A Fibrational Approach to Automata Theory Eilenberg-type Correspondences in One Liang-Ting Chen Henning Urbat TU Braunschweig CALCO 2015
Motivation: a zoo of Eilenberg’s variety theorems Algebraic description of regular languages: ⌣ Eilenberg’s variety theorem (Eilenberg, 1974), and a long list of variants ⌣ (Reutenauer, 1980) ⌣ (Pin, 1995) … (Polák, 2001) (Straubing, 2002) (Gehrke et al., 2009)
Motivation: a zoo of Eilenberg’s variety theorems Algebraic description of regular languages: ⌣ Eilenberg’s variety theorem (Eilenberg, 1974), and a long list of variants ⌣ (Reutenauer, 1980) ⌣ (Pin, 1995) … (Polák, 2001) (Straubing, 2002) (Gehrke et al., 2009)
Motivation: coalgebraic unifjcation Can we unify all of them? ⌣ General (Local) Variety Theorem (Adámek et al., 2014 & 2015) ⌢ Highly technical. ⌢ Two independent arguments. ⌢ An interesting instance (Straubing, 2002) is missing. Goal 1 General Local Variety Theorem = 2 Cover all interesting instances. ⇒ General Variety Theorem.
Motivation: coalgebraic unifjcation Can we unify all of them? ⌣ General (Local) Variety Theorem (Adámek et al., 2014 & 2015) ⌢ Highly technical. ⌢ Two independent arguments. ⌢ An interesting instance (Straubing, 2002) is missing. Goal 1 General Local Variety Theorem = 2 Cover all interesting instances. ⇒ General Variety Theorem.
Motivation: coalgebraic unifjcation Can we unify all of them? ⌣ General (Local) Variety Theorem (Adámek et al., 2014 & 2015) ⌢ Highly technical. ⌢ Two independent arguments. ⌢ An interesting instance (Straubing, 2002) is missing. Goal 1 General Local Variety Theorem = 2 Cover all interesting instances. ⇒ General Variety Theorem.
Background Defjnition A variety of regular languages is a set of regular languages closed under f Example 1 The variety of all regular languages. 2 The variety of star-free languages. Boolean ops. ∩ , ∪ , ( − ) ∁ , ∅ , Σ ∗ , ∆ ∗ , . . . Derivatives a − 1 L = { w ∈ Σ ∗ | aw ∈ L } and La − 1 Preimages f − 1 ( L ) for any monoid homomorphism ∆ ∗ → Σ ∗ . −
Background Defjnition A variety of regular languages is a set of regular languages closed under f Example 1 The variety of all regular languages. 2 The variety of star-free languages. Boolean ops. ∩ , ∪ , ( − ) ∁ , ∅ , Σ ∗ , ∆ ∗ , . . . Derivatives a − 1 L = { w ∈ Σ ∗ | aw ∈ L } and La − 1 Preimages f − 1 ( L ) for any monoid homomorphism ∆ ∗ → Σ ∗ . −
Pseudovarieties of monoids Defjnition A pseudovariety of monoids is a class of fjnite monoids closed under 1 fjnite products, 2 submonoids, and 3 quotients. Example 1 The pseudovariety of all fjnite monoids. 2 The pseudovariety of aperiodic monoids.
A centerpiece of algebraic automata theory ... Theorem (Eilenberg, 1974) varieties of regular languages pseudovarieties of monoids And, this is not the only interesting class of regular languages. ( ) ( ) ∼ =
Pin’s variety theorem Theorem (Pin, 1995) positive varieties of regular languages pseudovarieties of ordered monoids A positive variety is closed under ∩ , ∪ , derivatives, and preimages. ( ) ( ) ∼ =
Polák’s variety theorem Theorem (Polák, 2001) disjunctive varieties of regular languages pseudovarieties of idempotent semirings A disjunctive variety is closed under ∪ , derivatives, and preimages. ( ) ( ) ∼ =
Reutenauer’s variety theorem and preimages. Theorem (Reutenauer, 1980) xor varieties of regular languages pseudovarieties of An xor variety is closed under symmetric difgerences ⊕ , derivatives, ( ) ( ) ∼ = algebras over Z 2
Local Eilenberg theorems Theorem (Gehrke, Grigoriefg and Pin, 2008) local pseudovarieties local varieties of under quotients and subdirect products. 1 A local variety over Σ is a class of languages L ⊆ Σ ∗ closed under ∪ , ∩ , ( − ) ∁ , ∅ , Σ ∗ and derivatives. 2 A local pseudovariety over Σ is a class of M ↞ Σ ∗ closed For each alphabet Σ , ( ) ( ) ∼ = regular languages over Σ of monoid over Σ
Local Eilenberg theorems Theorem (Gehrke, Grigoriefg and Pin, 2008) local pseudovarieties local varieties of under quotients and subdirect products. 1 A local variety over Σ is a class of languages L ⊆ Σ ∗ closed under ∪ , ∩ , ( − ) ∁ , ∅ , Σ ∗ and derivatives. 2 A local pseudovariety over Σ is a class of M ↞ Σ ∗ closed For each alphabet Σ , ( ) ( ) ∼ = regular languages over Σ of monoid over Σ
Category theorists: Veni, vidi, vici Theorem (Adámek, Milius, Myers, and Urbat, 2014) local varieties of regular pseudovarieties varieties of Theorem (Adámek, Milius, Myers, and Urbat, 2015) local pseudovarieties General Variety Theorem: Let C and D be predual categories. General Local Variety Theorem: ( ) ( ) ∼ = C -languages over Σ of D -monoid over Σ ( ) ( ) ∼ = regular C -languages of D -monoid
Instances of General Local Variety Theorem DistLat/Pos Organising local varieties as an opfjbration to get non-local correspondences. Bool/Set local var. closed under C / D ¬ , ∩ , ∪ , ∅ , Σ ∗ ∩ , ∪ , ∅ , Σ ∗ ∨ -SLat/ ∨ -SLat ∪ , ∅ ⊕ , ∅ Z 2 -Vec/ Z 2 -Vec ⊕ , ∪ , ∅ BR / Set ∗
An opfjbration of local varieties, informally 1 f ∗ ( V ) is the “largest” local variety closed under f -preimages. 2 p is equivalent to a functor Free ( Mon D ) → Pos .
