(co-)algebraic automata theoryober:goncharov.pdf · (co-)algebraic automata theory sergey...

(Co-)algebraic Automata Theory

Sergey Goncharov, Stefan Milius, Alexandra Silva

Erlangen, 5.11.2013

Chair 8 (Theoretical Computer Science)

Short Histrory of Coalgebraic Invasion to Automata Theory

• Deterministic automata as coalgebras [Rutten, 1998].

• Generalized regular expressions and Kleene’s theorem for

(Kripke-)polinomial functors [Silva, 2010].

• Generalized powerset construction [Silva et al., 2010].

• Regular expressions for equationally presented functors and

monads [Myers, 2013].

• Context-free languages coalgebraically [Winter et al., 2013].

Erlangen, 5.11.2013 | Sergey Goncharov, Stefan Milius, Alexandra Silva | Chair 8 (Theoretical Computer Science) | 2

Base Case: Moore AutomataMoore automaton with input alphabet A and output alphabet B is given by

tm : X × A→ X (transition) and om : X → B (output)

Put differently, a Moore automaton is a coalgebra m : X → B × X A on Set.

A final LA,B -coalgebra is carried by the set BA∗ of formal power series on B.

Its coalgebra structure ι : BA∗ → B × (BA∗)A is given by

oι(σ : A∗ → B) = σ(ε) and t ι(σ : A∗ → B, a) = λw .σ(a · w).

Equivalently, the LA,B-coalgebra structure can be given in terms ifderivatives: given w ∈ A∗,

∂ε(σ) = oι(σ) and ∂a·w(σ) = t ι(∂w(σ), a)

Particular important case is B = 2. Then BA∗ ' P(A∗) is the set of all formal

languages over A.Erlangen, 5.11.2013 | Sergey Goncharov, Stefan Milius, Alexandra Silva | Chair 8 (Theoretical Computer Science) | 3










languages over A.

LA,BX := B × X A











languages over A.

LA,BX := B × X A We shall assume finite


Rationality of Formal Power Series

By the the universal property of the final coalgebra for any m : X → B × X A

there exists a unique LA,B-coalgebra homomorphism m̂ such that:

X

m��

m̂ // BA∗

ι

��

B × X AB×m̂A

// B × (BA∗)A

Given x ∈ X , JxKm := m̂(x) is the “language” recognized by m at x .

A formal power series σ : A∗ → B is rational if the set {∂w(σ) | w ∈ A∗} isfinite.

Theorem. A formal power series σ is rational iff σ = JxKm̂ for somem : X → B × X A and x ∈ X .Erlangen, 5.11.2013 | Sergey Goncharov, Stefan Milius, Alexandra Silva | Chair 8 (Theoretical Computer Science) | 6

Expressions for Moore Automata

Let σ : A∗ → B be rational, let A = {a1, . . . , an}, and let{σ1, . . . , σk} = {∂w(σ) | w ∈ A∗}. Then σ = σ1 can be found as the solutionof system of recursive equations

σi = a1.σi1 t · · · t an.σin t ci, (F)which should be read: for all i, j , ∂aj(σi) = σij and σi(ε) = ci .

System (F) uniquely determines the σ1, . . . , σn. Any recursive equationof (F) can be rewritten as

σi = µσi. (a1.σi1 t · · · t an.σin t ci)

where µ is the fixpoint operator binding the occurrences of σi in the rightterm. One can successively eliminate σi (i 6= 1) and obtain a “solution” σ = tof (F) in σ where t is a closed term given by

γ ::= µX . (a1.δ t · · · t an.δ t B) δ ::= X | γ


Expressions for Moore Automata

Let σ : A∗ → B be rational, let A = {a1, . . . , an}, and let{σ1, . . . , σk} = {∂w(σ) | w ∈ A∗}. Then σ = σ1 can be found as the solutionof system of recursive equations

σi = a1.σi1 t · · · t an.σin t ci, (F)which should be read: for all i, j , ∂aj(σi) = σij and σi(ε) = ci .

System (F) uniquely determines the σ1, . . . , σn. Any recursive equationof (F) can be rewritten as

σi = µσi. (a1.σi1 t · · · t an.σin t ci)

where µ is the fixpoint operator binding the occurrences of σi in the rightterm. One can successively eliminate σi (i 6= 1) and obtain a “solution” σ = tof (F) in σ where t is a closed term given by

γ ::= µX . (a1.δ t · · · t an.δ t B) δ ::= X | γ

Finite set of variables


Kleene’s Theorem for Moore Automata

Expressions

γ ::= µX . (a1.δ t · · · t an.δ t B) δ ::= X | γfaithfully represent all rational formal power series. We can introduceBrzozowski derivatives: given e = µx . (a1.e1 t · · · t an.e1 t c), let

∂ai(e) = ei[e/x ] o(e) = c

By induction, let ∂a·w = ∂a ◦ ∂w . Then, let JeK(w) = o(∂w(e)).

