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 ω S-AutStr Generalised Quantifiers for ω T-AutStr

Automata and Logical Definability Fact. For each type of object ♦ P t string , ω -string , tree , ω -tree ✉ there is a notion of synchronous automaton with robust closure properties. Classically 1. (W)MSO ♣ N , S q Ô automata on (finite) infinite words 2. (W)MSO ♣t 0 , 1 ✉ ✍ , s 0 , s 1 q 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 t string , ω -string , tree , ω -tree ✉ there is a notion of synchronous automaton with robust closure properties. Another point of view. 1. FO ♣t 0 , 1 ✉ ✝ , σ 0 , σ 1 , ➔ prefix , el q Ô automata on finite words. 2. FO ♣ trees , σ t l , r ✉ t 0 , 1 ✉ , ➔ ext , ✑ dom q Ô automata on infinite trees. Application: Decidability of Logical Theories.

Automatic Presentations Let ♦ P t string , ω -string , tree , ω -tree ✉ . A ♦ -automatic presentation of a relational structure A ✏ ♣ A , ♣ R i qq consists of 1. a tuple of ♦ -automata ♣ M , ♣ M i qq , 2. a bijection µ : L ♣ M q Ñ A , so that µ ♣ L ♣ M q , ♣ L ♣ M i qqq ✕ A . Say that A is an ♦ -automatic structure. [Hodgson 76] [Khoussainov,Nerode 95] [Blumensath, Gr¨ adel 00]

♣ ♣ q ❨ ❳ q P P t ✉ t P ⑤ r s ✏ ✉ ♣ ✂q P ③ ♣ ☎ ☎ ☎ q ➧ Examples Real Addition ♣r 0 , 1 q , �q P ω S-AutStr - µ : t 0 , 1 ✉ ω ③t 0 , 1 ✉ ✝ 1 ω Ñ r 0 , 1 q in base 2.

♣ ✂q P ③ ♣ ☎ ☎ ☎ q ➧ Examples Real Addition ♣r 0 , 1 q , �q P ω S-AutStr - µ : t 0 , 1 ✉ ω ③t 0 , 1 ✉ ✝ 1 ω Ñ r 0 , 1 q in base 2. Boolean Algebra ♣ P ♣ N q , ❨ , ❳ , c q P ω S-AutStr . - µ maps α P t 0 , 1 ✉ ω to t n P N ⑤ α r n s ✏ 1 ✉ .

Examples Real Addition ♣r 0 , 1 q , �q P ω S-AutStr - µ : t 0 , 1 ✉ ω ③t 0 , 1 ✉ ✝ 1 ω Ñ r 0 , 1 q in base 2. Boolean Algebra ♣ P ♣ N q , ❨ , ❳ , c q P ω S-AutStr . - µ maps α P t 0 , 1 ✉ ω to t n P N ⑤ α r n s ✏ 1 ✉ . Rational Multiplication ♣ Q , ✂q P T-AutStr ③ S-AutStr - µ maps ♣ u 1 , ☎ ☎ ☎ , u k q to the number ➧ p n i where p i is the i th i prime and u i is the integer n i written in base 2.

Open Questions Is Rational Addition ♣ Q , �q automatic? Is this Atomless Boolean Algebra ♣ P ♣ N q , ❨ , ❳ , c q④ ✒ e automatic? A ✒ e B : if ⑤ A △ B ⑤ ➔ 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 ♣ x q in the signature of A . The automatic presentation can be extended to ♣ A , Φ A q ie: µ ✁ 1 ♣ Φ A q is regular. (Induction on φ ) Ex. Every automatic presentation of ♣ N , �q can be expanded to one for ♣ N , � , ✁ , ➔ , S , 0 , 1 , ✑ r q .

Fundamental Properties Theorem. FO definable Ñ regular Given 1. ♦ -automatic presentation µ of A 2. FO-formula Φ ♣ x q in the signature of A . The automatic presentation can be extended to ♣ A , Φ A q ie: µ ✁ 1 ♣ Φ A q 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 ♣ x q in the signature of A . The automatic presentation can be extended to ♣ A , Φ A q ie: µ ✁ 1 ♣ Φ A q is regular. Goal Extend to more expressive logics

Overview Automatic Presentations Generalised Quantifiers for S-AutStr Generalised Quantifiers for ω S-AutStr Generalised Quantifiers for ω T-AutStr

Generalised Quantifiers Generalised Quantifier Q is a class of structures, over a fixed signature, closed under isomorphism. ♣ A , φ ♣✁ , z q A q P Q A ⑤ ù Q x φ ♣ x , z q :if Unary Examples. - ’There exists’: Q ✏ t♣ A , P q ⑤ ❍ ✘ P ⑨ A ✉ . - ’Counting quantifiers’: For C ⑨ N ❨ t✽✉ , define Q C ✏ t♣ A , P q ⑤ ⑤ P ⑤ P C ✉ . - ’Modulo quantifiers’ ❉ mod : Q ✏ t♣ A , P q ⑤ ⑤ P ⑤ ✑ k mod m ✉ .

