Generalised Quantifiers on Automatic Structures Sasha Rubin - - PowerPoint PPT Presentation

Generalised Quantifiers on Automatic Structures

Sasha Rubin

rubin@cs.auckland.ac.nz

Department of Computer Science University of Auckland

LSV 2008

Overview

Automatic Presentations Generalised Quantifiers for S-AutStr Generalised Quantifiers for ωS-AutStr Generalised Quantifiers for ωT-AutStr

Automata and Logical Definability

• Fact. For each type of object ♦ P tstring, ω-string, tree, ω-tree✉

there is a notion of synchronous automaton with robust closure properties. Classically

• 1. (W)MSO♣N, Sq Ô automata on (finite) infinite words
• 2. (W)MSO♣t0, 1✉✍, s0, s1q Ô automata on (finite) infinite trees

Weak: variables range over finite subsets of domain.

Automata and Logical Definability

• Fact. For each type of object ♦ P tstring, ω-string, tree, ω-tree✉

there is a notion of synchronous automaton with robust closure properties. Another point of view.

• 1. FO♣t0, 1✉✝, σ0, σ1, ➔prefix, elq Ô automata on finite words.
• 2. FO♣trees, σtl,r✉

t0,1✉, ➔ext, ✑domq Ô automata on infinite trees.

Application: Decidability of Logical Theories.

Automatic Presentations

Let ♦ P tstring, ω-string, tree, ω-tree✉. A ♦-automatic presentation of a relational structure A ✏ ♣A, ♣Riqq consists of

• 1. a tuple of ♦-automata ♣M, ♣Miqq,
• 2. a bijection µ : L♣Mq Ñ A,

so that ♣L♣Mq, ♣L♣Miqqq

µ

✕ A. Say that A is an ♦-automatic structure. [Hodgson 76] [Khoussainov,Nerode 95] [Blumensath, Gr¨ adel 00]

Examples

Real Addition ♣r0, 1q, q P ωS-AutStr

• µ : t0, 1✉ω③t0, 1✉✝1ω Ñ r0, 1q in base 2.

♣ ♣ q ❨ ❳ q P P t ✉ t P ⑤ r s ✏ ✉ ♣ ✂q P ③ ♣ ☎ ☎ ☎ q ➧

Examples

Real Addition ♣r0, 1q, q P ωS-AutStr

• µ : t0, 1✉ω③t0, 1✉✝1ω Ñ r0, 1q in base 2.

Boolean Algebra ♣P♣Nq, ❨, ❳,

cq P ωS-AutStr.

• µ maps α P t0, 1✉ω to tn P N ⑤ αrns ✏ 1✉.

♣ ✂q P ③ ♣ ☎ ☎ ☎ q ➧

Examples

Real Addition ♣r0, 1q, q P ωS-AutStr

• µ : t0, 1✉ω③t0, 1✉✝1ω Ñ r0, 1q in base 2.

Boolean Algebra ♣P♣Nq, ❨, ❳,

cq P ωS-AutStr.

• µ maps α P t0, 1✉ω to tn P N ⑤ αrns ✏ 1✉.

Rational Multiplication ♣Q, ✂q P T-AutStr③S-AutStr

• µ maps ♣u1, ☎ ☎ ☎ , ukq to the number ➧ pni

i

where pi is the ith prime and ui is the integer ni written in base 2.

Open Questions

Is Rational Addition ♣Q, q automatic? Is this Atomless Boolean Algebra ♣P♣Nq, ❨, ❳,

cq④ ✒e automatic?

A ✒e B : if ⑤A△B⑤ ➔ ω Is the free algebra on one generator and one binary operation automatic?

Fundamental Properties

• Theorem. FO definable Ñ regular

Given

• 1. ♦-automatic presentation µ of A
• 2. FO-formula Φ♣xq in the signature of A.

The automatic presentation can be extended to ♣A, ΦAq ie: µ✁1♣ΦAq is regular. (Induction on φ)

• Ex. Every automatic presentation of ♣N, q can be expanded to one

for ♣N, , ✁, ➔, S, 0, 1, ✑rq.

