# Polynomially Ambiguous Probabilistic Automata on Restricted Languages

We consider the computability and complexity of decision questions for Probabilistic Finite Automata (PFA) with sub-exponential ambiguity. We show that the emptiness problem for non strict cutpoints of polynomially ambiguous PFA remains undecidable even when the input word is over a bounded language and all PFA transition matrices are commutative. In doing so, we introduce a new technique based upon the Turakainen construction of a PFA from a Weighted Finite Automata which can be used to generate PFA of lower dimensions and of subexponential ambiguity. We also study freeness problems for polynomially ambiguous PFA and study the border of decidability and tractability for various cases.

## Authors

• 5 publications
01/17/2020

### Pumping lemmas for weighted automata

We present pumping lemmas for five classes of functions definable by fra...
02/18/2020

### Decidability of cutpoint isolation for letter-monotonic probabilistic finite automata

We show the surprising result that the cutpoint isolation problem is dec...
09/07/2020

### Ambiguity Hierarchy of Regular Infinite Tree Languages

An automaton is unambiguous if for every input it has at most one accept...
05/21/2019

### Approximating probabilistic models as weighted finite automata

Weighted finite automata (WFA) are often used to represent probabilistic...
05/27/2022

### Probabilistic Transformer: Modelling Ambiguities and Distributions for RNA Folding and Molecule Design

Our world is ambiguous and this is reflected in the data we use to train...
07/01/2020

### Minimal witnesses for probabilistic timed automata

Witnessing subsystems have proven to be a useful concept in the analysis...
05/18/2020

### On the Power of Unambiguity in Büchi Complementation

In this work, we exploit the power of unambiguity for the complementatio...
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## 1 Introduction

Probabilistic finite automata (PFA) are a simple yet expressive model of computation, obtained by extending nondeterministic finite automata so that transitions from each state (and for each input letter) form probability distributions. As input letters are read from some alphabet

, the automaton transitions among states according to these probabilities. The probability of accepting a word is given by the probability of the automaton being in one of its final states, denoted , where represents the initial state, represents the final state and each

is a row stochastic matrix representing the transition probabilities for letter

.

The PFA model has been studied extensively over the years, ever since its introduction by Rabin [27]; for example see [10] for a survey of research papers related to PFA in the eleven years since their introduction to just 1974. They have been used to study Arthur-Merlin games [2], space bounded interactive proofs [15], quantum complexity theory [32], the joint spectral radius and semigroup boundedness [8]

, Markov decision processes and planning questions

[9], and text and speech processing [24] among many others.

There are a variety of interesting questions that one may ask about PFA. A central question is that of the emptiness problem for cut-point languages, i.e. given some probability , does there exist some finite input word whose probability of acceptance is at least (i.e. does there exist such that , see Section 2.2 for formal details). This problem is known to be undecidable [26], even for a fixed number of dimensions or for two input matrices [7, 19]. A second natural question is the freeness problem for PFA, studied in [3] - given a PFA over alphabet determine whether the acceptance function is injective (i.e. does there exist two distinct words with the same acceptance probability).

When studying the frontiers of decidability of a problem, there are two competing objectives, namely, determine the most general version of the problem which is decidable, and the most restricted specialization which is undecidable; the latter being the focus of this paper.

Various classes of restrictions may be studied for PFA depending upon the structure of the PFA or on possible input words. Some restrictions relate to the number of states of the automaton, the alphabet size and whether one defined the PFA over the algebraic real numbers or the rationals. One may also study PFA with finite, polynomial or exponential ambiguity (in terms of the underlying NFA), PFA defined for restricted input words (for example those coming from regular, bounded or letter monotonic languages), PFA with isolated thresholds (a probability threshold is isolated if it cannot be approached arbitrarily closely) and PFA where all matrices commute, for which cut-point languages and non-free languages generated by such automata necessarily become commutative.

The cut-point emptiness problem for PFA is known to be undecidable for rational matrices [26], even over a binary alphabet when the PFA has dimension in [7]; later improved to dimension [19]. The authors of [6] show that the problem of determining if a threshold is isolated (resp. if a PFA has any isolated threshold) is undecidable and this was shown to hold even for PFA with (resp. ) states over a binary alphabet [7].

