#### Polynomially Ambiguous Probabilistic Automata on Restricted Languages

I C A L P
Polynomially Ambiguous Probabilistic Automata on Restricted Languages
Paul C. Bell 0 1
0 Department of Computer Science , Byrom Street , Liverpool John Moores University , Liverpool, L3-3AF , UK
1 Category Track B: Automata , Logic, Semantics, and Theory of Programming
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 cut-points 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/injectivity problems for polynomially ambiguous PFA and study the border of decidability and tractability for various cases. 2012 ACM Subject Classification Theory of computation ? Quantitative automata; Theory of computation ? Probabilistic computation Acknowledgements We thank the referees for their careful reading of this manuscript and their helpful improvements.
and phrases Probabilistic finite automata; ambiguity; undecidability; bounded language; emptiness
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 w ? ?? is given by the probability of the automaton being in
one of its final states, denoted fP (w) = xT Mw1 Mw2 ? ? ? Mwk y, where x represents the initial
state, y represents the final state and each Mwi is a row stochastic matrix representing the
transition probabilities for letter wi ? ?.
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 416 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
[33], 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 the emptiness problem for cut-point languages; given some probability ? ? [0, 1],
does there exist a finite input word whose probability of acceptance is greater than ? (i.e.
does there exist w ? ?? such that fP (w) > ?, see Section 2.2). This problem is known to be
undecidable [26], even for a fixed number of dimensions and for two input matrices [7, 19].
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A second natural question is the freeness problem (or injectivity problem) for PFA, studied
in [3] ? given a PFA P over alphabet ? determine whether the acceptance function fP (w) is
injective (i.e. do 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 46 in [7]; later improved to
dimension 25 [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 420 (resp. 2354) 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 L = a1?a2? ? ? ? a?k with
each ai ? ? (analogous to a 1.5 way PFA, whose read head may ?stay put? on an input
word letter but never moves left), then the problem remains undecidable. In other words,
does there exist w ? L such that fP (w) > ?? This restriction is inspired by the
wellknown 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 [31]. 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 k-ambiguous PFA for
every k > 0. 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.
I Theorem 1. The emptiness problem for polynomially ambiguous probabilistic finite
automata on letter monotonic languages is undecidable for non-strict cut-points, 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]1. Secondly, although other reductions
of undecidable computational problems to matrices are possible, the standard technique
of Turakainen (shown in [30]) 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)2. Finally, we note that
matrix problems for commutative matrices are often decidable; indeed there are polynomial
time algorithms for solving the orbit problem [22, 14] and the vector reachability problem
for commutative matrices [1]. Since the matrices commute, it is the Parikh vector of letters
of the input word which is important.
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/injectivity problem
to show the following two results.
I Theorem 2. The injectivity problem for linearly ambiguous four state probabilistic finite
automata is undecidable.
I Theorem 3. The injectivity 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 injectivity 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
2.1
Preliminaries
Linear Algebra
A ? B =
A
0n,m
0m,n ,
B
Given A = (aij ) ? Fm?m and B ? Fn?n, we define the direct sum A ? B and Kronecker
product A ? B of A and B by:
? a11B
A ? B = ?? a21B
?? ...
am1B
a12B
a22B
.
.
.
am2B
? ? ?
? ? ?
? ? ?
a1mB ?
a2mB
.
.
.
?
? ,
?
?
ammB
1 Although it is undecidable in general (i.e. not over a letter monotonic language) with an alphabet with
at least five letters [25].
2 This is due to an essential step of the Turakainen procedure that adds a positive constant offset to each
element of every generator matrix, thus making all matrices strictly positive [30].
where 0i,j denotes the zero matrix of dimension i ? j. Note that neither ? nor ? are
commutative in general. Given a finite set of matrices G = {G1, G2, . . . , Gm} ? Fn?n, hGi is
the semigroup generated by G. We will use the following notations:
m
M Gj = G1 ? G2 ? ? ? ? ? Gm,
j=1
m
O Gj = G1 ? G2 ? ? ? ? ? Gm
j=1
Given a single matrix G ? Fn?n, we inductively define G?k = G ? G?(k?1) ? Fnk?nk for
k > 0 with G?0 = 1 as the k-fold Kronecker power of G. Similarly, G?k = G ? G?(k?1) ?
