The Separation Problem: An Introduction and a Transfer Theorem Marc - PowerPoint PPT Presentation

The Separation Problem: An Introduction and a Transfer Theorem Marc Zeitoun Joint work with Thomas Place ACTS 2015, Chennai — February 9, 2015 1 / 30

Express Properties Words ababcbaa First-Order Logic ( FO ) Piecewise Testable ( ) Trees 2-Variables FO ( FO ) Fragments , Locally Threshold Testable ( LTT ) For this talk Objects we consider Descriptive Formalism Structures Words Words ababcbaa ababcbaa First-Order Logic ( FO ) First-Order Logic ( FO ) Piecewise Testable ( B Σ 1 ) Piecewise Testable ( B Σ 1 ) Trees Trees 2-Variables FO ( FO 2 ) 2-Variables FO ( FO 2 ) Fragments Σ i , B Σ i Fragments Σ i , B Σ i Locally Threshold Testable ( LTT ) Locally Threshold Testable ( LTT ) 2 / 30

Words ababcbaa First-Order Logic ( FO ) Piecewise Testable ( ) Trees 2-Variables FO ( FO ) Fragments , Locally Threshold Testable ( LTT ) For this talk Objects we consider Descriptive Formalism Structures Express Properties Words Words ababcbaa ababcbaa First-Order Logic ( FO ) First-Order Logic ( FO ) Piecewise Testable ( B Σ 1 ) Piecewise Testable ( B Σ 1 ) Trees Trees 2-Variables FO ( FO 2 ) 2-Variables FO ( FO 2 ) Fragments Σ i , B Σ i Fragments Σ i , B Σ i Locally Threshold Testable ( LTT ) Locally Threshold Testable ( LTT ) 2 / 30

Words ababcbaa First-Order Logic ( FO ) Piecewise Testable ( ) Trees 2-Variables FO ( FO ) Fragments , Locally Threshold Testable ( LTT ) Objects we consider Descriptive Formalism Structures Express Properties Words Words ababcbaa ababcbaa First-Order Logic ( FO ) First-Order Logic ( FO ) Piecewise Testable ( B Σ 1 ) Piecewise Testable ( B Σ 1 ) Trees Trees 2-Variables FO ( FO 2 ) 2-Variables FO ( FO 2 ) Fragments Σ i , B Σ i Fragments Σ i , B Σ i Locally Threshold Testable ( LTT ) Locally Threshold Testable ( LTT ) For this talk 2 / 30

A word is as a sequence of labeled positions that can be quantified. Unary predicates testing the label of a position. One binary predicate: the linear-order . Example: every comes after some First-order logic on words First-order logic, with only the linear order ’ < ’. a b b b c a a a c a 3 / 30

Example: every comes after some First-order logic on words First-order logic, with only the linear order ’ < ’. a b b b c a a a c a 0 1 2 3 4 5 6 7 8 9 ▶ A word is as a sequence of labeled positions that can be quantified. ▶ Unary predicates a ( x ) , b ( x ) , c ( x ) , . . . testing the label of a position. ▶ One binary predicate: the linear-order x < y . 3 / 30

First-order logic on words First-order logic, with only the linear order ’ < ’. a b b b c a a a c a 0 1 2 3 4 5 6 7 8 9 ▶ A word is as a sequence of labeled positions that can be quantified. ▶ Unary predicates a ( x ) , b ( x ) , c ( x ) , . . . testing the label of a position. ▶ One binary predicate: the linear-order x < y . Example: every a comes after some b ∀ x a ( x ) ⇒ ∃ y ( b ( y ) ∧ ( y < x )) 3 / 30

Membership Problem for a fragment INPUT A language . QUESTION Is expressible in ? Why look at fragments in addition to full FO? ▶ Simple formulas are better (aesthetically, algorithmically). ▶ Some parameters making formulas complex: ▶ Number of quantifier alternations, ▶ Allowed predicates, ▶ Number of variable names. 4 / 30

Why look at fragments in addition to full FO? ▶ Simple formulas are better (aesthetically, algorithmically). ▶ Some parameters making formulas complex: ▶ Number of quantifier alternations, ▶ Allowed predicates, ▶ Number of variable names. Membership Problem for a fragment F ▶ INPUT A language L . ▶ QUESTION Is L expressible in F ? 4 / 30

Can it be defined with an formula? Schützenberger’65, McNaughton and Papert’71 For a regular language, the following are equivalent: is FO -definable. The syntactic monoid of satisfies . First Problem: Membership Membership Problem for a fragment F ▶ INPUT A language L . ▶ QUESTION Is L expressible in F ? b b c c a a b b b b a a a a a a c c a a c c 5 / 30

Schützenberger’65, McNaughton and Papert’71 For a regular language, the following are equivalent: is FO -definable. The syntactic monoid of satisfies . First Problem: Membership Membership Problem for a fragment F ▶ INPUT A language L . ▶ QUESTION Is L expressible in F ? b b c c a a b b b b Can it be defined a a a a with an F formula? a a c c a a c c 5 / 30