A natural restriction on PFA was studied in [4], where possible input words of the PFA are restricted to be from some letter monotonic language of the form with each (analogous to a 1.5 way PFA acting on a fixed input word), then the problem remains undecidable. In other words, does there exist such that ? This restriction is inspired by the well- known property that many language-theoretic problems become decidable or tractable when restricted to bounded languages, and especially letter-monotonic languages [13]. Nevertheless, the emptiness problem for PFA on letter-monotonic languages was shown to be undecidable for high (but finite) dimensional matrices over the rationals via an encoding of Hilbert’s tenth problem on the solvability of Diophantine equations and the utilization of Turakainen’s method to transform weighted integer automata to a PFA [4].

The authors of [17] recently studied decision problems for PFA of various degrees of ambiguity in order to map the frontier of decidability for restricted classes of PFA. The degree of ambiguity of a PFA is defined as the maximum number of accepting runs over all possible words and can be used to give various classifications of ambiguity including finite, polynomial and exponential ambiguity. The ambiguity of a PFA is a property of the underlying NFA and is independent of the transition probabilities in so much as we only need care if the probability is zero or positive. The degree of ambiguity of automata is a well known and well studied property in automata theory [30]. The authors of [17] show that the emptiness problem for PFA remains undecidable even for polynomially ambiguous automata (quadratic ambiguity), before going on to show PSPACE-hardness results for finitely ambiguous PFA and that emptiness is in NP for the class of -ambiguous PFA for every . The emptiness problem for PFA was later shown to also be undecidable even for linearly ambiguous automata in [16].

### 1.1 Our Contributions

In this paper, we show that the emptiness problem is undecidable even for polynomially ambiguous PFA defined over letter monotonic languages when all matrices are rational and commutative. This combination of restrictions on the PFA significantly increases the difficulty of proving undecidability. The study of PFA over letter monotonic languages is a particularly interesting intermediate model, lying somewhere between single letter alphabets, for which we have decidability results, and PFA defined with multi-letter alphabets, for which most decision problems are undecidable.

The emptiness problem for polynomially ambiguous probabilistic finite automata on letter monotonic languages is undecidable for non-strict cutpoints, even when all matrices are commutative.

We note a few difficulties with proving this result. Firstly, Post’s correspondence problem, whose variants are often used for showing undecidability results in such settings, is actually decidable over letter monotonic languages [18] 111Although it is undecidable in general (i.e. not over a letter monotonic language) with an alphabet with at least five letters [25].. Secondly, although other reductions of undecidable computational problems to matrices are possible, the standard technique of Turakainen to modify such matrices to stochastic matrices introduces exponential ambiguity (indeed all such matrices are strictly positive, and thus we might think of such matrices as being maximally exponentially ambiguous). Finally, we note that matrix problems for commutative matrices are often decidable, indeed there is a polynomial time algorithms for solving the orbit problem [22, 14]

and the vector reachability problem for commutative matrices

[1].

We use a reduction of Hilbert’s tenth problem and various new encoding techniques to avoid the use of Turakainen’s method for converting from weighted to probabilistic automata, so as to retain polynomial ambiguity. We then move on to the freeness problem to show the following two results.

The freeness problem for linearly ambiguous four state probabilistic finite automata is undecidable.

The freeness problem for linearly ambiguous three-state probabilistic finite automata over letter-monotonic languages is NP-hard.

These results are proven via an encoding of the mixed modification PCP and our new encoding technique and the freeness problem for three state PFA over letter monotonic languages is NP-hard via an encoding of the variant of the subset sum problem and a novel encoding technique. We conclude with some open problems.

## 2 Preliminaries

### 2.1 Linear Algebra

Given and we define the direct sum and Kronecker product of and by:

where

denotes the zero matrix of dimension

. Note that neither nor are commutative in general. Given a finite set of matrices , is the semigroup generated by . We will use the following notations:

 m⨁j=1Gj=G1⊕G2⊕⋯⊕Gm,m⨂j=1Gj=G1⊗G2⊗⋯⊗Gm

Given a single matrix , we inductively define for with as the -fold Kronecker power of . Similarly, for with being a zero dimensional matrix. The rationalle for the base cases is that and that as expected.