Generalised Quantifiers Generalised Quantifier Q is a class of structures, over a fixed signature, closed under isomorphism. ♣ A , φ ♣✁ , z q A q P Q A ⑤ ù Q x φ ♣ x , z q :if Binary Examples. - ’Connectedness’: Q ✏ t♣ A , E q ⑤ graph is strongly connected ✉ . - ’Ramsey’: Q Ramsey ✏ t♣ A , E q 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 , φ ♣✁ , z q A q P Q A ⑤ ù Q x φ ♣ x , z q :if First order + quantifiers written FO � t Q i ✉

Generalised Quantifiers and Regularity Definition. Quantifier Q preserves ♦ -regularity (effectively) :if Given - ♦ -automatic presentation of A , - FO formula Φ ♣ x , z q in signature of A , the automatic presentation can be extended to include the relation defined in A by Q x Φ ♣ x , z q (and automaton be found effectively).

Quantifiers preserving regularity Examples on S-AutStr . - ❉ , ❅ : standard - ❉ ✽ : replace ❉ ✽ x φ ♣ x , z q by ♣❅ y ❉ x q r⑤ x ⑤ → ⑤ y ⑤ ❫ φ ♣ x , z qs . - ❉ k mod m : modified subset construction.

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 , E q from s to f (Configuration space of c.e but non-computable set). - Unary H¨ artig quantifier: ♣ A , P , Q q where P , Q ⑨ A and ⑤ P ⑤ ✏ ⑤ Q ⑤ (later).

Non-regularity preserving - Binary reachability quantifier: ♣ A , E , s , f q there is a path in graph ♣ A , E q from s to f (Configuration space of c.e but non-computable set). - Unary H¨ artig quantifier: ♣ A , P , Q q where P , Q ⑨ A and ⑤ P ⑤ ✏ ⑤ Q ⑤ (later). Quantifiers may bind more than one formula (adicity): A ⑤ ù Q x � φ 1 ♣ x , z q ☎ ☎ ☎ φ k ♣ x , z q � :if ♣ A , φ 1 ♣✁ , z q A , ☎ ☎ ☎ , φ k ♣✁ , z q A q P Q

Comparing expressive power A quantifier Q is definable from other quantifiers t Q i ✉ :if there is a FO � t Q i ✉ -sentence θ over signature of Q such that Q ✏ t A 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✽✉q k

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 N 2 ) can define the set W C of words w P 0 ✝ 1 ✝ with ♣ # 0 w , # 1 w q P C . 1. Use the formula (parameter w ) Qxy � pos ♣ w , x q ✏ 0 , pos ♣ w , y q ✏ 1 � interpreted over structure ♣ 0 ✝ 1 ✝ , pos q . 2. Q preserves regularity means that W C is regular. 3. So ♣ n , m q P C iff for some state q 0 n 1 m � q f � q q 0 4. So C is a finite union of N ✂ M where N , M are ultimately periodic. So Q is definable from ❉ mod .

✁ ♣☎ ☎ q P t ✉ ❉ ✽ r s ♣ q ✘ Binary quantifiers on S-AutStr Aim: Show that the set of tuples µ ✁ 1 ♣ z q such that A ⑤ ù Q Ramsey xy φ ♣ x , y , z q is regular.

Binary quantifiers on S-AutStr Aim: Show that the set of tuples µ ✁ 1 ♣ z q such that A ⑤ ù Q Ramsey xy φ ♣ x , y , z q is regular. Fix z . Idea: The graph µ ✁ 1 φ ♣☎ , ☎ , z q A contains an infinite clique iff there is an infinite sequence α P t 0 , 1 ✉ ω so that 1. ❉ ✽ n so that the word α r 0 , n s is a prefix of some word x n , 2. φ ♣ x n , x m , z q 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 , z q given that α ♣✁ , z q 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 ♣ H q 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 ♣ E q is regular.