Fnk?nk for k > 0 with G?0 being a zero dimensional matrix. The rationalle for the base
cases is that G ? G?0 = 1 ? G = G and that G ? G?0 = G as expected.
The following properties of ? and ? are well known, see [20] for proofs.
I Lemma 4. Let A, B, C, D ? Fn?n. We note that:
Associativity - (A ? B) ? C = A ? (B ? C) and (A ? B) ? C = A ? (B ? C), thus
A ? B ? C and A ? B ? C are unambiguous.
Mixed product properties: (A?B)(C ?D) = (AC ?BD) and (A?B)(C ?D) = (AC ?BD).
If A and B are stochastic matrices, then so are A ? B and A ? B.
It is trivial to prove that if A, B ? Fn?n are both upper-triangular then so are A ? B
and A ? B. This follows directly from the definition of the Kronecker sum and product.
2.2
Probabilistic Finite Automata (PFA)
Probabilistic Finite Automata (PFA) with n states over an alphabet ? are defined as
P = (x, {Ma|a ? ?}, y) where x ? Rn is the initial probability distribution; y ? {0, 1}n
is the final state vector and each Ma ? Rn?n is a (row) stochastic matrix. For a word
w = w1w2 ? ? ? wk ? ??, we define the acceptance probability fP : ?? ? R of P as:
fP (w) = xT Mw1 Mw2 ? ? ? Mwk y,
which denotes the acceptance probability of w.
For any ? ? [0, 1] and PFA A over alphabet ?, we define a cut-point language to be:
L??(A) = {w ? ??|fA(w) ? ?}, and a strict cut-point language L??(A) by replacing ? with
>. The (strict) emptiness problem for a cut-point language is to determine if L??(A) = ?
(resp. L>?(A) = ?).
Let ?` = {x1, x2, . . . , x`} be an `-letter alphabet for some ` > 0. A language L ? ??
`
is called a bounded language if and only if there exist words w1, w2, . . . , wm ? ?`+ such
that L ? w1?w2? ? ? ? wm?. A language L is called letter-monotonic if there exists letters
u1, u2, . . . , um ? ?` such that L ? u1?u2? ? ? ? u?m. 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 L, whereby all input words necessarily
come from L. In this case a cut-point language for a PFA P over bounded/letter monotonic
language L and a probability ? ? [0, 1] is defined as L??(A) = {w ? L|fA(w) ? ?}; similarly
for nonstrict cut point languages. We may then ask similar emptiness questions for such
languages, as before.
We also study the freeness/injectivity problem for PFA. Given a PFA P over alphabet ?
determine whether the acceptance function fP (w) is injective (i.e. do 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 [31]. We here define only those notions
required for our later proofs, see [31] for full details of these notions and a thorough discussion.
Let w ? ?? be an input word of an NFA N = (Q, ?, ?, QI , QF ). For each (p, w, q) ?
Q ? ?? ? Q, let daN (p, w, q) be defined as the number of all paths for w in N leading from
state p to state q. The degree of ambiguity of w in N , denoted daN (w), is defined as the
number of all accepting paths for w. The degree of ambiguity of N , denoted da(N ) is the
supremum of the set {daN (w)|w ? ??}. N is called infinitely ambiguous if da(N ) = ?,
finitely ambiguous if da(N ) < ?, and unambiguous if da(N ) ? 1. The degree of growth
of the ambiguity of N , denoted deg(M ) is defined as the minimum degree of a univariate
polynomial h with positive integral coefficients such that for all w ? ??, daN (w) ? h(|w|) if
such a polynomial exists, or infinity otherwise.
The above notions relate to NFA. We may derive an analogous notation of ambiguity
for PFA by considering an embedding of a PFA P to an NFA N with the property that for
each letter a ? ?, if the probability of transitioning from a state i to state j is nonzero under
P, then there is an edge from state i to j under N for letter a. The degree of (growth of)
ambiguity of P is then defined as the degree of (growth of) ambiguity of N .