The following properties of and are well known, see [20] for proofs.

Let . We note that:

• Associativity - and , thus and are unambiguous.

• Mixed product properties: and .

• If and are stochastic matrices, then so are and .

It is trivial to prove that if are both upper-triangular then so are and . This follows directly from the definition of the Kronecker sum and product.

### 2.2 Probabilistic Finite Automata (PFA)

Probabilistic Finite Automata (PFA) with states over an alphabet are defined as where is the initial probability distribution; is the final state vector and each is a (row) stochastic matrix. For a word , we define the acceptance probability of as:

 fP(w)=xTMw1Mw2⋯Mwky,

which denotes the acceptance probability of .

For any and PFA over alphabet , we define a cut-point language to be: , and a strict cut-point language by replacing with . The (strict) emptiness problem for a cut-point language is to determine if (resp. ).

Let be an -letter alphabet for some . A language is called a bounded language if and only if there exist words such that . A language is called letter-monotonic if there exists letters such that . One thus sees that letter monotonic languages are more restricted than bounded languages. We will be interested in PFA which are defined over a bounded language or a letter monotonic language , whereby all input words necessarily come from . In this case a cut-point language for a PFA over bounded/letter monotonic language and a probability is defined as ; similarly for nonstrict cut point languages. We may then ask similar emptiness questions for such languages, as before.

We will also be interested in the freeness problem for PFA. Given a PFA over alphabet determine whether the acceptance function is injective (i.e. does there exist two distinct words with the same acceptance probability). Such problems can readily be studied when the input words are necessarily derived from a bounded or letter-monotonic language.

### 2.3 PFA Ambiguity

The degree of ambiguity of a finite automaton is a structural parameter, roughly indicating the number of accepting runs for a given input word [30]. We here define only those notions required for our later proofs, see [30] for full details of these notions and a thorough discussion.

Let be an input word of an NFA . For each , let be defined as the number of all paths for in leading from state to state . The degree of ambiguity of in , denoted , is defined as the number of all accepting paths for . The degree of ambiguity of , denoted is the supremum of the set . is called infinitely ambiguous if , finitely ambiguous if , and unambiguous if . The degree of growth of the ambiguity of , denoted is defined as the minimum degree of a univariate polynomial with positive integral coefficients such that for all , if such a polynomial exists, or infinity otherwise. A state is called useful if there exists an accepting path which visits .

The above notions relate to NFA. We may derive an analogous notation of ambiguity for PFA by considering an embedding of a PFA to an NFA with the property that for each letter , if the probability of transitioning from a state to state is nonzero under , then there is an edge from state to under for letter . The degree of ambiguity of is then defined as the degree of ambiguity of , and similarly for the degree of growth of ambiguity of mutatis mutandis.

We may use the following notions to determine the degree of ambiguity of a given NFA (and thus a PFA) as is shown in the theorem which follows.

• [leftmargin=1.1cm]

• - There is a useful state such that, for some word , .

• - There are useful states and words such that for all , and are distinct and and for all , .

[[21, 28, 30]] An NFA (or PFA) having the EDA property is equivalent to it being exponentially ambiguous. For any , an NFA (or PFA) having property is equivalent to .

One may immediately notice that if agrees with for some , then it also agrees with . One must be careful with these notions of ambiguity when considering NFA/PFA , where inputs are necessarily from a bounded/letter-monotonic language . In such cases, the above criteria do not suffice to determine the ambiguity of , since the number of paths must be determined not over , but over all paths from . Of course, the degree of ambiguity of cannot increase by restricting to a bounded input language, but it may decrease.

As an example, if an NFA has property EDA, then there exists three words and such that is an accepting word and , thus has at least two distinct accepting runs. However, this implies that and thus has at least accepting runs. Now, if we are given some bounded language such that and then the same implication is not possible, unless is a single letter, otherwise there is no guarantee that . Nevertheless, in the results of this paper we will use the standard definitions of ambiguity since the distinction is not relevant in our results as will become clear.

We note the following trivial lemma, which will be useful later.

Probabilistic finite automata defined over upper-triangular matrices are polynomially ambiguous.