First Problem: Membership Membership Problem for a fragment F ▶ INPUT A language L . ▶ QUESTION Is L expressible in F ? b b c c a a b b b b Can it be defined a a a a with an F formula? a a c c a a c c Schützenberger’65, McNaughton and Papert’71 For L a regular language, the following are equivalent: ▶ L is FO -definable. ▶ The syntactic monoid of L satisfies u ω +1 = u ω . 5 / 30

Fragments of FO ▶ A fragment is obtained by restricting ▶ Number of quantifier alternations, ▶ Allowed predicates, ▶ Number of variable names. ▶ FO ( < ) , FO ( <, +1) and FO ( <, +1 , min, max ) : same expressiveness. ⇒ Allowing ‘ = ’ but not ‘ < ’ yields distinct fragments. Σ 1 ( <, +1) , and Σ 1 ( <, +1 , min, max ) Σ 1 ( < ) , ▶ We do not want to prove membership multiple times. 6 / 30

Some well-known fragments Weak variant Strong variant FO (=) FO (= , +1) FO 2 ( < ) FO 2 ( <, +1) Σ n ( < ) Σ n ( <, +1 , min, max ) B Σ n ( < ) B Σ n ( <, +1 , min, max ) ▶ Problem: Solve membership for strong variants without reproving everything nor mimicking the proof. 7 / 30

A generic result for membership ▶ Problem Solve membership for strong variants without reproving everything nor mimicking the proof. ▶ S. Eilenberg Each fragment is associated the class of finite monoids recognizing a language from the fragment. → [ x ω = x ω +1 ] . Example: FO ← ▶ H. Straubing 1985 + M. Kulfleitner & A. Lauser 2014: generic result. Weak Fragment F Strong Fragment F + Variety V Variety V ∗ D 8 / 30

Remarks One need to establish the correspondence 1. That V D preserves decidability is a difficult result. V Straubing’s Theorem Weak Fragment F Strong Fragment F + 1 2 Variety V Variety V ∗ D 3 1. Show the correspondence between F and algebraic variety V . 2. In most cases, the enriched fragment F + corresponds to V ∗ D . 3. In most cases, V �→ V ∗ D preserves decidability. 9 / 30

Straubing’s Theorem Weak Fragment F Strong Fragment F + 1 2 Variety V Variety V ∗ D 3 1. Show the correspondence between F and algebraic variety V . 2. In most cases, the enriched fragment F + corresponds to V ∗ D . 3. In most cases, V �→ V ∗ D preserves decidability. Remarks ▶ One need to establish the correspondence 1. ▶ That V �→ V ∗ D preserves decidability is a difficult result. 9 / 30

An alternative approach B. Steinberg 2001 ▶ All fragments share a property entailing decidability of membership. ▶ This property is preserved through enrichment. Even if we are interested in the membership problem for F , it does not give sufficient information to reason about F . 10 / 30

Why we want more than membership If the membership answer for L ▶ is YES ▶ All “subparts” of the minimal automaton of L are F -definable. ▶ is NO , then even if F can talk about L : ▶ We have little information. ▶ Eg, defining L in FO would require differentiating some u ω and u ω +1 . 11 / 30

2 examples of “transfer results”: decidability of separation is preserved when enriching with successor. decidability of separation for level of the quantifier alternation hierarchy entails decidability of membership for . We shouldn’t restrict ourselves to membership Motivations for Separation ▶ Need more general techniques to extract information for all languages. ▶ Cannot start from canonical object for the separator, which is unknown. ▶ Therefore, may give insight to solve harder problems. 12 / 30

We shouldn’t restrict ourselves to membership Motivations for Separation ▶ Need more general techniques to extract information for all languages. ▶ Cannot start from canonical object for the separator, which is unknown. ▶ Therefore, may give insight to solve harder problems. ▶ 2 examples of “transfer results”: ▶ decidability of separation is preserved when enriching F with successor. ▶ decidability of separation for level Σ i of the quantifier alternation hierarchy entails decidability of membership for Σ i +1 . 12 / 30

Motivations for Separation ▶ Need more general techniques to extract information for all languages. ▶ Cannot start from canonical object for the separator, which is unknown. ▶ Therefore, may give insight to solve harder problems. ▶ 2 examples of “transfer results”: ▶ decidability of separation is preserved when enriching F with successor. ▶ decidability of separation for level Σ i of the quantifier alternation hierarchy entails decidability of membership for Σ i +1 . ⇒ We shouldn’t restrict ourselves to membership 12 / 30

-definable -definable -separable from complement -definable Can Can be separated from be separated from with an with an formula? formula? Beyond membership: Separation Decide the following problem: Take two regular languages L 1 , L 2 Take two regular languages L 1 , L 2 a a L 1 a a a a a L 2 a a b b b b b b a a b 13 / 30