We may use the following notions to determine the degree of ambiguity of a given NFA
(and thus a PFA) A as is shown in the theorem which follows. A state q ? Q is called useful
if there exists an accepting path which visits q.
EDA. There is a useful state q ? Q such that, for some word v ? ??, daA(q, v, q) ? 2.
IDAd. There are useful states r1, s1, . . . , rd, sd ? Q and words v1, u2, v2, . . . , ud, vd ? ?? such
that for all 1 ? ? ? d, r? and s? are distinct and (r?, v?, r?), (r?, v?, s?), (s?, v?, s?) ? ?
and for all 2 ? ? ? d, (s??1, u?, r?) ? ?.
I Theorem 5 ([21, 28, 31]). An NFA (or PFA) A having the EDA property is equivalent to
it being exponentially ambiguous. For any d ? N, an NFA (or PFA) A having property IDAd
is equivalent to deg(A) ? d.
Clearly, if N agrees with IDAd for some d > 0, then it also agrees with IDA1, . . . , IDAd?1.
One must be careful with these notions of ambiguity when considering NFA/PFA A, where
inputs are necessarily from a bounded language L. In such cases, the above criteria do not
suffice to determine the ambiguity of A, since the number of paths must be determined not
over ??, but over all paths from L. Of course, the degree of ambiguity of A 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 w1, w2 and w3
such that w1w2w3 is an accepting word and daA(q, w2, q) ? 2, thus w1w2w3 has at least two
distinct accepting runs. However, this implies that daA(w1w2kw3) ? 2k and thus w1w2kw3
has at least 2k accepting runs. Now, if we are given some bounded language L such that
w1w2w3 ? L and daA(q, w2, q) ? 2 then the same implication is not possible, unless w2 ? ?
is a single letter, otherwise there is no guarantee that w1w2kw3 ? L. 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.
I Lemma 6. Probabilistic finite automata defined over upper-triangular matrices are
polynomially ambiguous.
Proof. This lemma is immediate from Theorem 5 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 ?a?) which is upper-triangular only defines transitions of the form ?(i, a) =
j where i ? j and thus the states entered for any run are monotonically nondecreasing. J
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).
I Problem 7 (Mixed Modification PCP (MMPCP)). Given a binary alphabet ?2, a finite set
of letters ? = {s1, s2, . . . , s|?|}, and a pair of homomorphisms h, g : ?? ? ??, the MMPCP
2
asks to decide whether there exists a word w = x1 . . . xk ? ?+, xi ? ? such that
h1(x1)h2(x2) . . . hk(xk) = g1(x1)g2(x2) . . . gk(xk),
where hi, gi ? {h, g}, and there exists at least one j such that hj 6= gj.
I Theorem 8 ([12]). The Mixed Modification PCP is undecidable for |?| ? 9.
A second useful undecidable problem is Hilbert?s tenth problem: Let P (n1, n2, . . . , nk)
be an integer polynomial with k variables - determine if there exists a procedure to find if
there exist x1, x2, . . . , xk ? Z such that: P (x1, x2, . . . , xk) = 0. 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 Z-rational formal power series
S ? ZhhAii, whether there exists any word w ? A? such that (S, w) = 0. The undecidability
of this problem was shown in 1970 by Y. Matiyasevich (building upon work of Davis, Putman,
Robinson and others). For more details, 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 L, it is undecidable to determine if {w ? L|fP (w)??} is
empty for ? ? {?, <, >, ?}. The constructed PFA P 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.
I Lemma 9. Let A1, . . . , A` ? Fn?n. Then for any index sequence (i1, j1), . . . , (i`, j`) with
each (is, js) ? [1, n] ? [1, n] then there exists 1 ? i, j ? n` such that
`
Y (Am)im,jm =
m=1
Proof. The proof proceeds by induction. For the base case when ` = 1, we just set
(i, j) = (i1, j1) and we are done. Assume then that the result holds for some ` ? 1, then for
sequence (i1, j1), (i2, j2), . . . , (i`?1, j`?1) there exists 1 ? i0, j0 ? n`?1 such that:
`?1
Y (Am)im,jm =
m=1
`?1
O Am
m=1
!
i0,j0
By the definition of Kronecker product,
`?1
O Am
m=1
!