###### Proof.

This lemma is immediate from Theorem 2.3 and property (EDA), since a PFA defined over upper-triangular matrices clearly does not have property (EDA). This is since a transition matrix (for a letter ‘’) which is upper-triangular only defines transitions of the form where and thus the states entered for any run are monotonically nondecreasing. ∎

### 2.4 Reducible Undecidable Problems

We will require the following undecidable problems for proving later results. The first is a variant of the famous Post’s Correspondence Problem (PCP).

###### Problem (Mixed Modification PCP (MMPCP)).

Given a binary alphabet , a finite set of letters , and a pair of homomorphisms the MMPCP asks to decide whether there exists a word such that

 h1(x1)h2(x2)…hk(xk)=g1(x1)g2(x2)…gk(xk),

where and there exists at least one such that

[12] - The Mixed Modification PCP is undecidable for .

A second userful undecidable problem is Hilbert’s tenth problem. Let be an integer polynomial with variables. Hilbert’s tenth problem is to determine if there exists a procedure to find if there exist such that: . It is well known that this may be reduced to a problem in formal power series. It was shown in [29, p.73] that the above problem can be reduced to that of determining for a -rational formal power series , whether there exists any word such that .

A “negative solution” to the problem was shown in 1970 by Y. Matiyasevich building upon earlier work of many mathematicians, including M. Davis, H. Putman and J. Robinson. For more details of the history of the problem as well as the full proof of the undecidability of this theorem, see the excellent reference [23]. We may, without loss of generality, restrict the variables to be natural numbers [23, p.6].

## 3 Cut-point languages for polynomially ambiguous PFA over letter monotonic languages

It was proven in [4] that the emptiness problem is undecidable for probabilistic finite automata even when input words are given over a letter-monotonic language, i.e., given a letter-monotonic language , it is undecidable to determine if is empty for . The constructed PFA of [4] has exponential ambiguity, due to the well-known Turakainen conversion of arbitrary integer matrices into stochastic matrices. Here, we show that the emptiness problem for PFA over letter-monotonic languages can also be achieved even when all matrices have polynomial ambiguity by a modified Turakainen procedure.

The following property of the Kronecker product will also be required for the proof of Theorem 1.1.

Let . Then for any index sequence with each then there exists such that

 ℓ∏m=1(Am)im,jm=(ℓ⨂m=1Am)i,j
###### Proof.

The proof proceeds by induction. For the base case when , we just set and we are done. Assume then that the result holds for some , then for sequence there exists such that:

 ℓ−1∏m=1(Am)im,jm=(ℓ−1⨂m=1Am)i′,j′

By the definition of Kronecker product,

 ((ℓ−1⨂m=1Am)⊗Aℓ)ni′+iℓ,nj′+jℓ=ℓ−1∏m=1(Am)im,jm×(Aℓ)iℓ,jℓ

as required. ∎

Note that we can of course work out the particular value of and , but in general the formula for does not have a nice form when , and anyway will not be necessary for us, so we settle for an existential proof of such and .

### 3.1 Proof of Theorem 1.1

###### Proof.

We will construct a polynomially ambiguous probabilistic finite automaton , a cutpoint and a letter monotonic language .

We begin by encoding an instance of Hilbert’s tenth problem into a set of integer matrices. Let be a Diophantine equation. Homogenenization of polynomials is a well known technique, as is used for example in the study of Gröbner bases [11], which allows us to convert such a Diophantine equation to

with a new dummy variable

such that is a homogeneous polynomial (each term having the same degree ) and for which when . We thus assume a homogeneous Diophantine equation with implied constraint which will be dealt with later. Furthermore, we assume that gives nonnegative values, which may be assumed by redefining , which clearly does not affect whether a zero exists for such a polynomial.

Notice that given , then . We will generalise this property to a set of matrices so that given any tuple , then appears as an element on the superdiagonal of for each . We will also have the property that each has the same row sum of for every row, which will be useful when we later convert to stochastic matrices.

