructures of A extends to an automorphism of the whole structure. Example: ( Q , ≤ ) 24
Homogeneous structures A relational structure A is homogeneous if every isomorphism of finite induced substructures of A extends to an automorphism of the whole structure. Example: ( Q , ≤ ) 24
Homogeneous structures A relational structure A is homogeneous if every isomorphism of finite induced substructures of A extends to an automorphism of the whole structure. Example: ( Q , ≤ ) Theorem: [Freisse 1953] A homogeneous structure is uniquely determined by its finite induced substructures (age). 24
Homogeneous structures 25
Homogeneous structures equality atoms ( A , =) total order atoms ( Q , <) dense-time atoms ( Q , <, +1) Z 25
Homogeneous structures equality atoms ( A , =) total order atoms ( Q , <) dense-time atoms ( Q , <, +1) Z 25
Homogeneous structures equality atoms ( A , =) total order atoms ( Q , <) dense-time atoms ( Q , <, +1) discrete-time atoms ( Z , <, +1) 25
Homogeneous structures equality atoms ( A , =) total order atoms ( Q , <) dense-time atoms ( Q , <, +1) discrete-time atoms ( Z , <, +1) equivalence atoms universal (random) graph universal partial order universal directed graph universal tournament ... 25
Homogeneous is oligomorphic Theorem: Every homogeneous relational structure is oligomorphic 26
Homogeneous is oligomorphic Theorem: Every homogeneous relational structure is oligomorphic Proof: 26
Homogeneous is oligomorphic Theorem: Every homogeneous relational structure is oligomorphic Proof: Theorem: Homogeneous = oligomorphic + quantifier elimination 26
Recommend
More recommend