? A`
!
ni0+i`,nj0+j`
`?1
= Y (Am)im,jm ? (A`)i`,j`
m=1
as required.
Note that we can of course work out the particular value of i and j, but in general the
formula for i, j does not have a nice form when ` > 2, and anyway will not be necessary for
us, so we settle for an existential proof of such i and j.
3.1
Proof of Theorem 1
Proof. We will construct a polynomially ambiguous probabilistic finite automaton P, a
cut-point ? ? [0, 1] and a letter monotonic language L.
We begin by encoding an instance of Hilbert?s tenth problem into a set of integer matrices.
Let P (x1, x2, . . . , xt) = 0 be a Diophantine equation. Homogenenization of polynomials is
a well known technique, as is used for example in the study of Gr?bner bases [11], which
allows us to convert such a Diophantine equation to P h(x0, x1, x2, . . . , xt) = 0 with a new
dummy variable x0 such that P h is a homogeneous polynomial (each term having the same
degree d) and for which P h(x0, x1, . . . , xt) = P (x1, x2, . . . , xt) when x0 = 1. We thus assume
a homogeneous Diophantine equation P h(x0, x1, . . . , xt) = 0 with implied constraint x0 = 1
which will be dealt with later. Furthermore, we assume that P h gives nonnegative values,
which may be assumed by redefining P h = (P h)2, which clearly does not affect whether a
zero exists for such a polynomial.
Notice that given A = 10 11 , then Ak = 10 k1 . We will generalise this property to a
set of t + 1 matrices A0, A1, . . . , At ? Z(t+3)?(t+3) so that given any tuple (x0, x1, x2, . . . , xt),
then xi appears as an element on the superdiagonal of A0x0 A1x1 ? ? ? Atxt for each 0 ? i ? t. We
will also have the property that each Ai has the same row sum of 2 for every row, which will
be useful when we later convert to stochastic matrices.
We define each matrix Ai for 0 ? i ? t + 1 in the following way:
?1 ?0,i
?0 1
??0 0
Ai = ???? ... ...
??0 0
??0 0
0 0
(1)
where 0 ? j 6= i ? t and ?`,i ? {0, 1} is the Kronecker delta (thus ?i,i = 1 and ?`,i = 0 for
` 6= i). We also denote J = At+1, noting that this is the matrix (1) when all ?`,i have the
value 0. Notice then that every row sum of Ai and J is 2. This structure is retained under
matrix powers and it is easy to see that:
All row sums of Aik are 2k and exactly one element of the superdiagonal is equal to k,
with all other elements on the superdiagonal (excluding that on row t + 2) zero. Taking
powers of Ai will allow us to choose any positive value of variable xi. Note that J k has the
same form as the matrix of (2) with all ?`,i = 0 and acts as a kind of identity matrix, (in its
upperleft block) while retaining the 2k row sum. Indeed, one sees that for all 0 ? i, j ? t + 1,
then AiAj = AjAi, i.e. these matrices commute (as does J since J = At+1). We now show
how to compute terms of P h.
We may write P h(x0, x1, . . . , xt) = Pjr=1 Tj(x0, x1, . . . , xt), where Tj denotes the j?th
term of P h, with P h having r terms. Since P h is a homogeneous polynomial, each term has
the same degree d. We may thus write each term as:
Tj(x0, x1, . . . , xt) = cjRj(x0, x1, . . . , xt),
with cj ? Z and Rj(x0, x1, . . . , xt) = Qt`=0 x`rj,` with rj,` ? 0 and Pt`=0 rj,` = d. For
convenience, we define a d-dimensional vector sj = Nt`=0 `?rj,` ? [0, t]d. For example, if
t = 4, d = 8 and Tj(x0, x1, x2, x3, x4) = 6x21x53x4, then Rj(x0, x1, x2, x3, x4) = x0x1x2x3x4
0 2 0 5 1
and thus sj = (1, 1, 3, 3, 3, 3, 3, 4)T ? [0, 4]8. By sj[i] we denote the i?th element of vector sj.
