a coalgebraic view on context-free languages and streamshzantema/strsemjw.pdf · introduction...

IntroductionRegular expressions and languages

Context-free languagesGeneralizing context-freenessConclusions and further work

A Coalgebraic View on Context-Free Languagesand Streams

Joost Winter

Centrum Wiskunde & Informatica

May 10, 2011

Joost Winter A Coalgebraic View on Context-Free Languages and Streams



OverviewFormal power series, formal languages, and streamsF-CoalgebrasCoalgebras representing formal power seriesHomomorphisms and bisimulationsFinal coalgebras

Overview

I The coalgebraic picture of regular languages and expressions,and likewise that of rational streams and power series, iswell-known.

I It is interesting to see how this work can be extended to, infirst instance, context-free languages.

I In this presentation, a coalgebraic treatment of context-freelanguages through systems of behavioural differentialequations is given.

I This definition format can be generalized to arbitrary formalpower series (in noncommuting variables), including streams,yielding a notion of context-free power series and streams.

I Some examples of streams that are found to be ‘context-free’in this sense are given.





Formal power series, formal languages, and streams

I Given a finite set A called the alphabet, and a semiring R, aformal power series on A with coefficients in R is a function

A∗ → R.

I When R is the Boolean semiring {0, 1} (with 1 + 1 = 1),formal power series on A with coefficients in R correspond toformal languages over the alphabet A.

I When A is a singleton set, formal power series on A withcoefficients on R correspond to streams over R.





F-Coalgebras

Given a functor F , an F -coalgebra consists of a tuple (X , f ):

I X is a set, the carrier set.

I f is a function from X to FX .

Diagrammatically:

X

FX

f

?





Coalgebras representing formal power series (1)

In the this talk, we will be concerned with coalgebras over functorsof the type R × (−)A. But what does this mean?

I R is some semiring.

I × is the cartesian product.

I A is a finite set called the alphabet.

I (−) is a placeholder for the carrier set.

I XA denotes the function space from A to X .






So: a coalgebra (X , f ) over the functor R × (−)A consists of a setX and a function f that maps every x ∈ X to an elementf (x) ∈ R × XA.In this talk, we will use the following notation:

I Given x ∈ X , o(x) (called the output value of x) will be thefirst component of f (x).

I Given x ∈ X , xa (called the a-derivative of x) will be thesecond component of f (x), applied to a.

So: for every x , o(x) is an element of the semiring R, and forevery x and a, xa is an element of X again.When (in the case of streams) the alphabet is a singleton set {a},we usually write x ′ instead of xa.






We can extend the notion of derivatives from alphabet symbols(i.e. elements of A) to words (i.e. elements of A∗) inductively:

I xλ = x

I xa·w = (xa)w .





Homomorphisms and bisimulations (1)

Given two R × (−)A-coalgebras (X , f ) and (Y , g), a functionh : X → Y is a homomorphism if the following hold:

1. For every x ∈ X , o(x) = o(h(x)).

2. For every x ∈ X and a ∈ A, h(xa) = (h(x))a.





Homomorphisms and bisimulations (2)

Given two R × (−)A-coalgebras (X , f ) and (Y , g), a relationR ⊆ X × Y is a bisimulation if the following hold:

1. If (x , y) ∈ R, then o(x) = o(y).

2. If (x , y) ∈ R, then for all a ∈ A, (xa, ya) ∈ R.





Final coalgebras (1)

Consider the 2× (−)A-coalgebra (L, l) defined as follows:

I L is the set of all languages on the alphabet A.I For any L ∈ L:

I o(L) is 1 iff the empty word is in L.I Fa = {w | a · w ∈ L}.

This is a final coalgebra: for every 2× (−)A-coalgebra (X , f ),there is a unique homomorphism h from (X , f ) to (L, l).Given a 2× (−)A-coalgebra (X , f ), and an element x ∈ X , we letJxK denote the value of x under this unique homomorphism.