Fundamental Properties

• Theorem. FO definable Ñ regular

Given

• 1. ♦-automatic presentation µ of A
• 2. FO-formula Φ♣xq in the signature of A.

The automatic presentation can be extended to ♣A, ΦAq ie: µ✁1♣ΦAq is regular. Corollary The following problem is decidable: Input: The automata forming an automatic presentation of some structure A, and a FO-sentence σ. Output: Whether or not A ⑤ ù σ. Parameters: No problem in the finite cases. In the ω-cases, as long as they are ultimately-periodic strings / regular trees.

Fundamental Properties

• Theorem. FO definable Ñ regular

Given

• 1. ♦-automatic presentation µ of A
• 2. FO-formula Φ♣xq in the signature of A.

The automatic presentation can be extended to ♣A, ΦAq ie: µ✁1♣ΦAq is regular. Goal Extend to more expressive logics

Overview

Automatic Presentations Generalised Quantifiers for S-AutStr Generalised Quantifiers for ωS-AutStr Generalised Quantifiers for ωT-AutStr

Generalised Quantifiers

Generalised Quantifier Q is a class of structures, over a fixed signature, closed under isomorphism. A ⑤ ù Qx φ♣x, zq :if ♣A, φ♣✁, zqAq P Q Unary Examples.

• ’There exists’: Q ✏ t♣A, Pq ⑤ ❍ ✘ P ⑨ A✉.
• ’Counting quantifiers’: For C ⑨ N ❨ t✽✉, define

QC ✏ t♣A, Pq ⑤ ⑤P⑤ P C✉.

• ’Modulo quantifiers’ ❉mod: Q ✏ t♣A, Pq ⑤ ⑤P⑤ ✑ k mod m✉.

Generalised Quantifiers

Generalised Quantifier Q is a class of structures, over a fixed signature, closed under isomorphism. A ⑤ ù Qx φ♣x, zq :if ♣A, φ♣✁, zqAq P Q Binary Examples.

• ’Connectedness’: Q ✏ t♣A, Eq ⑤ graph is strongly connected✉.
• ’Ramsey’: QRamsey ✏ t♣A, Eq ⑤ ❉ infinite X ⑨ A : X 2 ⑨ E✉.
• Any property of graphs.

Generalised Quantifiers

Generalised Quantifier Q is a class of structures, over a fixed signature, closed under isomorphism. A ⑤ ù Qx φ♣x, zq :if ♣A, φ♣✁, zqAq P Q First order + quantifiers written FO tQi✉

Generalised Quantifiers and Regularity

• Definition. Quantifier Q preserves ♦-regularity (effectively) :if

Given

• ♦-automatic presentation of A,
• FO formula Φ♣x, zq in signature of A,

the automatic presentation can be extended to include the relation defined in A by Qx Φ♣x, zq (and automaton be found effectively).

Quantifiers preserving regularity

Examples on S-AutStr.

• ❉, ❅: standard
• ❉✽: replace

❉✽xφ♣x, zq by ♣❅y❉xq r⑤x⑤ → ⑤y⑤ ❫ φ♣x, zqs.

• ❉k mod m: modified subset construction.

Non-regularity preserving

• Binary reachability quantifier: ♣A, E, s, f q there is a path in

graph ♣A, Eq from s to f (Configuration space of c.e but non-computable set).

• Unary H¨

artig quantifier: ♣A, P, Qq where P, Q ⑨ A and ⑤P⑤ ✏ ⑤Q⑤ (later). ⑤ ù ♣ q ☎ ☎ ☎ ♣ q ♣ ♣✁ q ☎ ☎ ☎ ♣✁ q q P

Non-regularity preserving

• Binary reachability quantifier: ♣A, E, s, f q there is a path in

graph ♣A, Eq from s to f (Configuration space of c.e but non-computable set).

• Unary H¨

artig quantifier: ♣A, P, Qq where P, Q ⑨ A and ⑤P⑤ ✏ ⑤Q⑤ (later). Quantifiers may bind more than one formula (adicity): A ⑤ ù Qx φ1♣x, zq ☎ ☎ ☎ φk♣x, zq :if ♣A, φ1♣✁, zqA, ☎ ☎ ☎ , φk♣✁, zqAq P Q

Comparing expressive power

A quantifier Q is definable from other quantifiers tQi✉ :if there is a FO tQi✉-sentence θ over signature of Q such that Q ✏ tA ⑤ A ⑤ ù θ✉.

