The Class of µ -Continuous Chomsky Algebras is Closed under Matrix Rings Hans Leiß leiss@cis.uni-muenchen.de Universit¨ at M¨ unchen Centrum f¨ ur Informations- und Sprachverarbeitung CSL 2016, Aug.29 – Sept.1, Universit´ e Marseille 1 / 29
Content 1. Chomsky-algebras: idemp. semirings “with least fixed points” 2. Reducing n -ary µ to unary µ without ω -completeness 3. µ -continuity of + and · and the equational theory of CFGs 4. if M is a Chomsky algebra, so is Mat n , n ( M ) 5. if M is a µ -continuous Chomsky algebra, so is Mat n , n ( M ) 2 / 29
Chomsky Algebras A Chomsky algebra ( M , + , · , 0 , 1) is an idempotent semiring where every finite system of polynomial inequations x 1 ≥ p 1 ( x 1 , . . . , x n , y 1 , . . . , y m ) , . . (1) . or x ≥ ¯ ¯ p (¯ x , ¯ y ) , x n ≥ p n ( x 1 , . . . , x n , y 1 , . . . , y m ) , b ∈ M m there is a (unique) least has least solutions, i.e. for all ¯ a = a 1 , . . . , a n ∈ M n such that ¯ a , ¯ a i ≥ p M i (¯ b ) for i = 1 , . . . , n . Here ≤ is the natural partial order on M defined by a ≤ b iff a + b = b . The system ¯ x ≥ ¯ p (¯ x , ¯ y ) is a context-free grammar with nonterminals ¯ x = x 1 , . . . , x n and terminals ¯ y = y 1 , . . . , y m . 3 / 29
Example Let ( X ∗ , · , ǫ ) be the monoid of all finite words of elements of X . Its power set is an idempotent semiring: ∅ , 1 := { ǫ } , 0 := ( P X ∗ , + , · , 0 , 1) , with A + B := A ∪ B , A · B := { a · b | a ∈ A , b ∈ B } . For a vector ¯ B of m languages, ¯ x ≥ ¯ p (¯ x , ¯ y ) leads to an increasing sequence ¯ A k = ( A k , 1 , . . . , A k , n ) of language vectors by A k +1 , i := p P X ∗ (¯ A k , ¯ A 0 , i := ∅ , B ) , i = 1 , . . . , n . i The least solution of the inequation system, relative to ¯ B , is ¯ � { ¯ A := A k | k ∈ N } . Therefore, ( P X ∗ , + , · , 0 , 1) is a Chomsky algebra. 4 / 29
The inclusion ¯ p P X ∗ (¯ A , ¯ A ⊇ ¯ B ) follows from the fact that + and · are compatible with arbitrary unions (sup-continuous): for all A , B ⊆ X ∗ and ∅ � = C ⊆ P X ∗ , � � A + C = { A + C | C ∈ C} , � � A · C · B = { A · C · B | C ∈ C} . Example Rel ( X ) := ( P ( X × X ) , + , · , 0 , 1), the set of all binary relations on X with union as +, relation composition ; as · , the empty relation as 0 and the identity relation as 1. 5 / 29
Example The set C X ∗ of context-free languages over X is the smallest L ⊆ P X ∗ such that (i) each finite subset of X ∪ { ǫ } is in L , and y ) is a polynomial system and ¯ B ∈ L m , then the x ≥ ¯ (ii) if ¯ p (¯ x , ¯ A ∈ ( P X ∗ ) n with ¯ least ¯ p P X ∗ (¯ A , ¯ B ) belongs to L n . A ⊇ ¯ With the operations inherited from P X ∗ , ( C X ∗ , + , · , 0 , 1) is a Chomsky algebra. Likewise: ◮ the set C ( X 2 ) ⊆ P ( X 2 ) of context-free relations on the set X , ◮ the set C M ⊆ P M of context-free subsets of a monoid M . The regular languages over X do not form a Chomsky algebra, as they don’t have solutions for inequations like axb + 1 ≤ x . 6 / 29
