Weighted automata and weighted logics with discounting

Weighted Automata and Weighted Logicswith Discounting

Manfred Droste1 and George Rahonis2

1Leipzig University, Germany2Aristotle University of Thessaloniki, Greece

CIAA 2007Prague, July 16-18, 2007

CIAA 2007 (Prague) M. Droste, G. Rahonis July 16, 2007 1 / 19

A alphabet, Aω = fall in�nite words over Ag

Theorem (Büchi �62)

Let L � Aω. The following are equivalent:

L is Büchi recognizable

L is MSO-de�nable

L is ω-rational

L is Muller recognizable (McNaughton �66)

Goal of this work:Generalization to weighted automata

Question: Is this task just a simple generalization?Answer: No






L is MSO-de�nable

L is ω-rational









L is MSO-de�nable

L is ω-rational









L is MSO-de�nable

L is ω-rational









L is MSO-de�nable

L is ω-rational









L is MSO-de�nable

L is ω-rational









L is MSO-de�nable

L is ω-rational



Question: Is this task just a simple generalization?

Answer: No






L is MSO-de�nable

L is ω-rational





A alphabet, K semiring

Weighted automaton over A and K (Schützenberger �61)

A = (Q, in,wt, out)

Q : �nite state setin : Q ! K initial distributionwt : Q � A�Q ! K weight transition mappingout : Q ! K �nal distribution

a path of A over w = a0a1 . . . an�1 2 A� :Pw = (ti )0�i�n�1, ti = (qi , ai , qi+1) (0 � i � n� 1)weight(Pw ) := in(q0) � ∏

0�i�n�1wt(ti ) � out(qn)

behavior of A: kAk : A� ! Kw 7! ∑

Pwweight(Pw )

a path of A over w = a0a1 . . . 2 Aω :Pw = (ti )i�0, ti = (qi , ai , qi+1) (i � 0)Problem: weight(Pw ) =?




A = (Q, in,wt, out)





Pwweight(Pw )





A = (Q, in,wt, out)

Q : �nite state set

in : Q ! K initial distributionwt : Q � A�Q ! K weight transition mappingout : Q ! K �nal distribution




Pwweight(Pw )





A = (Q, in,wt, out)

Q : �nite state setin : Q ! K initial distribution

wt : Q � A�Q ! K weight transition mappingout : Q ! K �nal distribution




Pwweight(Pw )





A = (Q, in,wt, out)

Q : �nite state setin : Q ! K initial distributionwt : Q � A�Q ! K weight transition mapping

out : Q ! K �nal distribution




Pwweight(Pw )





A = (Q, in,wt, out)





Pwweight(Pw )





A = (Q, in,wt, out)


a path of A over w = a0a1 . . . an�1 2 A� :

Pw = (ti )0�i�n�1, ti = (qi , ai , qi+1) (0 � i � n� 1)weight(Pw ) := in(q0) � ∏



Pwweight(Pw )





A = (Q, in,wt, out)


a path of A over w = a0a1 . . . an�1 2 A� :Pw = (ti )0�i�n�1, ti = (qi , ai , qi+1) (0 � i � n� 1)

weight(Pw ) := in(q0) � ∏0�i�n�1

wt(ti ) � out(qn)


Pwweight(Pw )





A = (Q, in,wt, out)





Pwweight(Pw )





A = (Q, in,wt, out)




behavior of A: kAk : A� ! K

w 7! ∑Pwweight(Pw )





A = (Q, in,wt, out)





Pwweight(Pw )





A = (Q, in,wt, out)





Pwweight(Pw )

a path of A over w = a0a1 . . . 2 Aω :

Pw = (ti )i�0, ti = (qi , ai , qi+1) (i � 0)Problem: weight(Pw ) =?




A = (Q, in,wt, out)





Pwweight(Pw )

a path of A over w = a0a1 . . . 2 Aω :Pw = (ti )i�0, ti = (qi , ai , qi+1) (i � 0)

Problem: weight(Pw ) =?