We define each matrix for in the following way:

 Ai=⎛⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜⎝1δ0,i0⋯001−δ0,i01δ1,i⋯001−δ1,i001⋯001−δ2,i⋮⋮⋮⋱⋮⋮⋮000⋯1δt,i1−δt,i000⋯011000⋯002⎞⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟⎠∈N(t+3)×(t+3), (1)

where and is the Kronecker delta (thus and for ). We also denote , noting that this is the matrix (1) when all have the value . Notice then that every row sum of and is . This structure is retained under matrix powers and it is easy to see that:

 Aki=⎛⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜⎝1kδ0,i0⋯002k−kδ0,i−101kδ1,i⋯002k−kδ1,i−1001⋯002k−kδ2,i−1⋮⋮⋮⋱⋮⋮⋮000⋯1kδt,i2k−kδt,i−1000⋯012k−1000⋯002k⎞⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟⎠∈N(t+3)×(t+3) (2)

All row sums of are and exactly one element of the superdiagonal is equal to , with all other elements on the superdiagonal (excluding that on row ) zero. Taking powers of will allow us to choose any positive value of variable . Note that has the same form as the matrix of (2) with all

and acts as a kind of identity matrix, (in its upperleft block) while retaining the

row sum. Indeed, one sees that for all , then , i.e. these matrices commute (as does since ). We now show how to compute terms of .

We may write , where denotes the ’th term of , with having terms. Since is a homogeneous polynomial, each term has the same degree . We may thus write each term as:

 Tj(x0,x1,…,xt) = cjRj(x0,x1,…,xt), (3)

with and with and . For convenience, we define a -dimensional vector . For example, if and , then and thus . By we denote the ’th element of vector .

We now define matrices corresponding to term :

 Xj,i=i−1⨂ℓ=0J⊗rj,ℓ⊗A⊗rj,ii⊗d⨂ℓ=i+1J⊗rj,ℓ,

where . The dimension of such matrices is since each submatrix has dimension and we take the -fold Kronecker product. Similarly, we see that the row sum of each is since the row sum of each and is and we take a -fold Kronecker product. Clearly then, by the mixed product property (see Lemma 2.1):

 Xkj,i=i−1⨂ℓ=0(Jk)⊗rj,ℓ⊗(Aki)⊗rj,i⊗d⨂ℓ=i+1(Jk)⊗rj,ℓ,

for any . In the example when , , , and , then . We then see that .

Now, we see that:

 Xx0j,0Xx1j,1⋯Xxtj,t = t∏i=0(i−1⨂ℓ=0(Jxi)⊗rj,ℓ⊗(Axii)⊗rj,i⊗d⨂ℓ=i+1(Jxi)⊗rj,ℓ) (4) = d⨂ℓ=0(Dx0ℓ,0Dx1ℓ,1⋯Dxtℓ,t), (5)

where for . The derivation of Eqn (5) from Eqn (4) follows by the mixed product property of the Kronecker product (Lemma 2.1). For each product , wee see that and for all with . As discussed earlier, matrices and commute, for any and thus we may rewrite (5) as:

 Xx0j,0Xx1j,1⋯Xxtj,t=d⨂ℓ=0(Axsj[ℓ]sj[ℓ]⊗J¯¯¯¯¯¯¯¯¯¯xsj[ℓ]), where ¯¯¯¯¯¯¯¯¯¯xsj[ℓ]=∑0≤q≤dq≠sj[l]xq (6)

By Lemma 3, we see that some element of is thus equal to as required, since there is an element on the superdiagonal of equal to for each . Let us assume that appears at row and column . Now, we may define a vector and where is the coefficient of term as in Eqn (3) and are basis vectors. We may now see that

 (u′j)TXx0j,0Xx1j,1⋯Xxtj,tv′j=cjRj(x0,x1,…,xt)=Tj(x0,x1,…,xt) (7)

In order to derive the sum of the such terms , we will utilise the direct sum. For , we define by:

 Y′ℓ=r⨁j=1Xj,ℓ∈Nr(t+3)d×r(t+3)d

We shall now modify each so that they are row stochastic. We recall that the row sum of each and is . Therefore, the row sum of each is , since is a -fold Kronecker product of and matrices. Then the row sum of each is also since direct sums do not modify the row sum. We thus see that is row stochastic.