We now define t + 1 matrices corresponding to term Tj:
(2)
(3)
(4)
(5)
for any k ? 0. In the example when rj,0 = 0, rj,1 = 2, rj,2 = 0, rj,3 = 5 and rj,4 = 1,
then Xj,3 = J ?0 ? J ?2 ? J ?0 ? Ai?5 ? J ?1 = J ?2 ? Ai?5 ? J . We then see that Xjk,3 =
(J k)?2 ? (A3k)?5 ? J k.
Now, we see that:
Xjx,00Xjx,11 ? ? ? Xjx,tt =
t i?1
Y O(J xi )?rj,` ? (Aixi )?rj,i ?
i=0 `=0
d
= O
D`x,00D`x,11 ? ? ? D`x,tt ,
d
O (J xi )?rj,`
!
`=i+1
i?1 d
Xj,i = O J ?rj,` ? Ai?rj,i ? O J ?rj,` ,
`=0 `=i+1
i?1
Xjk,i = O(J k)?rj,` ? (Aik)?rj,i ?
`=0
d
O (J k)?rj,` ,
`=i+1
where 0 ? i ? t. The dimension of such matrices is (t + 3)d ? (t + 3)d since each submatrix
has dimension (t + 3) ? (t + 3) and we take the d-fold Kronecker product. Similarly, we see
that the row sum of each Xj,i is 2d since the row sum of each Ai and J is 2 and we take a
d-fold Kronecker product. Clearly then, by the mixed product property (see Lemma 4):
where D`,i ? {J, Ai} for 0 ? i ? t. The derivation of Eqn (5) from Eqn (4) follows
by the mixed product property of the Kronecker product (Lemma 4). For each product
D`x,00D`x,11 ? ? ? D`x,tt, we see that D`,sj[`] = Asj[`] and D`,j = J for all 0 ? j ? d with j 6= sj[`].
As discussed earlier, matrices Ai and J commute, for any 0 ? i ? t and thus we may
rewrite (5) as:
Asxjs[j`[]`] ? J xsj[`] , where xsj[`] =
X xq
By Lemma 9, we see that some element of Xjx,00Xjx,11 ? ? ? Xjk,tt is thus equal to Rj(x0, x1, . . . , xt)
as required, since there is an element on the superdiagonal of Asxjs[j`[]`] equal to xsj[`] for each
0 ? ` ? d. Let us assume that Rj(x0, x1, . . . , xt) appears at row i1 and column i2. Now,
we may define a vector u0j = cjei1 and vj0 = ei2 where cj is the coefficient of term Tj as in
Eqn (3) and ei1 , ei2 ? Z(t+3)d are basis vectors. We may now see that
(u0j)T Xjx,00Xjx,11 ? ? ? Xjx,ttvj0 = cjRj(x0, x1, . . . , xt) = Tj(x0, x1, . . . , xt)
In order to derive the sum of the r such terms Pjr=1 Tj(x0, x1, . . . , xt), we will utilise the
direct sum. For 0 ? ` ? d, we define Y`0 by:
(6)
(7)
r
Y`0 = M Xj,` ? Nr(t+3)d?r(t+3)d
j=1
We shall now modify each Y`0 so that they are row stochastic. We recall that the row sum of
each A` and J is 2. Therefore, the row sum of each Xj,` is 2d, since Xj,` is a d-fold Kronecker
product of Ai and J matrices. Then the row sum of each Y`0 is also 2d since direct sums do
not modify the row sum. We thus see that Y` = 21d Y`0 is row stochastic.
We now consider the coefficients of each term. We previously multiplied each initial
vector uj by cj and we may consider taking the Kronecker sum of each uj before normalising
the resulting vector (normalising according to L1 norm). We face an issue however, since
some coefficients cj 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 uT Xv = ? ? [0, 1], then by defining v0 = 1 ? v where 1 is the
all-one vector of appropriate dimension (i.e. swapping between final and non final states),
then uT Xv = 1 ? ? ? [0, 1].