Final coalgebras (2)

Generalizing the picture from the previous slide to arbitrarysemirings R, the final coalgebras will sets of formal power series:

I S is the set of all formal power series on A with coefficients inR, i.e. the function space from A∗ to R.

I For any F ∈ S:I o(F ) = F (λ).I Fa = G : G (x) = F (a · x).

Again, we let JxK denote the value of x under the finalhomomorphism.





Languages and streams

The presentation given above is a generalization of both coalgebrasrepresenting languages and coalgebras representing streams:

I When we choose the Boolean semiring {0, 1}, we getcoalgbras representing languages over the alphabet A.

I When we choose a singleton alphabet A, we get coalgebrasrepresenting streams over the semiring R.

The next few slides will be about coalgebras for the functorD := 2× (−)A of formal languages.




The coalgebra of regular expressionsKleene’s theorem, coalgebraically

The coalgebra of regular expressions

The set E of regular expressions over a finite alphabet A and thesemiring (2,+, ·, 0, 1) can be defined as follows:

t ::= a ∈ A | r ∈ 2 | t + t | t · t | t∗

We can assign a D-coalgebra structure to this set of regularexpressions by specifying the output values and derivatives for eachexpression, giving us a D-coalgebra (E , e):

t o(t) tar ∈ 2 r 0b ∈ A 0 if b = a then 1 else 0u + v o(u) + o(v) ua + vau · v o(u) · o(v) ua · v + o(u) · vau∗ 1 ua · u∗




The coalgebra of regular expressionsKleene’s theorem, coalgebraically

Kleene’s theorem, coalgebraically

I For any language L ∈ L, we define the subcoalgebra generatedby L as

〈L〉 := {Lw |w ∈ A∗}

It is easy to see that this indeed generates a subcoalgebra:given any K ∈ 〈L〉, it is easy to see that for every a ∈ A, alsoKa ∈ 〈L〉. In other words, 〈L〉 is closed under takingderivatives to alphabet symbols.

I Kleene’s theorem, coalgebraically (Rutten, 1998): For anyL ∈ L, 〈L〉 is finite iff there is a regular expression t such thatL = JtK.




Introduction: context-free grammars and languagesSystems of equationsFrom CFGs to systems of equationsFrom systems of equations to CFGs

Introduction: context-free grammars and languages

I The ‘next step up’ from regular expressions and languages,and finite automata, in the Chomsky hierarchy, are thecontext-free languages and grammars, and pushdownautomata.

I We will present a format of coinductively defined systems ofequations: it turns out that these systems of equationschracterize precisely the context-free languages.





Systems of equations (1)

We will use terms t specified as follows:

t ::= a ∈ A | x ∈ X | r ∈ 2 | t + t | t · t

where X is a finite set of variables, and A, as before, is a finitealphabet. Given X , we let TX denote the set of terms over X .A well-formed system of equations, for a set of variables X ,consists of:

1. For every x ∈ X , exactly one equation of the form o(x) = v ,where v ∈ {0, 1}.

2. For every x ∈ X and a ∈ A, exactly one equation of the formxa = t, where t ∈ TX .






Alternatively, we can consider a well-formed system of equations asa mapping

f : X → 2× TXA

We can extend such a mapping f to the D-coalgebra (TX , f̄ )generated by (X , f ) as follows:

t o(t) tax ∈ X o(x) xa (as specified by f )r ∈ 2 r 0b ∈ A 0 if b = a then 1 else 0u + v o(u) + o(v) ua + vau · v o(u) · o(v) ua · v + o(u) · va






I This construction can be summarized diagrammatically:

X ⊂

i- TX

J K- L

2× TXA

f

?-

�

f̄

2× LA

l

?

I Proposition: A language L is context-free iff there is awell-formed system of equations (X , f ) and an x ∈ X , suchthat JxK = L w.r.t. the coalgebra (TX , f̄ ) generated by it.