• Example. ’there exists infinitely many’ is definable by

’there are an even number or there are an odd number’

Unary quantifiers on S-AutStr: characterisation

In general: The only unary quantifiers that preserve regularity on S-AutStr are those definable from ❉mod. Recall unary quantifiers (on countable structures) are determined by a relation C ❸ ♣N ❨ t✽✉qk

Unary quantifiers on S-AutStr: characterisation

In general: The only unary quantifiers that preserve regularity on S-AutStr are those definable from ❉mod. Idea: Di-adic unary quantifier Q (determined by C ❸ N2) can define the set WC of words w P 0✝1✝ with ♣#0w, #1wq P C.

• 1. Use the formula (parameter w)

Qxy pos♣w, xq ✏ 0, pos♣w, yq ✏ 1 interpreted over structure ♣0✝1✝, posq.

• 2. Q preserves regularity means that WC is regular.
• 3. So ♣n, mq P C iff for some state q

q0

0n

q

1m qf

• 4. So C is a finite union of N ✂ M where N, M are ultimately
• periodic. So Q is definable from ❉mod.

Binary quantifiers on S-AutStr

Aim: Show that the set of tuples µ✁1♣zq such that A ⑤ ù QRamseyxy φ♣x, y, zq is regular.

✁

♣☎ ☎ q P t ✉ ❉✽ r s ♣ q ✘

Binary quantifiers on S-AutStr

Aim: Show that the set of tuples µ✁1♣zq such that A ⑤ ù QRamseyxy φ♣x, y, zq is regular. Fix z. Idea: The graph µ✁1 φ♣☎, ☎, zqA contains an infinite clique iff there is an infinite sequence α P t0, 1✉ω so that

• 1. ❉✽n so that the word αr0, ns is a prefix of some word xn,
• 2. φ♣xn, xm, zq for all n ✘ m.

Express this using ω-word automata.

Applications of Ramsey quantifier on S-AutStr

The proof shows more:

• Ramsey preserves regularity effectively.
• Yields quantifiers of the form

there exists an infinite set X such that α♣X, zq given that α♣✁, zq always defines a family of sets closed under subset.

Applications of Ramsey quantifier on S-AutStr

Applications:

• Automatic Ramsey Theorem: For a graph G P S-AutStr:

There exists an infinite monochromatic set H ⑨ G such that µ✁1♣Hq is regular.

• Extendible nodes are regular: For a tree ♣T, ➝q P S-AutStr:

The set E ⑨ T of nodes on infinite paths has that µ✁1♣Eq is regular.

Other quantifiers

A quantifier Q is robust :if

• for every structure A in signature of Q and every finite F ⑨ A,

A P Q if and only if A restricted to ♣A③Fq is in Q. Robust

• Ramsey quantifier: ’contains an infinite clique’
• ’contains an even number of disjoint infinite cliques’
• ’contains an infinite complete bipartite graph’

Not Robust

• exists, for all,
• modulo quantifiers.

Other quantifiers

A quantifier Q is 1-partitionable :if

• for every structure A and every finite partition A ✏ ❨Ai,

A P Q iff A restricted to some Ai is in Q. 1-paritionable

• ❉✽ and Ramsey quantifier

Not 1-paritionable

• ’contains an infinite complete bipartite graph’.

Robust + Partionable

• Prop. Every robust 1-paritionable quantifier Q preserves regularity

for S-AutStr. Given: Automatic A, formula φ♣x, y, zq, robust 1-paritionable Q. Aim: Show that the set of z with ♣A, φ♣✁, ✁, zqAq P Q is regular. Idea: Fix z. ♣✁ ✁ q t ✉✍ ⑤ ⑤ ✏ ⑤ ⑤ t ✉➔⑤ ⑤ ♣ ♣✁ ✁ q q P P ♣ q

Robust + Partionable

• Prop. Every robust 1-paritionable quantifier Q preserves regularity

for S-AutStr. Given: Automatic A, formula φ♣x, y, zq, robust 1-paritionable Q. Aim: Show that the set of z with ♣A, φ♣✁, ✁, zqAq P Q is regular. Idea: Fix z.

