Representations of Algebraic Moore-Machines

Review of the last talkWho is the Moore from Moore-Automata?

More about SimulationsRepresentations

Summary

Categorial and Algebraical Methods inAutomata Theory

IV. Representations of Algebraic Moore-Machines

Ch. Pech

2006-11-29 / SLADIM

Ch. Pech Categorial and Algebraical Methods in Automata Theory 2006-11-29 / SLADIM 1 / 34



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Outline

1 Who is the Moore from Moore-Automata?

2 More about SimulationsAn Excursion to Categorical AlgebraSimulations in the language of Lawvere-Theories

3 Representations




Summary

Review of the last talk

We discussed the proper definition of alternating automata.

We saw two different classical models of alternatingautomata,We argued that these two models should be calledBoolean- and alternating automata (as is done by manyother authors)We argued that the proper definition of alternating automatauses free bounded distributive lattices as state-algebras.

We compared types of algebraic Moore-machines.We defined a concept of simulation,We showed how simulations preserve the algebraicbehavior.We showed how simulations preserve the full observablebehavior.

We studied a bit the order between optimal typesWe noted that term-equivalent output-algebras generateequivalent types.We noted that non-deterministic automata can be simulatedby weighted automata over idempotent semirings.




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Edward F. Moore

Born: November 23,1925 in Baltimore.Died: June 14, 2003 in Madison Wisconsin.1950: PhD in Mathematics.

1950-1951: Work on the electronic computer project ILIAC.1951-1956: Work for Bell-Labs, New Jersey.1961-1962: Visiting Professor at MIT and Visiting Lecturer at

Harvard.1966-1985: Professor of Mathematics and Computer Science

at University of Wisconsin

He was one of the founders of Automata Theory.The sequential machines that are now named after him heintroduced in“Gedanken-experiments on Sequential Machines.” pp129–153, Automata Studies, Annals of MathematicalStudies, no. 34, Princeton University Press, Princeton, N.J., 1956




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An Excursion to Categorical AlgebraSimulations in the language of Lawvere-Theories

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Simulations

Simulations were defined as pairs (λ, e) whereλ monad-embedding,e embedding of the output-algebras,λ and e have to be “compatible”.

In our context the types of algebraic Moore-machines areessentially given by varieties of finitary algebras.Monads are “more powerful” than such varieties.It makes sense to give a definition of simulations on thecorrect level of abstraction.Let us use the language of Lawvere’s algebraic theories.




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Note:

Algebraic Moore-Machines can be defined also using varietiesof infinitary algebras (such as, e.g., complete sup-semilattices).




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Lawvere-Theories

A Lawvere-Theory is a category with:

countably many objects C0, C1, . . . , Cn . . .

distinguished families of morphisms (πni : Cn → C1)1≤i≤n.

Cn is the n-th power of C1,

(πni : Cn → C1)1≤i≤n are the product-projections.

Morphisms:

Let T , T ′ be Lawvere-theories. A homomophism from T to T ′ isa functor from T to T ′ that preserves the distinguishedproducts.




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Four Ways to Represent a Variety

Equational Theory Variety Adjunction

Lawvere-Theory Monad (Triple)

Let us recall the chain of constructions:

Variety Adjunction Monad Lawvere − Theory




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Variety −→ Adjunction

Given a variety V by:

a signature F ,a set Eq of equations.

V has a natural underlying functor U : V → Sets : A 7→ A.

UV has a left-adjoint FV : Sets → V mapping a set X to analgebra in V freely generated by X .

Let us agree about the construction of FV(X ):

Let TF (X ) be the term-algebra generated by X .Take FV(X ) as the quotient-algebra of TF (X ) w.r.t. thesmallest fully invariant congruence-relation generated byEq.For a term t , we denote [t ] the corresponding element fromFV(X ).




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Adjunction −→ Monad

Given:

The adjunction (FV , UV , η, ε) for the variety V.

Define:

TV : Sets → Sets ; X 7→ UV(FV(X )).

η : I → TV ; ηX : x 7→ [x ].

µ : T 2V → TV ; µX := U(νX )

where νX : FV(TV(X )) → FV(X ) is the free extension of theidentity on TV(X )

TV = (TV , η, µ) is the monad associated with V.




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The Kleisli-Category of a Monad

Given:

T = (T , η, µ) a monad in the category Sets.

The Kleisli-Category KT

Ob(KT) := Ob(Sets),

homKT(X , Y ) := homSets(X , T (Y )).

Composition

Given f ∈ homKT(X , Y ), g ∈ homKT

(Y , Z ).