Goals of this talk: For any Chomsky algebra M : ◮ The matrix ring Mat n , n ( M ) is also a Chomsky algebra. ◮ If least solutions of systems ¯ x ≥ ¯ t can be computed in M iteratively, they can so be computed in Mat n , n ( M ). Chomsky algebras were introduced by Grathwohl, Henglein, Kozen 2013 in providing an infinitary complete axiom system for the equational theory of context-free grammars as fixed-point terms. Theorem (Grathwohl,Henglein,Kozen FICS 2013) The axioms of idempotent semirings and µ -continuity are sound and complete for the equational theory of the context-free languages. 7 / 29
µ -terms and Park µ -semirings Let X be an infinite set of variables. The µ -terms over X are defined by t := x | 0 | 1 | ( s · t ) | ( s + t ) | µ xt . A term t is is algebraic or a polynomial if it does not contain µ . ◮ free ( t ) is the set of variables having a free occurrence in t . t ( x 1 , . . . , x n ) indicates free ( t ) ⊆ { x 1 , . . . , x n } . ◮ t [ x / s ] is the result of substituting all free occurrences of x in t by s , renaming bound variables of t to avoid variable capture. ◮ The µ -depth of a term is: 0 for x , 0, 1; is µ -depth( t ) + 1 for µ xt ; and is max { µ -depth( s ) , µ -depth( t ) } for ( s + t ), ( s · t ). Note: We’ll write µ x . t [ y / s ] for µ x ( t [ y / s ]), using . to save the brackets of the metalanguage. (We prefer µ x ( t + s ) over µ x . t + s .) 8 / 29
A partially ordered µ -semiring ( M , + , · , 0 , 1 , ≤ ) is a semiring ( M , + , · , 0 , 1) with a partial order ≤ on M , where every µ -term t defines a function t M : ( X → M ) → M , so that for all variables x ∈ X , terms s , t and valuations g , h : X → M we have: 0 M ( g ) ( s + t ) M ( g ) s M ( g ) + t M ( g ) , 1. = 0 , = 1 M ( g ) ( s · t ) M ( g ) s M ( g ) · g M ( g ) , = 1 , = ( µ -rule) if s M ≤ t M , then µ xs M ≤ µ xt M , x M ( g ) = g ( x ) , 2. (monotonicity) if g ≤ h pointwise, then t M ( g ) ≤ t M ( h ), 3. (coincidence) if g , h agree on free ( t ), then t M ( g ) = t M ( h ), 4. (substitution) t [ x / s ] M ( g ) = t M ( g [ x / s M ( g )]). For t M ( g ) we also write t M [ x 1 / a 1 , . . . , x n / a n ] or t M ( a 1 , . . . , a n ). A first-order formula built from (in)equations holds in M if it is true for every valuation g : X → M . 9 / 29
A Park µ -semiring is a partially ordered µ -semiring M where for all terms t and variables x , y , the following “Park axioms” hold in M : t [ x /µ xt ] ≤ µ xt , (2) t [ x / y ] ≤ y → µ xt ≤ y . (3) Then the following also hold in M : t [ x /µ xt ] = µ xt , µ y . t [ x / y ] = µ xt , for y / ∈ free ( t ). Park’s axioms imply that µ xt M ( g ) is the least solution of t ≤ x in M , g , i.e. the least a ∈ M such that t M ( g [ x / a ]) ≤ a . 10 / 29
Lemma (c.f. Grathwohl/Henglein/Kozen 2013) Every Chomsky algebra M is an idempotent, partially ordered µ -semiring, if for all terms t, variables x and valuations g : X → M µ xt M ( g ) := the least a ∈ M such that t M ( g [ x / a ]) ≤ a . (4) Moreover, every inequation system ¯ t (¯ x , ¯ y ) ≤ ¯ x with µ -terms y ) has least solutions in M, i.e. for all parameters ¯ ¯ t (¯ x , ¯ b from M a , ¯ a in M such that ¯ t M (¯ b ) ≤ ¯ there is a least tuple ¯ a. Proof: Simultaneously by induction on the µ -depth of terms. Corollary Every Chomsky algebra M, in particular C X ∗ , is a Park µ -semiring. For every µ -term and g : X → C X ∗ , t C X ∗ ( g ) = t P X ∗ ( g ) . 11 / 29