• cut the domain of φ♣✁, ✁, zqA by sets of the form wt0, 1✉✍

(where ⑤w⑤ ✏ ⑤z⑤) and the finite set t0, 1✉➔⑤z⑤.

• Let Gw be the induced subgraphs.
• ♣A, φ♣✁, ✁, zqAq P Q iff some Gw P Q.
• But the isomorphism type of Gw depends only on the state

reached after reading ♣w, w, zq.

Other quantifiers

A quantifier Q is k-partitionable :if

• for every structure A and every finite partition A ✏ ❨Ai,

A P Q if and only if there exist ijs so that A restricted to Ai1 ❨ ☎ ☎ ☎ ❨ Aik is in Q. Examples.

• ❉✽ and Ramsey quantifier are 1-paritionable.
• ’contains an infinite complete bipartite graph’ is 2-partitionable

but not 1.

• Prop. Every robust k-paritionable quantifier Q preserves regularity

for S-AutStr.

beyond partionable

The binary quantifier there are an even number of connected components that are isomorphic to the complete infinite graph Kω. preserves regularity on S-AutStr, is not partitionable.

beyond partionable

• Definition. A (binary) quantifier Q is k-decomposable if
• there exists finitely many properties P1, ☎ ☎ ☎ , Pm,
• there exists a non-robust quantifier Qf ,
• for every A and finite parition A ✏

n

❨Ai, A P Q iff the contracted graph is in Qf . Contracted graph G♣A, Ai, Pj, kq (k P N parameter):

• domain is rns.
• vertex i is coloured P if the the graph AæAi ⑤

ù P.

• pair ti, j✉ coloured P if the graph AæAi ❨ Aj ⑤

ù P.

• . . . set of size k is coloured . . .

beyond partionable

• Definition. A (binary) quantifier Q is k-decomposable if
• there exists finitely many properties P1, ☎ ☎ ☎ , Pm,
• there exists a non-robust quantifier Qf ,
• for every A and finite parition A ✏

n

❨Ai, A P Q iff the contracted graph is in Qf .

• Prop. Every robust + decomposable quantifier preserves regularity
• n S-AutStr.

Overview

Automatic Presentations Generalised Quantifiers for S-AutStr Generalised Quantifiers for ωS-AutStr Generalised Quantifiers for ωT-AutStr

Quantifiers on ωS-AutStr

❉✽: Replace ❉✽x φ♣x, zq by There is an infinite string α and ❉✽n so that αr0, ns can be extended to some x satisfying φ♣x, zq. ❉ ❉→ ❉→ ♣ q ♣✁ q ⑨ t ✉

Quantifiers on ωS-AutStr

❉✽: Replace ❉✽x φ♣x, zq by There is an infinite string α and ❉✽n so that αr0, ns can be extended to some x satisfying φ♣x, zq. ❉k mod m: as before. ❉→ℵ0: Replace ❉→ℵ0x φ♣x, zq by the set X of ω-strings satisfying φ♣✁, zq contains an infinite perfect subset, ie: no isolated points in Cantor Topology. (Suslin: Every uncountable analytic set has a perfect subset) Use tree-automata to express X ⑨ t0, 1✉ω having a perfect subset by the set of prefixes of X contains a complete binary tree under prefix order.

Quantifiers on ωS-AutStr

Let A be ♦-automatic and let ✓ be a ♦-regular congruence on A. Then the FO theory of A④ ✓ is decidable. However, Quotient Problem: Is the quotient structure ♦-automatic?

Quantifiers on ωS-AutStr

Let A be ♦-automatic and let ✓ be a ♦-regular congruence on A. Then the FO theory of A④ ✓ is decidable. However, Quotient Problem: Is the quotient structure ♦-automatic? S-AutStr, T-AutStr: Yes [Blumensath99] [Colcombet, L¨

• ding07].

ωS-AutStr: No [Hjorth, Khoussainov, Montalban, Nies 07]. ωT-AutStr: ? Which quantifiers preserve regularity on quotient structures?