A = (Q, in,wt, out)





Pwweight(Pw )

a path of A over w = a0a1 . . . 2 Aω :Pw = (ti )i�0, ti = (qi , ai , qi+1) (i � 0)Problem: weight(Pw ) =?CIAA 2007 (Prague) M. Droste, G. Rahonis July 16, 2007 3 / 19

SolutionConsider complete semirings

S. Eilenberg, Z. Ésik & W. Kuich, M. Droste & G. Rahonis

Very particular automata

K. Culik II & J. Karhumäkireal functions, digital image processing

Discounting as inmathematical economics, game theory, Markov processes

M. Droste & D. Kuske

We adopt the discounting method

Underlying semirings:

max-plus: Rmax = (R+ [ f�∞g,_,+,�∞, 0)min-plus: Rmin = (R+ [ f∞g,min,+,∞, 0)

























K. Culik II & J. Karhumäki

real functions, digital image processing

































































max-plus: Rmax = (R+ [ f�∞g,_,+,�∞, 0)

min-plus: Rmin = (R+ [ f∞g,min,+,∞, 0)












p 2 R+, p � (�∞) = �∞

the mapping p : Rmax ! Rmax, x 7! p � xis an endomorphism of Rmax

Each endomorphism of Rmax is of the above form(Droste and Kuske �06)


p 2 R+, p � (�∞) = �∞the mapping p : Rmax ! Rmax, x 7! p � xis an endomorphism of Rmax



p 2 R+, p � (�∞) = �∞the mapping p : Rmax ! Rmax, x 7! p � xis an endomorphism of Rmax



De�nitionA weighted Büchi automaton over A and Rmax : A = (Q, in,wt,F )

Q : �nite state setin : Q ! Rmax initial distributionwt : Q � A�Q ! Rmax weight transition mappingF � Q : �nal state set

A weighted Muller automaton over A and Rmax : A = (Q, in,wt,F )

Q, in, wt as aboveF � P(Q) �nal state sets.

a path of A over w = a0a1 . . . 2 Aω :Pw = (ti )i�0, ti = (qi , ai , qi+1) (i � 0)let 0 � p < 1weight(Pw ) := in(q0) + ∑

i�0pi � wt(ti )

C = maxfin(q),wt(t) j q 2 Q, t 2 Q � A�Qgweight(Pw ) � C + C � 1

1�p < ∞



Q : �nite state set

in : Q ! Rmax initial distributionwt : Q � A�Q ! Rmax weight transition mappingF � Q : �nal state set




i�0pi � wt(ti )


1�p < ∞



Q : �nite state setin : Q ! Rmax initial distribution

wt : Q � A�Q ! Rmax weight transition mappingF � Q : �nal state set




i�0pi � wt(ti )


1�p < ∞



Q : �nite state setin : Q ! Rmax initial distributionwt : Q � A�Q ! Rmax weight transition mapping

F � Q : �nal state set




i�0pi � wt(ti )


1�p < ∞







i�0pi � wt(ti )


1�p < ∞







i�0pi � wt(ti )


1�p < ∞




A weighted Muller automaton over A and Rmax : A = (Q, in,wt,F )Q, in, wt as above

F � P(Q) �nal state sets.


i�0pi � wt(ti )


1�p < ∞




A weighted Muller automaton over A and Rmax : A = (Q, in,wt,F )Q, in, wt as aboveF � P(Q) �nal state sets.


i�0pi � wt(ti )


1�p < ∞





a path of A over w = a0a1 . . . 2 Aω :Pw = (ti )i�0, ti = (qi , ai , qi+1) (i � 0)

let 0 � p < 1weight(Pw ) := in(q0) + ∑

i�0pi � wt(ti )


1�p < ∞





a path of A over w = a0a1 . . . 2 Aω :Pw = (ti )i�0, ti = (qi , ai , qi+1) (i � 0)let 0 � p < 1

weight(Pw ) := in(q0) + ∑i�0pi � wt(ti )


1�p < ∞






i�0pi � wt(ti )


1�p < ∞






i�0pi � wt(ti )

C = maxfin(q),wt(t) j q 2 Q, t 2 Q � A�Qg

weight(Pw ) � C + C � 11�p < ∞






i�0pi � wt(ti )


1�p < ∞


InQ (Pw ) = fq 2 Q j 9ωi : ti = (q, ai , qi+1)g

Pw is successful if�InQ (Pw ) \ F 6= ∅ BüchiInQ (Pw ) 2 F Muller

The p-behavior of A : kAk : Aω ! Rmax

w 7!_

Pw successful

weight(Pw )

Example a 2 A, k 2 Rmax, Büchi automaton A = (fqg, in,wt, fqg)in(q) = 0

wt(q, a, q) = k, wt(q, a0, q) = 0 (a0 2 A, a0 6= a)Then w 7! k � ∑

i2ω,ai=api

gives the discounted cost of a�s in w





w 7!_

Pw successful

weight(Pw )



i2ω,ai=api






w 7!_

Pw successful

weight(Pw )



i2ω,ai=api






w 7!_

Pw successful

weight(Pw )



i2ω,ai=api






w 7!_

Pw successful

weight(Pw )