An opfjbration of local varieties, informally 1 f ∗ ( V ) is the “largest” local variety closed under f -preimages. 2 p is equivalent to a functor Free ( Mon D ) → Pos .
An opfjbration of local varieties of regular languages closed under f -preimages � � V Defjnition � � W � f f The category LAN consists of objects (Σ , V ) , a local variety V of regular languages of Σ ; → (∆ , W ) , a morphism Σ ∗ → ∆ ∗ s.t. V is − − morphisms (Σ , V ) ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ Reg (∆) � Reg (Σ) f − 1 with a projection p : LAN → Free ( Mon D ) .
Opfjbrations of local pseudovarieties of monoids M The category LPV consists of f f � N �� Defjnition � �� objects (Σ , P ) , a local pseudovariety V of monoids over Σ morphisms (Σ , P ) − → (∆ , Q ) , a monoid morphism f such that: ΨΣ ∗ Ψ∆ ∗ ∃ e M ∈ P ∀ e N ∈ Q ❴ ❴ ❴ ❴ ❴ ❴ with a projection p : LPV → Free ( Mon D ) .
An opfjbrations of local varieties and related structures LAN p � PFMon the opfjbration of fjnitely generated profjnite � q LPV ❑ ❑ ❑ ❑ ❑ ❑ ❑ ❑ ❑ ❑ ❑ ❑ ❑ ❑ ❑ ❑ ❑ ❑ Free ( Mon D ) FLan the opfjbration of local varieties of languages in C . LPV the opfjbration of local pseudovarieties of D -monoids. D -monoids .
An opfjbrations of local varieties and related structures LPV q � Lim PFMon the opfjbration of fjnitely generated profjnite ∼ = � PFMon � qqqqqqqqqqqqqqqqqqq q ′ Free ( Mon D ) FLan the opfjbration of local varieties of languages in C . LPV the opfjbration of local pseudovarieties of D -monoids. D -monoids .
The connection between local and global LAN Theorem (Fibrational Variety Isomorphism) � q Opfjbrations LAN and LPV are isomorphic. p � ∼ = � LPV ❑ ❑ ❑ ❑ ❑ ❑ ❑ ❑ ❑ ❑ ❑ ❑ ❑ ❑ ❑ ❑ ❑ ❑ Free ( Mon D )
The key observation profjnite equational varieties of LAN Global sections of � Corollary ∼ ∼ = = � LPV � PFMon � ❑❑❑❑❑❑❑❑❑❑❑❑❑❑❑❑❑❑ q q q q q q q q q q q q q q q q q q q Free ( Mon D ) LAN varieties of regular languages in C PFMon profjnite equational theories of D -monoids ( ) ( ) ∼ = regular C -languages theories of D -monoid
The key observation profjnite equational varieties of LAN Global sections of � Corollary ∼ ∼ = = � LPV � PFMon � ❑❑❑❑❑❑❑❑❑❑❑❑❑❑❑❑❑❑ q q q q q q q q q q q q q q q q q q q Free ( Mon D ) LAN varieties of regular languages in C PFMon profjnite equational theories of D -monoids ( ) ( ) ∼ = regular C -languages theories of D -monoid
Correspondence between pseudovarieties and profjnite equations Modifjcation of (Reiterman, 1982) & (Banaschewski, 1983): Theorem profjnite equational pseudovarieties ( ) ( ) ∼ = of D -monoids theories of D -monoid
General Variety Theorem as a corollary Corollary varieties of regular pseudovarieties Proof. By Fibrational Isomorphism and Reiterman’s Correspondence. ( ) ( ) ∼ = C -languages of D -monoids
Change of base, for free! j � For each subcategory S , take the pullback along the inclusion PFMon S-varieties of � profjnite equational The missing case (Straubing, 2002) is a special instance when � p LAN � p C � Corollary LAN C � � PFMon C � � ❴✤ ❴✤ q ′ q ′ C � S � � j � Free ( Mon D ) S � � � Free ( Mon D ) ( ) ( ) ∼ = regular C -languages S-theories of D -monoids 1 C / D = Set / BA and 2 S is a non-full subcategory on all free D -monoids.
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