Brzozowski’s theorem: for any expression e, {∂w(e) | w ∈ A∗} is finite.

Kleene’s theorem: for any Moore automaton m and any state x ∈ X thereis an expression e such that JxKm = JeK and converselly, for any expressione there are m and x such that JxKm = JeK.


Expressions for Deterministic FSM: Example

B = 2

q0start q1 q2

ba

ba

a

b

e0 = a.e0 t b.e1 t 0e1 = a.e0 t b.e2 t 0e2 = a.e0 t b.e0 t 1

Hence: e0 = µx .(a.x t b.µy .(a.x tb.µz.(a.x t b.x t 1) t 0) t 0)

Equivalently: e0 = (a + ba + b)∗b.


Powerset Construction

Let m : X → B × (PωX )A where B is a join-semilattice with bottom.We can extend it to m ] : m : PωX → B × (PωX )A by defining

tm ]

(s ∈ PωX , a) = {tm(x , a) | x ∈ s} om ]

(s ∈ PωX ) =∨x∈s

o(x)

X{--}

//

m��

PωX m̂ ]//

m ]xxrrrrrrrrrrrrrrrrrr

BA∗

ι

��

B × (PωX )A id×(m̂ ])A// B × (BA∗)A

Let JxKm = m̂{x}.Theorem: JxKm is the formal power series accepted by m at state x ∈ X .


Powerset Construction

Let m : X → B × (PωX )A where B is a join-semilattice with bottom.We can extend it to m ] : m : PωX → B × (PωX )A by defining

tm ]

(s ∈ PωX , a) = {tm(x , a) | x ∈ s} om ]

(s ∈ PωX ) =∨x∈s

o(x)

X{--}

//

m��

PωX m̂ ]//

m ]xxrrrrrrrrrrrrrrrrrr

BA∗

ι

��

B × (PωX )A id×(m̂ ])A// B × (BA∗)A

Equivalently given by tm : X × A→ PωX and om : X → B

Let JxKm = m̂{x}.Theorem: JxKm is the formal power series accepted by m at state x ∈ X .


Generalized Powerset Construction and T-automata

Definition: Let T be a monad. A T-automaton is triple of maps

om : X → B, tm : X × A→ TX , am : TB → B

where the latter map satisfies the axioms of T-algebra.

A T-automaton can be identified as a coalgebra m : X → B × (TX )A. Let

tm ]

(p, a) = do x ← p; tm(x , a) om ]

(p) = am(do x ← p; η(om(x))).

ThenX

η//

m��

TX m̂ ]//

m ]yytttttttttttttttttt

BA∗

ι

��

B × (TX )A id×(m̂ ])A// B × (BA∗)A

and we can define JxKm = Jη(x)Km ].


Algebraic Theories

An algebraic theory is given by a signature Σ and a set of equations.

Any algebraic theory E defines a monad: TX = ‘set of Σ-terms over Xmodulo E ’; η coerces a variable to a term; do x ← p; q substituteseach x in p : TX with q(x).

Definition: We call a monad T finitely presented if it is generated by afinitely axiomatizable algebraic theory E over finite Σ.

Example: Finite powerset monad Pω ⇐⇒ join semilattices with bottom.

Finite state monad TX = (X × S)S ⇐⇒ finite mnemoids(lookupl : X v → X , updatel,v : X → X ).

Non-Example: Distribution monad (TX = ‘set of finitely supportedprobability distributions’).


Delimited Stores

Definition: Given a set S, let L be a nonempty meet-semilattice ofequivalence relations of finite index on S. An L-delimited store monad is asubmonad T of the store monad on S such that: for any ∼ ∈ L and any〈r , t〉 : S → (X × S) ∈ TX , there exists ≈ ∈ L finer than ∼ such that tpreserves ≈ and r sends ≈-equivalent elements to equal.

Examples: Stack monad: L consists of equivalences ∼n on Γ ∗ identifyingany s and s′ with the same n-prefixes.

Dequeue monad: L consists of equivalences ∼n,m on Γ∗ identifying any sand s′ with the same n-prefixes and m-postfixes.

