Conway and Iteration Semirings Stephen L. Bloom Stevens Institute - PDF document

Conway and Iteration Semirings Stephen L. Bloom Stevens Institute of Technology Hoboken, NJ International Category Theory Conference 2008

Joint work with Zoltan ´ Esik

Outline • Some categories with fixed points • Conway and iteration theories • Matrix theories and star semirings • A Kleene type theorem • Some characterizations of N rat � � A ∗ � � and � A ∗ � N ∞ rat � � • Open problems

Some categories with fixed point operators

Suppose C is a category with coproducts. A parameterized fixed point operation is a func- tion † � C ( X, Y ) C ( X, X + Y ) such that � f † , 1 Y � f f † � X + Y � Y. = X † � C ( Y, X ) such In product form: C ( X × Y, X ) that � f † , 1 Y � f f † � X × Y � X = Y

Pfn ( A ) • A is a collection of sets closed under finite coproducts + • Pfn ( A ) is category with objects X ∈ A � Y is partial function • an arrow f : X � X + Y , f † : X � Y is “do • if f : X f while value is in X ”. � f † , 1 Y � f f † = X � X + Y � Y.

F ω ( A ) • A is a collection of ω -complete posets closed under finite products × • F ω ( A ) is category with objects X ∈ A � Y is continuous func- • an arrow f : X tion � X , f † : Y � X is least • if f : X × Y x such that f ( x, y ) = x . � f † , 1 Y � f f † � X × Y � X = Y f † ( y ) f ( f † ( y ) , y ) . =

� � Fn ( A ) • A a collection of ω -complete categories closed under finite products • objects in Fn ( A ) are categories in A • an arrow in Fn ( A ) is ω -continuous functor � B f : A � A , for each b ∈ B , • if f : A × B f b : A A is = f ( − , b ) f b • f † : B � A on b is initial f b -algebra f b ( f † ( b )) f † ( b ) .

Trees Σ ⊥ TR • Σ a ranked set ; ⊥ ∈ Σ 0 . � p is node labeled Σ- • a morphism 1 tree with internal nodes with n -successors labeled by letter in Σ n ; leaves labeled either by letter in Σ 0 or “variable” x 1 , . . . , x p � p is n -tuple of trees 1 • a morphism n � p . • composition: tree substition � 1 + p , f † : 1 � p is unique • if f : 1 tree such that � f † , 1 p � f f † = 1 � 1 + p � p.

Matrices The subsets of words on an alphabet X form a semiring 2 X ∗ where A + B = A ∪ B and A · B = { uv : u ∈ A, v ∈ B } . Matrices over this semiring form a category Mat (2 X ∗ ): � p is n × p matrix • an arrow f : n • f · g matrix product � n + p , f † = [ a ∗ b ], where • if f = [ a b ] : n = 1 n + a + a 2 + . . . . a ∗

Theories with fixed points Each of the above examples are categories T with • finite coproducts or products • a parameterized fixed point operation • In Pfn, Mat, F ω , f † is a least fixed point • In Σ ⊥ TR , f † is a unique fixed point • In Fn , f † is an initial fixed point

Main Results • (Bloom, ´ Esik) A large number of the com- putationally interesting structures have one of two equational theories IT, IT 0 . • IT := Id ( F ω ( A )) = Id ( Fn ( A )) = Id (Σ ⊥ TR ) IT is the iteration theory identities. • IT 0 = IT + { x = y : x, y : 1 → 0 } = Id ( Pfn ( A )) = Id ( Mat (2 X ∗ )), • (Simpson, Plotkin) Every consistent itera- tion theory is either IT or IT 0 .

Axiomatization Axioms for iteration theory identities fall into two groups. • The “Conway axioms”: fixed point, pa- rameter, simple composition and double- dagger • The group axioms.

Representing f † X X Y X = f f Y

Parameter identity f = f g g f † · g ( f · ( 1 X ⊕ g )) † = � X + Y , g : Y � Z . f : X

� � � Another formulation of the parameter identity † � C ( X, Y ) C ( X, X + Y ) C ( X, X + Y ) C ( X, Y ) C ( 1 X , 1 X ⊕ g ) C ( 1 X ,g ) C ( X, X + Z ) C ( X, X + Z ) C ( X, Z ) C ( X, Z ) † � Z . commutes, for any g : Y

Simple Composition identity f g = f g f ( f · g ) † f · ( g · f ) † = � Y , g : Y � X . f : X

Double dagger Identity f f = f:X −> X+X+Y f †† ( f · ( � 1 X , 1 X � ⊕ 1 Y )) † =

Traced monoidal categories Conway theories are essentially the same as traced monoidal categories of Joyal, Street and Verity. (Feedback replaces dagger.)

Iteration Theories An iteration theory is a Conway theory satisfying all group identities. For the two element group G 2 : f = f f 1 2 · � f · ρ 1 , f · ρ 2 � † f †† . = The G 2 -identity

Star semirings A semiring, or rig, S consists of • a commutative monoid ( S, + , 0), and • a monoid ( S, · , 1), such that • multiplication distributes over addition x ( y + z ) = xy + xz ( y + z ) x = yx + zx 0 · x = x · 0 = 0 . • A star semiring is semiring with ∗ : S → S .