From CFGs to systems of equations (1)

I We say a context-free grammar is in weak Greibach normalform, if every production rule has a right hand side eitherequal to the empty word λ, or of the form a · t.

I As the name implies, this is a weakening of the more familiarGreibach normal form. As a direct result, every CFG can berepresented in weak Greibach normal form.





From CFGs to systems of equations (2)

We transform a CFG G in weak Greibach normal form into asystem of equations as follows:

I We let the set X of variables be equal to the set ofnonterminals in the grammar.

I Given a x ∈ X , we set o(x) = 1 iff the grammar contains aproduction rule x → λ.

I Given a x ∈ X and an a ∈ A, we set

xa =∑{w | x → a · w}

Given an initial symbol x0 ∈ X , we now have o((x0)w ) = 1 (and,hence, w ∈ Jx0K) iff w is in the language generated by G .





From systems of equations to CFGs

Conversely, given a system of equations, we can construct a CFGin weak Greibach normal form:

I We first transform the system of equations to a new,equivalent, system, in which all derivatives are in disjunctivenormal form, and do not contain any superfluous 0s or 1s.

I Derivatives in this new system are disjunctions of sequences ofalphabet symbols and variables.

I We let the grammar include a rule x → λ whenever o(x) = 1.

I We let the grammar include a rule x → a · w , whenever w is asequence of alphabet symbols and variables occurring as adisjunct in xa.




Context-free streams

Generalizing context-freeness

I Note that most of the definitions on the earlier slides do notnecessarily require the underlying semiring to be the Booleansemiring!

I Taking the definition of regular expressions and applying it toarbitrary semirings, we obtain rational streams and powerseries, which are well-known.

I Furthermore, we can easily generalize the notion above, ofcontext-free languages, to context-free streams andcontext-free power series.





Context-free streams (1)

When we take the semiring of natural numbers as underlyingsemiring, the Catalan numbers

1, 1, 2, 5, 14, 42, 132, 429, 1430, . . .

are context-free, and are generated by the following system ofequations:

o(x) = 1 x ′ = x · x






When we take the Boolean field (with 1 + 1 = 0) as underlyingsemiring, the Prouhet-Thue-Morse sequence

10 01 0110 01101001 . . .

is context-free, and is generated by the following system ofequations:

o(x) = 1 x ′ = y

o(y) = 0 y ′ = z

o(z) = 0 z ′ = x + y + z + z · wo(w) = 1 w ′ = y · w






Still using the Boolean field as underlying semiring, thecontext-free system of equations

o(x) = 1 x ′ = y

o(y) = 1 y ′ = z

o(z) = 0 z ′ = w

o(w) = 1 w ′ = v

o(v) = 1 v ′ = v + w + v · x + x · x

gives us the paperfolding sequence, or Dragon curve sequence. . .




Conclusions and further work

I There is a very neat coalgebraic representation of regularexpressions, and Kleene’s theorem can be expressed succinctlyin a coalgebraic fashion.

I We have extended this work towards context-free languagesand grammars, and provided a coalgebraic characterizationusing systems of equations.

I The first steps towards a generalization to other functors ofthe type R × (−)A havs been made, and there are some neatexamples of context-free streams.

I Future work: further investigate this notion of ‘generalizedcontext-freeness’ of power series, and see how this relates toother, existing notions.




Bibliography

[Jacobs/Rutten, 1997] Bart Jacobs, Jan Rutten, A Tutorial on(Co)Algebras and (Co)Induction

[Rutten, 1998] Jan Rutten, Automata and Coinduction (AnExercise in Coalgebra)

[Rutten, 2005] Jan Rutten, A Coinductive Calculus of Streams

[Silva, 2010] Alexandra Silva, Kleene Coalgebra

[Winter/Bonsangue/Rutten, 2011] Joost Winter, MarcelloBonsangue, Jan Rutten, Context-free Languages,Coalgebraically


a coalgebraic view on context-free languages and streamshzantema/strsemjw.pdf · introduction...

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