Define g ◦ f ∈ homKT(X , Z ) by

X T (Y ) T 2(Z ) T (Z )f T (g) µZ




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Kleisli-Categories of Monads from VarietiesKV as full subcategory of V

Given:a variety V,its adjunction (FV , UV),its monad TV = (TV , η, µ),the corresponding Kleisli-category KV

1 the arrows from X to Y in KV are function from X to thefree algebra generated by Y ,

2 hence we can identify them with their free extensions tohomomorphisms from FV(X ) to FV(Y )

3 therefore the KV is isomorphic to the full subcategory of Vgenerated by all FV(X ).




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Kleisli-Categories of Monads from VarietiesKV as extension of Sets

Given:a variety V,

its adjunction (FV , UV),

its monad TV = (TV , η, µ),

the corresponding Kleisli-category KV

1 every function from f : X → Y defines a morphismf # : X → Y of KV according to f # := ηY ◦ f ,

2 the functor X 7→ X , f 7→ f # is an embedding of Sets to KV .




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Given:a variety V,









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Monad −→ Lawvere-Theory


Let:

0, 1, 2, . . . denote the finite cardinals,LV be the full subcategory of KV generated by the finitecardinals,en

i : 1 → n be the function that maps 0 to i − 1πn

i := (eni )#.

ThV := Lop

Then ThV together with the morphisms (πni )∗ is a

Lawvere-Theory.




Summary


Lawvere-Theories isomorphic to ThV

1 The dual of the full subcategory of V generated by all FV(i)for finite cardinals i ,

2 The Lawvere-Theory determined by the clone ofterm-functions of FV(ω) (morphisms from Cn to Cm arem-tuples of n-ary term-functions).




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Models of Lawvere-Theories

Given:

a Lawvere-Theory Th with objects (Ci)i<ω, and withdistinguished projections (πn

i : Cn → C1)1≤i≤n.

Models

A functor A : Th → Sets is called model of Th if it preservesproducts. I.e.

1 A(Ci) = A(C1)n,

2 A(πni ) : A(C1)

n → A(C1) : (x1, . . . , xn) 7→ xi .

The category of all models of Th is denoted by Mod(Th).(Arrows are natural transformations)

Note:1 Lawvere-Theories are essentially the same like abstract

clones,2 Models of Lawvere-Theories are essentially the same like

concrete clones.




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Lawvere-Theory −→ Variety

Given:a variety V,its adjunction (FV , UV),its monad TV = (TV , η, µ),the corresponding Kleisli-category KV ,the corresponding Lawvere-theory ThV .

Mod(ThV) is a concrete category together with the forgetfulfunctor U : Mod(ThV) → Sets : A 7→ A(1),V is a concrete category with the functor UV : V → Sets.

Connection between ThV and V [Lawvere ’66]

Mod(ThV) and V are concretely isomorphic. (i.e. there is anisomorphism the preserves the underlying sets)




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An alternative Definition of Simulations

Given:

Two types T1 = (F1,V1, K1) and T2 = (F2,V2, K2).

Let:ThV1 be the Lawvere-theory corresponding to V1,ThV2 be the Lawvere-theory corresponding to V2,K1 be the model of ThV1 corresponding to K1,K2 be the model of ThV2 corresponding to K2,

Definition of simulations

A Simulation of T1 by T2 is a pair (λ, e) such that1 λ : ThV1 →֒ ThV2 is a theory-embedding, and2 e : K1 →֒ K2 ◦ λ is a natural embedding.




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Simulations in the Language of Universal Algebra

Given:

two types T1 = (F1,V1, K1) and T2 = (F2,V2, K2),a simulation (λ, e) (in terms of Lawvere-theories).

λ : ThV1 →֒ ThV2 induces a concrete functor R : V2 → V1

according to A 7→ A ◦ λ,such functors are known in UA under the nameinterpretation,since λ is an embedding, R is essentially a reduction(though not necessarily from F2 to F1),FV1(X ) embeds via λ into R(FV2),K1 embeds via e into R(K2),




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Σ-Actions

Given:

a type T = (F ,V, K)an alphabet Σ

Let:

FΣ := F ∪ {a(1) | a ∈ Σ}.VΣ the class of all FΣ-algebras such that

1 the F-reduct is in V,2 for a ∈ Σ : a · x is an endomorphism of the F-reduct.

Definition (Σ-action)

The elements of VΣ are called Σ-actions of V.

Remark

We can extend the action of Σ to an action of Σ∗ such that:1 ε · x = x,2 v · (w · x) = (vw) · x.




Summary

Σ-homomorphisms, Σ-subalgebras

1 Homomorphisms between Σ-actions are calledΣ-homomorphisms.

2 Sub-Σ-actions of Σ-actions are called Σ-subalgebras(denoted by A ≤Σ B).




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Representations

Given:

a type T = (F ,V, K),an alphabet Σ

Definition

A representation is a triple A = (ι, A, κ) where1 A is a Σ-action of V,2 ι ∈ A,3 K ∈ V,4 κ : A → K a homomorphism.