Quantifiers on ωS-AutStr

The unary quantifiers ❉✽, ❉mod, ❉→ℵ0, ❉↕ℵ0 all preserve regularity

• n quotient structures A④ ✓ where ♣A, ✓q P ωS-AutStr.

♣ q ✓ ♣✁ q ☎ ☎ ☎ ♣✁ q ♣✁ q ✓ ✒

Quantifiers on ωS-AutStr

The unary quantifiers ❉✽, ❉mod, ❉→ℵ0, ❉↕ℵ0 all preserve regularity

• n quotient structures A④ ✓ where ♣A, ✓q P ωS-AutStr.

In fact

• Given FO-formula ϕ♣

x, zq

• there is a constant k (computable from the presentation), so

that for all tuples z of infinite words the following are equivalent:

• 1. ✓ restricted to domain ϕ♣✁,

zq has countably many equivalence classes.

• 2. there exist k-many words x1, ☎ ☎ ☎ xk each satisfying ϕ♣✁,

zq so that every x satisfying ϕ♣✁, zq is ✓-equivalent to some y ✒e xi.

Applications for ωS-AutStr

This condition is FO expressible (using ✒eq so we can eliminate the quantifiers ❉↕ℵ0 and ❉→ℵ0. ❉➔✽ ☎ ☎ ☎ ♣✁ q ✓ ✓ ✓ ✒

Applications for ωS-AutStr

This condition is FO expressible (using ✒eq so we can eliminate the quantifiers ❉↕ℵ0 and ❉→ℵ0. Also, ❉➔✽ can be replaced (for suitable constant M) by ’there exists x1, ☎ ☎ ☎ , xk so that every x satisfying ϕ♣✁, zq is ✓-equivalent to some y that is equal to one of the xi from position M onwards’ ✓ ✓ ✒

Applications for ωS-AutStr

This condition is FO expressible (using ✒eq so we can eliminate the quantifiers ❉↕ℵ0 and ❉→ℵ0. Also, ❉➔✽ can be replaced (for suitable constant M) by ’there exists x1, ☎ ☎ ☎ , xk so that every x satisfying ϕ♣✁, zq is ✓-equivalent to some y that is equal to one of the xi from position M onwards’ Also, if equivalence ✓ has countable index then every ✓-class contains an element ✒e-equivalent to some xi. Consequently, ωS-AutStr is closed under quotient. Why? Can select xi to be ultimately periodic.

Overview

Automatic Presentations Generalised Quantifiers for S-AutStr Generalised Quantifiers for ωS-AutStr Generalised Quantifiers for ωT-AutStr

Quantifiers on ωT-AutStr

The unary quantifiers ❉✽, ❉mod, ❉→ℵ0, ❉↕ℵ0 preserve regularity for the class ωT-AutStr. ❉✽ ♣ q ⑨ ✘ ♣✁ q

Quantifiers on ωT-AutStr

The unary quantifiers ❉✽, ❉mod, ❉→ℵ0, ❉↕ℵ0 preserve regularity for the class ωT-AutStr.

• Example. Replace

❉✽x ϕ♣x, zq by there is a tree T such that for infinitely many finite prefix-closed subsets P ⑨ T there exists some tree x ✘ T satisfying ϕ♣✁, zq and extending P.

Questions and attributions

• 1. Which binary quantifiers preserve regularity on S-AutStr?
• 2. Which unary quantifiers preserve regularity on

quotient structures A④ ✓ where ♣A, ✓q P ωT-AutStr? ♣ ✓q P

Questions and attributions

• 1. Which binary quantifiers preserve regularity on S-AutStr?
• 2. Which unary quantifiers preserve regularity on

quotient structures A④ ✓ where ♣A, ✓q P ωT-AutStr? S-AutStr : [Blumensath 99], [Khoussainov, Stephan, R 03] T-AutStr : [Colcombet 04] ωS-AutStr: [Kuske, Lohrey 06] ωT-AutStr: [Kaiser, B´ ar´ any, Rabinovich, R 07] ♣A, ✓q P ωS-AutStr: [Kaiser, B´ ar´ any, R 07]