Tape monad: L consists of equivalences ∼n,m on Γ∗ × (Γ + 1)× Γ∗

identifying any 〈s, x , t〉 and 〈s′, x , t ′〉 for which s and s′ have the samen-postfixes and t and t ′ have the same m-prefixes.


Delimited Stores

Definition: Given a set S, let L be a nonempty meet-semilattice ofequivalence relations of finite index on S. An L-delimited store monad is asubmonad T of the store monad on S such that: for any ∼ ∈ L and any〈r , t〉 : S → (X × S) ∈ TX , there exists ≈ ∈ L finer than ∼ such that tpreserves ≈ and r sends ≈-equivalent elements to equal.

Examples: Stack monad: L consists of equivalences ∼n on Γ ∗ identifyingany s and s′ with the same n-prefixes.

Dequeue monad: L consists of equivalences ∼n,m on Γ∗ identifying any sand s′ with the same n-prefixes and m-postfixes.

Tape monad: L consists of equivalences ∼n,m on Γ∗ × (Γ + 1)× Γ∗

identifying any 〈s, x , t〉 and 〈s′, x , t ′〉 for which s and s′ have the samen-postfixes and t and t ′ have the same m-prefixes.

Finite tape alphabet


Stack Automata

For a stack monad T, TX ⊆ (X × Γ∗)Γ∗ consists of those state transformers〈r , t〉 : Γ∗ → (X × Γ∗), for which

r(u · w) = r(u) t(u · w) = t(u) · wwhenever |u| ≥ n for some n.

The stack monad is finitely presentable w.r.t. pop : X Γ+1 → X andpushi : X → X (i ≤ |Γ|):

pushi(pop(x1, . . . , xn, y)) = xi

pop(push1(x), . . . , pushn(x), x) = xpop(x1, . . . , xn, pop(y1, . . . , yn, z)) = pop(x1, . . . , xn, z)

For a deterministic stack automaton m : X → 2L[Γ∗] ×TX am is the

w. p. transformer: for any s ∈ S and 〈r , t〉 : (2L[Γ∗] × Γ∗)Γ∗ ∈ T (2L[Γ∗])

am(f )(s) = 1 iff ∃s′ ∈ Γ∗. s ∈ t−1(s′) ∧ r(s)(s′) = 1Erlangen, 5.11.2013 | Sergey Goncharov, Stefan Milius, Alexandra Silva | Chair 8 (Theoretical Computer Science) | 17

Stack Automata

For a stack monad T, TX ⊆ (X × Γ∗)Γ∗ consists of those state transformers〈r , t〉 : Γ∗ → (X × Γ∗), for which

r(u · w) = r(u) t(u · w) = t(u) · wwhenever |u| ≥ n for some n.

The stack monad is finitely presentable w.r.t. pop : X Γ+1 → X andpushi : X → X (i ≤ |Γ|):

pushi(pop(x1, . . . , xn, y)) = xi

pop(push1(x), . . . , pushn(x), x) = xpop(x1, . . . , xn, pop(y1, . . . , yn, z)) = pop(x1, . . . , xn, z)

For a deterministic stack automaton m : X → 2L[Γ∗] ×TX am is the

w. p. transformer: for any s ∈ S and 〈r , t〉 : (2L[Γ∗] × Γ∗)Γ∗ ∈ T (2L[Γ∗])

am(f )(s) = 1 iff ∃s′ ∈ Γ∗. s ∈ t−1(s′) ∧ r(s)(s′) = 1

p ∈ 2L[Γ∗] iff it is stable under some ∼n∈ L


(Deterministic) Real-time CFL

Deterministic real-time context-free languages are exactly those, which arerecognized by deterministic real-time (with no internal transitions)push-down automata.

Theorem. Let m be a stack automaton. Then for any x and any w ∈ Γ∗,JxKm(w) is a deterministic real-time CFL; and conversely: any deterministicreal-time CFL can be obtained in this way.

Conjecture. Real-time CFL (=proper CFL) are recognized exactly bynon-deterministic stack automata. That is, automata overTX ⊆ Pω(X × Γ∗)Γ∗, such that 〈r , t〉 : Γ∗ → Pω(X × Γ∗) ∈ TX iff

r(u · w) = r(u) t(u · w) = {u′ · w | u′ ∈ t(u)}whenever |u| ≥ n for some n.

Theorem. Given x ∈ X and m : X → B× TX A with finite B, JxKm is rational.


(Deterministic) Real-time CFL

Deterministic real-time context-free languages are exactly those, which arerecognized by deterministic real-time (with no internal transitions)push-down automata.