Semantics:

The semantics of a representation A is a formal power seriesSA defined by:

(SA, w) := κ(w · ι).

We say that A represents SA.




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Representation from Algebraic Moore-Machines

Given:

a type T = (F ,V, K),an algebraic Moore-machine, M = (Q, Q, Σ, Kδ, κ),q ∈ Q,S the algebraic behavior of M in q.

Proposition

1 Q can be turned into a Σ − action:

a · q := δ(a)(q).

2 M := (q, Q, κ) is a representation.3 the fps SM represented by M is equal to S.




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Given:


Proposition


a · q := δ(a)(q).





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Given:


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a · q := δ(a)(q).





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Given:


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a · q := δ(a)(q).





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Division-Preorder on Representations

Given:

a type T = (F ,V, K),an alphabet Σtwo representations A1 = (ι1, A1, κ1) and A2 = (ι2, A2, κ2).

Definition (Division)

We say that A1 divides A2 (written A1 � A2) if1 ∃B ≤Σ A1,2 ∃ϕ : B ։ A2 surjective Σ-homomorphism

such that the following diagram commutes:

A1 B A2

Kκ1 κ2

ϕ ≤




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Finite and finitary Representations

Given:

a type T = (F ,V, K),

an alphabet Σ,

a representation A = (ι, A, κ)

Definition1 A is called finite representation if A is finitely generated as

F-algebra.2 A is called finitary representation if it divides a finite

representation.




Summary

Algebraic Moore-Machines from finitaryrepresentations

Given:

a type T = (F ,V, K), an alphabeth Σ, and a finitaryrepresentation A = (ι, A, κ)

Let:

A′ = (ι′, A′, κ′) a finite representation such that A � A′,Q be a finite generating set of the F-reduct of A′,Q be an algebra in V freely generated by Q.χ : Q → A′ the canonical epimorphism

define δ : Σ → End(Q) by δ(a)(q) := [t ] where [t ] ∈ Q issuch that χ([t ]) = a · χ(q).Take q0 ∈ Q such that χ(q0) = ι′ define κ′′ : Q → K byκ′′(q) := κ′(χ(q)).

The algebraic behavior of (Q, Q, Σ, K, δ, κ′′) in q0 is equal to thesemantics of A.




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Algebraic Moore-Machines from finitaryrepresentations

Given:

a type T = (F ,V, K), an alphabeth Σ, and a finitaryrepresentation A = (ι, A, κ)

Let:

A′ = (ι′, A′, κ′) a finite representation such that A � A′,Q be a finite generating set of the F-reduct of A′,Q be an algebra in V freely generated by Q.χ : Q → A′ the canonical epimorphism

define δ : Σ → End(Q) by δ(a)(q) := [t ] where [t ] ∈ Q issuch that χ([t ]) = a · χ(q).Take q0 ∈ Q such that χ(q0) = ι′ define κ′′ : Q → K byκ′′(q) := κ′(χ(q)).

The algebraic behavior of (Q, Q, Σ, K, δ, κ′′) in q0 is equal to thesemantics of A.




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Minimal Representations

Given:

a type T = (F ,V, K),an alphabet Σ,S : Σ∗ → K a formal power series.

Theorem

There exists a minimal representation A of S. I.e. for allrepresentations B of S we have that A � B.

Proof-idea.

KΣ∗

be the algebra of formal power-series.KΣ∗

forms a Σ-action with (a · T , w) = (T , aw).πε : KΣ∗

→ K : T 7→ (T , ε) is a homomorphism (its aprojection)take representation of S formed by the triple (S, AS, πε)where AS is the Σ-subalgebra of KΣ∗

generated by S.(S, AS, πε) is minimal




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State-Minimal Representations

There are well-known state-minimimal canonicalrepresentations for

Moore-Machines,

Weighted automata over fields,

deterministic weighted automata over subsemirings ofsemifields

The reason is that in all these varieties1 finitary representations are finite,2 the algebras of the finite representations have

element-minimal generating systems that are all free.




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The problem of State-Minimization

1 In general minimal presentations do not give rise tostate-minimal algebraic Moore-machines,

2 The problem of finding a state-minimal algebraicMoore-machine recognizing a given fps S is equivalent tofinding a set of as few as possible recognizable fps whosegenerated subalgebra contains the Σ-action of the minimalrepresentation of S.

3 even for weighted automata over Qmax nominimization-algorithm is known.




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We discussed several ways of representing varieties.

We translated the definition of simulation into the languageof Lawvere-Theories.

We defined representations and syntactic algebras offormal power series.

We discussed some classical problems from algebraicautomata theory in the context of algebraicMoore-machines.


Representations of Algebraic Moore-Machines

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