Conway and iteration matrix theories Mat ( S ), the theory of matrices over a semiring S , is a Conway theory iff S is a star semiring satisfying • the sum star identity : ( x + y ) ∗ ( x ∗ y ) ∗ x ∗ , = • the product star identity : ( xy ) ∗ 1 + x ( yx ) ∗ y. = Special cases: zero and fixed point identity : 0 ∗ = 1 1 + xx ∗ = 1 + x ∗ x. x ∗ =

Definition. A Conway semiring is star semiring satisfying the sum and product star identities. Examples • Language semirings: (2 X ∗ , + , · , 0 , 1 , ∗ ), where X is an alphabet, and for A, B ⊆ X ∗ , A + B = A ∪ B ; 0 = ∅ ; 1 = { ǫ } A · B = { uv : u ∈ A, v ∈ B } ∞ A ∗ A k . � = k =0

More Examples • N ∞ = { 0 , 1 , . . . , } ∪ {∞} , with 0 ∗ = 1 , x ∗ = ∞ , otherwise. • The boolean semiring B = { 0 , 1 } , with x ∗ = 1 = 1 + x .

Which Rings are reducts of Conway semirings? • The star fixed point identity: x ∗ 1 + x · x ∗ = implies x ∗ · (1 − x ) = 1 . • Letting x = 1: 0 = 1 .

Inductive definition of M ∗ If M is n × n , n > 1, write � � a b = M c d with a, d square. The Conway identities imply that M ∗ is ( a + bd ∗ c ) ∗ ( a + bd ∗ c ) ∗ bd ∗ � � M ∗ = ( d + ca ∗ b ) ∗ ca ∗ ( d + ca ∗ b ) ∗

Group identities for matrix theories Suppose G is a group with underlying set { 1 , 2 , . . . , n } . The G identity is ( x 1 + . . . + x n ) ∗ e 1 M ∗ = G u n M G is n × n matrix with entries in { x 1 , . . . , x n } M G [ i, j ] = x i − 1 · j i − 1 · j is computed in G . = [1 0 . . . 0] . e 1 u n is n × 1 matrix of 1’s.

Example When G = G 2 is the 2-element group, � � x 1 x 2 = M G x 2 x 1 The G 2 -identity is ( x 1 + x 2 ) ∗ ( x 1 + x 2 x ∗ 1 x 2 ) ∗ (1 + x ∗ = 1 x 2 ) . Special case: x 1 = 0 and x 2 = 1, 1 ∗ (1 + 1) = 1 ∗ + 1 ∗ . 1 ∗ =

Iteration semirings • Suppose S is a Conway semiring. Mat ( S ) is an iteration theory iff S satisfies all group identities. • Definition. An iteration semiring is a Con- way semiring satisfying all group identities.

Closure Properties If S is Conway or iteration semiring, so is S n × n for n ≥ 0, the semiring of n × n matrices over S , with matrix product as product, pointwise sum, and star computed using the inductive formula.

Power series semirings If S is Conway or iteration semiring, so is � A ∗ � S � � the semiring of formal power series. Elements are functions A ∗ � S . • Notational convention: � = ( f, u ) u, f u ∈ A ∗ where ( f, u ), the coefficient of u , is f ( u ). • Examples. For a ∈ A, s ∈ S : τ a ( u ) = 1 if u = a, 0 otherwise. σ s ( u ) = s if u = ǫ, 0 otherwise. • The notational convention implies = a τ a = s σ s .

� A ∗ � Semiring operations in S � � • If � = ( f, u ) u f u ∈ A ∗ � = ( g, u ) u g u ∈ A ∗ then � f + g := (( f, u ) + ( g, u )) u u ∈ A ∗ � � f · g := ( ( f, x ) · ( g, y )) u. xy = u u ∈ A ∗ • Example If f = 2 + 3 a, g = 2 b + ab , 4 b + 8 ab + 3 a 2 b. = fg ab = ǫ · ( ab ) or a · b .

f ∗ in S � � A ∗ � � ? • If f ( ǫ ) = 0, there is unique f ∗ satisfying f ∗ 1 + f · f ∗ = f ∗ ( f, x )( f ∗ , y )) u. � � = 1 + ( xy = u u • If f = s + g , g ( ǫ ) = 0, the sum star identity implies ( s + g ) ∗ = ( s ∗ g ) ∗ s ∗ . f ∗ =

Some iteration and Conway semirings • Each of B = { 0 , 1 } , N ∞ , 2 X ∗ is an iteration semiring. • The initial Conway semiring S 0 has ele- ments 0 , 1 , 2 , . . . , k (1 ∗ ) p , and 1 ∗∗ . • In any iteration semiring 1 ∗ + 1 ∗ 1 ∗ . = • The initial iteration semiring S 1 has ele- ments 0 , 1 , 2 , . . . , (1 ∗ ) p , 1 ∗∗ .

(Countably) complete semirings S is (countably) complete if for any (count- able) set I , � i ∈ I s i exists, and has the usual properties. We may define 1 + s + s 2 + . . . s ∗ := and the resulting star semiring is an iteration semiring. Example: N ∞ is countably complete. In N ∞ ,  1 x = 0 x ∗  = ∞ otherwise. 

For each n ≥ 1, • S n × n is a Conway, but not an iteration 0 semiring, and � A ∗ � • For each alphabet A , S 0 � � is a Conway, but not an iteration semiring.

Rational power series For a Conway semiring S and alphabet A , � A ∗ � • S rat � � , the rational series, is the least � A ∗ � sub star semiring of S � � containing each series σ s , and τ a , for s ∈ S, a ∈ A .