Theorem. Let m be a stack automaton. Then for any x and any w ∈ Γ∗,JxKm(w) is a deterministic real-time CFL; and conversely: any deterministicreal-time CFL can be obtained in this way.

Conjecture. Real-time CFL (=proper CFL) are recognized exactly bynon-deterministic stack automata. That is, automata overTX ⊆ Pω(X × Γ∗)Γ∗, such that 〈r , t〉 : Γ∗ → Pω(X × Γ∗) ∈ TX iff

r(u · w) = r(u) t(u · w) = {u′ · w | u′ ∈ t(u)}whenever |u| ≥ n for some n.

Theorem. Given x ∈ X and m : X → B× TX A with finite B, JxKm is rational.

T = Pω⊗ “stack monad over Γ∗”


Tensors

Given two algebraic theories E1 and E2 their tensor E1 ⊗ E2 is obtained byjoining operations and equations and forcing tensor laws of the form

f (g(x11 , . . . , x

1m), . . . , g(xn

1 , . . . , xnm)) = g(f (x1

1 , . . . , xn1 ), . . . , f (x1

m, . . . , xnm))

for any f : X n → X from E1 and any g : X m → X from E2.

This induces tensor of monads; and if T1 and T2 are finitely presentable thenso is T1 ⊗ T2.

Properties: • T⊗ (−× S)S yields state monad transformer T (X × S)S.

• Tape monad is the tensor of stack monad with itself.

• Finite powermonad T⊗ Pω can be axiomatized in terms of unary

operations, + and 0 due to tensor law

f (x1, . . . , xn) = f (x1, . . . , 0) + . . . + f (0, . . . , xn)Erlangen, 5.11.2013 | Sergey Goncharov, Stefan Milius, Alexandra Silva | Chair 8 (Theoretical Computer Science) | 21

Effectful Fixpoint Expressions

Let E be an algebraic theory over signature Σ corresponding to T. Given afinite A = {a1, . . . , an} and a T-algebra B effectful fixpoint expressions EB

are closed terms γ given by the following grammar

γ ::= µX . (a1.δ t · · · t an.δ t B) δ ::= X | γ | f (δ, . . . , δ) (f ∈ Σ)

These form a Σ-algebra: given ek = µX .(a1.ek

1 t · · · t an.ekn t bk

),

f (e1, . . . , em) = µX .(a1.f (e1

1, . . . , em1 ) t · · · t an.f (e1

n, . . . , emn ) t f (b1, . . . , bm)

)but also a LA,B-coalgebra:

o(µX . (a1.e1 t · · · t an.en t b)) = b∂ai(µX . (a1.e1 t · · · t an.en t b)) = ei[e/X ]

Let e ∼ e′ iff for all w ∈ A∗, o(∂w(e)) = o(∂w(e′)).


Kleene’s Theorem for Effectful Fixpoint Expressions

Given e ∈ ET,B and w ∈ A∗, let JeK(w) = o(∂w(e)).

Conjecture (Kleene’s Theorem): for any T-automaton with a finitelypresentable T and any x ∈ X there is e ∈ ET,B such that JxKm = JeK.Conversely, for any e ∈ ET,B there is m and x ∈ X , such that JxKm = JeK.

Conjecture: ∼ is properly Π01 (by reduction to CFL).

Conjecture: Let T be the tensor product of n instances of the stack monadand let m be a delimited store monad over T. Then deciding if w ∈ JeKm islinear time. Deciding if w ∈ JeKm with T replaced by T⊗ Pω isnondeterministic linear time.

Open Question: Does it generalize to any delimited store monad?


Thank You for Your Attention!

Myers, R. (2013).

Rational Coalgebraic Machines in Varieties: Languages, Completeness

and Automatic Proofs.

PhD thesis, Imperial College London.

Rutten, J. J. M. M. (1998).

Automata and coinduction (an exercise in coalgebra).

In Sangiorgi, D. and de Simone, R., editors, CONCUR, volume 1466 of

Lecture Notes in Computer Science, pages 194–218. Springer.

Silva, A. (2010).

Kleene coalgebra.

PhD thesis, Radboud Universiteit Nijmegen.


Silva, A., Bonchi, F., Bonsangue, M. M., and Rutten, J. J. M. M. (2010).

Generalizing the powerset construction, coalgebraically.

In FSTTCS, volume 8 of LIPIcs, pages 272–283.

Winter, J., Bonsangue, M. M., and Rutten, J. J. M. M. (2013).

Coalgebraic characterizations of context-free languages.

Logical Methods in Computer Science, 9, abs/1308.1228(3).


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