We now consider the coefficients of each term. We previously multiplied each initial vector by and we may consider taking the Kronecker sum of each before normalising the resulting vector (normalising according to norm). We face an issue however, since some coefficients may be negative and thus the resulting vector is not stochastic (it must be nonnegative). Fortunately we may modify a technique utilised by Bertoni [5] to solve this issue. Given a PFA for which , then by defining where is the all-one vector of appropriate dimension (i.e. swapping between final and non final states), then .

Now, since each has a row sum of and is of unit length ( norm), then Eqn. (7) can be adapted to the following:

 (u′j)TXx0j,0Xx1j,1⋯Xxtj,t(1−v′j) = 2d(x0+x1+…+xt)−cjRj(x0,x1,…,xt) (8) = 2d(x0+x1+…+xt)−Tj(x0,x1,…,xt)

Let us assume, without loss of generality, that we have arranged the terms of such that those terms with a positive coefficient (positive terms) appear first, followed by those with a negative coefficient (negative terms). Since we have terms in , there exists some such that we have postive and negative terms. Let us define , which is similar to defined previously, but using the absolute value of the corresponding coefficient.

We define as the final vector, so that we take the Kronecker sum of all final vectors, but we swap final and non-final states for the negative terms.

We now define the initial vector , which must be a probability distribution. Let be the sum of absolute values of coefficients and define . Note that is stochastic (a probability distribution).

We now see that:

 uTY0Ya11⋯Yattv = ∑r′j=1uj(⨂dℓ=0Axsj[ℓ]sj[ℓ]⊗J¯¯¯¯¯¯¯¯¯¯xsj[ℓ])vj+∑rj=r′+1uj(⨂dℓ=0Axsj[ℓ]sj[ℓ]⊗J¯¯¯¯¯¯¯¯¯¯xsj[ℓ])(1−vj)g2d(1+a1+⋯+at)

Here we used the definition of matrices and Eqn. (6) to rewrite the expressions for . Notice that the power of is set at , since that constraint is required by the conversion from a standard Diophantine polynomial to a homogeneous one as explained previously. Now, using Eqn. (7) and Eqn. (8), we can rewrite Eqn. (3.1) as:

 ∑r′j=1Tj(x0,…,xt)+∑rj=r′+1(2d(1+a1+…+at)−|Tj(x0,…,xt)|)g2d(1+a1+⋯+at) (10) = (r−r′+1)g+∑r′j=1Tj(x0,…,xt)+∑rj=r′Tj(x0,…,xt)g2d(1+a1+⋯+at) (11) = (r−r′+1)g+Ph(x0,x1,…,xt)g2d(1+a1+⋯+at) (12)

We therefore define and as our PFA, with letter monotonic language and as the cutpoint. There exists some word such that if and only if . Therefore the strict emptiness problem for is undecidable on letter monotonic languages. Since is upper-triangular, then it is polynomially ambiguous. We may note the surprising fact that all matrices thus generated are in fact commutative (each is commutative and direct sums do not affect commutativity), which leads to undecidability of non-strict points for polynomially ambiguous PFA defined over commutative matrices. In this case, the order of the input word in irrelevant, only the Parikh vector of alphabet characters is important. ∎

## 4 Freeness problems for polynomially ambiguous PFA

We now study the freeness of acceptance probabilities of polynomially ambiguous PFA. The next result begins with a proof technique from [4], where the undecidability of the freeness problem was shown for exponentially ambiguous PFA over five states. Here we show that the freeness problem remains undecidable even when the PFA is polynomially ambiguous and over four states by using our new encoding technique (avoiding the Turakainen procedure which increases the matrix dimensions by two and generates an exponentially ambiguous PFA).

### 4.1 Proof of Theorem 1.1

###### Proof.

Let and be distinct alphabets and be an instance of the mixed modification PCP. The naming convention will become apparent below. We define two injective mappings by:

and . Thus represents as a reverse -adic number and represents as a fractional number (e.g. if , then is represented as and , where subscript denotes base ). Note that and . It is not difficult to see that and .

Define by

 γ′′(u,v)=⎛⎜⎝(n+1)|u|0α(u)0(n+1)−|v|β(v)001⎞⎟⎠.

It is easy to verify that i.e.,