Vector versions and matrix ring of Park µ -semirings x ≥ ¯ To name the least solution of a system ¯ t of n inequations, one x ¯ might introduce terms µ ¯ t using an n -ary fixed-point operator µ . As is well-known, a unary µ normally suffices: Theorem (Beki´ c 1984) Let ( M , ≤ ) be a partially ordered set in which every countable increasing chain A = { a i | i ∈ N } has a least upper bound, � A. Suppose f , g : M 2 → M are continuous in each component, i.e. f ( � A , b ) = � { f ( a , b ) | a ∈ A } for countable chains A, etc. Then the least solution of the system ( x , y ) ≥ ( f ( x , y ) , g ( x , y )) can be obtained from least solutions of single inequations: µ ( x , y )( f ( x , y ) , g ( x , y )) (5) = ( µ x . f ( x , µ y . g ( x , y )) , µ y . g ( µ x . f ( x , y ) , y ) . 12 / 29
x ≥ ¯ For an n -dimensional inequation system ¯ t , we define an x ¯ n -tuple µ ¯ t of µ -terms by recursively using Beki´ c’s equations (5). ◮ If n = 1, then µ ¯ x ¯ t := µ x 1 t 1 . ◮ If n > 1, ¯ z ) and ¯ x = (¯ y , ¯ t = (¯ r , ¯ s ) with term vectors ¯ r , ¯ s of t is 1 x ¯ lengths | ¯ y | , | ¯ z | < n , then µ ¯ µ (¯ y , ¯ z )(¯ r , ¯ s ) := ( µ ¯ y . ¯ r [¯ z /µ ¯ z ¯ s ] , µ ¯ z . ¯ s [¯ y /µ ¯ y ¯ r ]) . (6) x ¯ However, Beki´ c’s theorem does not imply that µ ¯ t denotes the x ≥ ¯ t in Chomsky algebras, like C X ∗ , as these least solution of ¯ need not be closed under unions of countable increasing chains. x ¯ x ≥ ¯ To see that µ ¯ t denotes the least solution of ¯ t in a Chomsky algebra M , we show that Park’s axioms for term vectors hold in M . 1 Recall that µ xt [ y / s ] differs from µ x . t [ y / s ] := µ x ( t [ y / s ]). 13 / 29
s , ¯ s = ¯ s ≤ ¯ For term vectors ¯ t of the same dimension, let ¯ t resp. ¯ t be the conjunction of all s i = t i resp. s i ≤ t i . For ¯ t = ( t 1 , . . . , t n ) we write ¯ t M ( g ) for ( t M 1 ( g ) , . . . , t M n ( g )). Lemma Let M be a Park µ -semiring. For all vectors ¯ t of terms and ¯ x , ¯ y of variables, of the same dimension, the vector versions of (2) and (3), ¯ x ¯ x ¯ ≤ t [¯ x /µ ¯ t ] µ ¯ t , (7) ¯ x ¯ t [¯ x / ¯ y ] ≤ ¯ y → µ ¯ t ≤ ¯ y , (8) hold in M. Moreover, for any g : X → M, s ] M ( g ) = µ ¯ t M ( g [¯ s M ( g )]) , x ¯ x ¯ µ ¯ t [¯ y / ¯ y / ¯ (9) if no variable of ¯ x is free in the terms ¯ s. Proof: induction on dimension, simultaneously for (7), (8) and (9). 14 / 29
Corollary x ¯ t M ( g ) For any Chomsky algebra M and valuation g : X → M, µ ¯ a such that ¯ t M ( g [¯ is the least ¯ x / ¯ a ]) ≤ ¯ a. Corollary If M is a Park µ -semiring, the vector version of the µ -rule holds: s, ¯ for vectors ¯ t of terms and ¯ x of different variables, all of the s M ≤ ¯ s M ≤ µ ¯ t M , then µ ¯ x ¯ t M . same dimension, if ¯ x ¯ 15 / 29
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