Example a 2 A, k 2 Rmax, Büchi automaton A = (fqg, in,wt, fqg)

in(q) = 0


i2ω,ai=api






w 7!_

Pw successful

weight(Pw )



i2ω,ai=api






w 7!_

Pw successful

weight(Pw )


wt(q, a, q) = k, wt(q, a0, q) = 0 (a0 2 A, a0 6= a)

Then w 7! k � ∑i2ω,ai=a

pi






w 7!_

Pw successful

weight(Pw )



i2ω,ai=api



Rp�ω�recmax hhAωii = fall Büchi (or ω-) recognizable seriesg

Rp�M�recmax hhAωii = fall Muller recognizable seriesg

Theorem (�rst main result)

Rp�ω�recmax hhAωii = Rp�M�rec

max hhAωii

Rp�ω�recmax hhAωii is closed under max, sum and scalar sum and

application of strict alphabetic homomorphisms h : Aω ! Bω andtheir inverses

Let L 2 ω� Rec(A). Then 1L 2 Rp�ω�recmax hhAωii.






max hhAωii









max hhAωii









max hhAωii









max hhAωii





Weighted MSO logic with discounting

De�nitionThe syntax of the weighted MSO-formulas over A and Rmax is given byϕ := k j Pa(x) j :Pa(x) j S(x , y) j :S(x , y) j x � y j :(x � y)

j x 2 X j :(x 2 X ) j ϕ _ ψ j ϕ ^ ψ j 9x � ϕ j 9X � ϕ j 8x � ϕk 2 Rmax, a 2 A.

MSO(Rmax,A) = fall weighted MSO-formulas over A and Rmaxg.

Representation of w = a0a1 . . . 2 Aω, by w :=�ω,�, (Ra)a2A

�where Ra := fi 2 ω j ai = ag (a 2 A)Consider ϕ 2 MSO(Rmax,A), Aϕ = A� Free(ϕ)w 2 Aω, σ : assignment of free variables of ϕ to positions /sets of positions in w(w , σ) 2 Aω

ϕ





MSO(Rmax,A) = fall weighted MSO-formulas over A and Rmaxg.Representation of w = a0a1 . . . 2 Aω, by w :=

�ω,�, (Ra)a2A

�where Ra := fi 2 ω j ai = ag (a 2 A)

Consider ϕ 2 MSO(Rmax,A), Aϕ = A� Free(ϕ)w 2 Aω, σ : assignment of free variables of ϕ to positions /sets of positions in w(w , σ) 2 Aω

ϕ






�ω,�, (Ra)a2A

�where Ra := fi 2 ω j ai = ag (a 2 A)Consider ϕ 2 MSO(Rmax,A), Aϕ = A� Free(ϕ)

w 2 Aω, σ : assignment of free variables of ϕ to positions /sets of positions in w(w , σ) 2 Aω

ϕ






�ω,�, (Ra)a2A

�where Ra := fi 2 ω j ai = ag (a 2 A)Consider ϕ 2 MSO(Rmax,A), Aϕ = A� Free(ϕ)w 2 Aω, σ : assignment of free variables of ϕ to positions /sets of positions in w(w , σ) 2 Aω

ϕ


De�nition (Semantics of MSO(Rmax,A)-formulas)

Semantics of ϕ 2 MSO(Rmax,A) : kϕk : Aωϕ ! Rmax.

The coe¢ cient (kϕk , (w , σ)) 2 Rmax is de�ned inductively (σ a validassignment) by:

(kkk , (w , σ)) = k

(kPa(x)k , (w , σ)) =�

0 if ai = a for i = σ(x)�∞ otherwise

(kS(x , y)k , (w , σ)) =�

0 if σ(x) + 1 = σ(y)�∞ otherwise

(kx � yk , (w , σ)) =�

0 if σ(x) � σ(y)�∞ otherwise

(kx 2 Xk , (w , σ)) =�

0 if σ(x) 2 σ(X )�∞ otherwise

(k:ϕk , (w , σ)) =�

0 if (kϕk , (w , σ)) = �∞�∞ if (kϕk , (w , σ)) = 0 , provided that

ϕ is of the form Pa(x), S(x , y), x � y or x 2 X





(kkk , (w , σ)) = k

(kPa(x)k , (w , σ)) =�


(kS(x , y)k , (w , σ)) =�


(kx � yk , (w , σ)) =�


(kx 2 Xk , (w , σ)) =�


(k:ϕk , (w , σ)) =�







(kkk , (w , σ)) = k

(kPa(x)k , (w , σ)) =�


(kS(x , y)k , (w , σ)) =�


(kx � yk , (w , σ)) =�


(kx 2 Xk , (w , σ)) =�


(k:ϕk , (w , σ)) =�







(kkk , (w , σ)) = k

(kPa(x)k , (w , σ)) =�


(kS(x , y)k , (w , σ)) =�


(kx � yk , (w , σ)) =�


(kx 2 Xk , (w , σ)) =�


(k:ϕk , (w , σ)) =�







(kkk , (w , σ)) = k

(kPa(x)k , (w , σ)) =�


(kS(x , y)k , (w , σ)) =�


(kx � yk , (w , σ)) =�


(kx 2 Xk , (w , σ)) =�


(k:ϕk , (w , σ)) =�







(kkk , (w , σ)) = k

(kPa(x)k , (w , σ)) =�


(kS(x , y)k , (w , σ)) =�


(kx � yk , (w , σ)) =�


(kx 2 Xk , (w , σ)) =�


(k:ϕk , (w , σ)) =�


ϕ is of the form Pa(x), S(x , y), x � y or x 2 XCIAA 2007 (Prague) M. Droste, G. Rahonis July 16, 2007 10 / 19

De�nition (Semantics continued)

(kϕ _ ψk , (w , σ)) = (kϕk , (w , σ)) _ (kψk , (w , σ))

(kϕ ^ ψk , (w , σ)) = (kϕk , (w , σ)) + (kψk , (w , σ))(k9x � ϕk , (w , σ)) =

_i2ω

(kϕk , (w , σ[x ! i ]))

(k9X � ϕk , (w , σ)) =_I�ω

(kϕk , (w , σ[X ! I ]))

(k8x � ϕk , (w , σ)) = ∑i2ω

pi � (kϕk , (w , σ[x ! i ])) .

Example Let ϕ = 8x � (Pa(x)! k), k 2 Rmax.

Then (kϕk ,w) = k � ∑i2ω,ai=a

pi

returns the discounted cost of a�s in w .



(kϕ _ ψk , (w , σ)) = (kϕk , (w , σ)) _ (kψk , (w , σ))(kϕ ^ ψk , (w , σ)) = (kϕk , (w , σ)) + (kψk , (w , σ))

(k9x � ϕk , (w , σ)) =_i2ω

(kϕk , (w , σ[x ! i ]))

(k9X � ϕk , (w , σ)) =_I�ω

(kϕk , (w , σ[X ! I ]))

(k8x � ϕk , (w , σ)) = ∑i2ω

pi � (kϕk , (w , σ[x ! i ])) .



pi




(kϕ _ ψk , (w , σ)) = (kϕk , (w , σ)) _ (kψk , (w , σ))(kϕ ^ ψk , (w , σ)) = (kϕk , (w , σ)) + (kψk , (w , σ))(k9x � ϕk , (w , σ)) =

_i2ω

(kϕk , (w , σ[x ! i ]))

(k9X � ϕk , (w , σ)) =_I�ω

(kϕk , (w , σ[X ! I ]))

(k8x � ϕk , (w , σ)) = ∑i2ω

pi � (kϕk , (w , σ[x ! i ])) .



pi





_i2ω

(kϕk , (w , σ[x ! i ]))

(k9X � ϕk , (w , σ)) =_I�ω

(kϕk , (w , σ[X ! I ]))

(k8x � ϕk , (w , σ)) = ∑i2ω

pi � (kϕk , (w , σ[x ! i ])) .



pi





_i2ω

(kϕk , (w , σ[x ! i ]))

(k9X � ϕk , (w , σ)) =_I�ω

(kϕk , (w , σ[X ! I ]))

(k8x � ϕk , (w , σ)) = ∑i2ω

pi � (kϕk , (w , σ[x ! i ])) .



pi





_i2ω

(kϕk , (w , σ[x ! i ]))

(k9X � ϕk , (w , σ)) =_I�ω

(kϕk , (w , σ[X ! I ]))

(k8x � ϕk , (w , σ)) = ∑i2ω

pi � (kϕk , (w , σ[x ! i ])) .



pi





_i2ω

(kϕk , (w , σ[x ! i ]))

(k9X � ϕk , (w , σ)) =_I�ω

(kϕk , (w , σ[X ! I ]))

(k8x � ϕk , (w , σ)) = ∑i2ω

pi � (kϕk , (w , σ[x ! i ])) .



pi



De�nitionA formula ϕ 2 MSO(Rmax,A) is almost existential if whenever ϕcontains a subformula 8x � ψ,then ψ does not contain any universal quanti�er.

Purely syntactic de�nition!

Rp�aemsomax hhAωii) : all series in Rmax hhAωii

which are de�nable by some almost existential sentence inMSO(Rmax,A)












Theorem (second main result)

Rp�ω�recmax hhAωii = Rp�aemso

max hhAωii


Proof.By induction on the structure of weighted MSO-formulaswe show R

p�aemsomax hhAωii � R

p�ω�recmax hhAωii .

Crucial steps:If kϕk takes on only �nitely many values, then

so do k9x � ϕk and k9X � ϕk , andk8x � ϕk is ω-recognizable.

Conversely, given any weighted Muller automaton Awe can e¤ectively construct an almost existential sentence ϕ suchthat kAk = kϕk .

In the paper we have shown corresponding results for �nitary series(over �nite words).














so do k9x � ϕk and k9X � ϕk , and

k8x � ϕk is ω-recognizable.




























Semiring (K+, �, 0, 1): additively locally �niteif for all x 2 K , fnx j n � 0g is �nite

Examplesany idempotent semiring

any �eld of prime characteristic

the semiring of polynomials (K [X ],+, �, 0, 1) over a variable X andan additively locally �nite semiring K .

CorollaryLet K be a computable, additively locally �nite, commutative semiring, orlet K = Rmax or K = Rmin. Let 0 � p < 1. Given an almost existentialMSO(K ,A)-formula ϕ whose atomic entries from K are e¤ectively given,we can e¤ectively compute a weighted automaton, resp. a weighted Mullerautomaton, A such that kϕk = kAk.


Semiring (K+, �, 0, 1): additively locally �niteif for all x 2 K , fnx j n � 0g is �niteExamples

any idempotent semiring





Semiring (K+, �, 0, 1): additively locally �niteif for all x 2 K , fnx j n � 0g is �niteExamplesany idempotent semiring




















Other workWeighted automata and weighted logics on

�nite words

M. Droste, P. Gastin, Weighted automata and weighted logics,Theoret. Comput. Sci. 380(2007) 69-86; extended abstract in:32nd ICALP, LNCS 3580(2005) 513-525.

in�nite words over complete semirings

M. Droste, G. Rahonis, Weighted automata and weighted logics onin�nite words. Special issue on "Workshop on words and automata,WOWA�2006" (M. Volkov, ed.) Russian Mathematics (Iz. VUZ), toappear; extended abstract in:Proceedings of DLT�06, LNCS 4036(2006) 49-58.



�nite words






�nite words






�nite words





Other work (continued)Weighted automata and weighted logics on

�nite trees

M. Droste, H. Vogler, Weighted tree automata and weighted logics,Theoret. Comput. Sci. 366(2006) 228-247.

in�nite trees

G. Rahonis, Weighted Muller tree automata and weighted logics.Special issue on "Weighted automata" (M. Droste, H. Vogler, eds.)J. of Automata Languages and Combinatorics, accepted.

pictures

I. Mäurer, Weighted picture automata and weighted logics, in:Proceedings of STACS 2006, LNCS 3884(2006).I. Fichtner, Characterizations of Recognizable Picture Series,PhD Thesis, Leipzig University, 2006.



�nite trees


in�nite trees


pictures




�nite trees


in�nite trees


pictures




�nite trees


in�nite trees


pictures




�nite trees


in�nite trees


pictures




�nite trees


in�nite trees


pictures

I. Mäurer, Weighted picture automata and weighted logics, in:Proceedings of STACS 2006, LNCS 3884(2006).

I. Fichtner, Characterizations of Recognizable Picture Series,PhD Thesis, Leipzig University, 2006.



�nite trees


in�nite trees


pictures




traces

I. Meinecke, Weighted logics for traces, in: Proceedings of CSR 2006,LNCS 3967(2006) 235-246.

texts

C. Mathissen, De�nable transductions and weighted logics for texts,11th International Conference on Developments in Language Theory(DLT) 2007, Turku.



traces


texts




traces


texts




traces


texts



Thank you


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Weighted automata and weighted logics with discounting

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