Now, since each Xj,` has a row sum of 2d and u0j is of unit length (L1 norm), then Eqn. (7)
can be adapted to the following:
(u0j)T Xjx,00Xjx,11 ? ? ? Xjx,tt(1 ? vj0) = 2d(x0+x1+...+xt) ? cjRj(x0, x1, . . . , xt)
= 2d(x0+x1+...+xt) ? Tj(x0, x1, . . . , xt)
(8)
Let us assume, without loss of generality, that we have arranged the terms of P h such
that those terms with a positive coefficient (positive terms) appear first, followed by those
with a negative coefficient (negative terms). Since we have r terms in P h, there exists
some 1 ? r0 < r such that we have r0 postive and r ? r0 negative terms. Let us define
uj = |cj|ei1 , which is similar to u0j defined previously, but using the absolute value of the
corresponding coefficient.
j=1 vj ? Ljr=r0+1(1 ? vj) ? {0, 1}r(t+3)d as the final vector, so that we
We define v = Lr0
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 u, which must be a probability distribution. Let g =
Pjr=1 |cj| be the sum of absolute values of coefficients and define u = g1 Ljr=1 uj ? [0, 1]r(t+3)d .
Note that u is stochastic (a probability distribution).
We now see that:
=
=
=
uT Y0Y1a1 ? ? ? Ytat v
Pjr=01 uj
N`d=0 Asxjs[j`[]`] ? J xsj[`] vj + Pjr=r0+1 uj
N`d=0 Asxjs[j`[]`] ? J xsj[`] (1 ? vj)
g2d(1+a1+???+at)
Here we used the definition of matrices Yi and Eqn. (6) to rewrite the expressions for
Xj,0 ? ? ? Xj,t. Notice that the power of Y0 is set at 1, 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. (9) as:
Pjr=01 Tj(x0, . . . , xt) + Pjr=r0+1 2d(1+a1+...+at) ? |Tj(x0, . . . , xt)|
g2d(1+a1+???+at)
(r ? r0 + 1) +
g
(r ? rg0 + 1) + Pgh2(dx(01+,xa11+,.??.?+.,axt)t)
Pjr=01 Tj(x0, . . . , xt) + Pjr=r0 Tj(x0, . . . , xt)
g2d(1+a1+???+at)
We therefore define P = (u, {Ya|a ? ?t}, v) and ?t = {0, 1, . . . , t} as our PFA, with letter
monotonic language L = 01?2? ? ? ? t? and ? = r?rg0+1 ? [0, 1]?Q as the cut-point. There exists
some word w = 01x1 2x2 ? ? ? txt ? L such that fP (w) ? ? if and only if P h(1, x1, x2, . . . , xt) = 0.
Therefore the strict emptiness problem for P is undecidable on letter monotonic languages.
Since P is upper-triangular, then it is polynomially ambiguous. We note the surprising fact
that all generator matrices are in fact commutative (each Xj,i is commutative and direct
sums do not affect commutativity), which leads to the undecidability of non-strict cut-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 letters is important. In
fact we may redefine u = uY0 and L = 1?2? ? ? ? t? to remove Y0 and all constraints on L. J
4
Injectivity problems for polynomially ambiguous PFA
We now study the injectivity of acceptance probabilities of polynomially ambiguous PFA. The
next result begins with a proof technique from [4], where the undecidability of the injectivity
problem (called the freeness problem in [4], although we here rename it injectivity) was shown
for exponentially ambiguous PFA over five states. We show that the injectivity 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 2
Proof. Let ? = {x1, x2, . . . , xn?2} and ? = {xn?1, xn} be distinct alphabets and h, g :
?? ? ?? be an instance of the mixed modification PCP. The naming convention will become
apparent below. We define two injective mappings ?, ? : (? ? ?)? ? Q by:
?(xi1 xi2 ? ? ? xim ) = ?jm=1ij(n + 1)j?1,
?(xi1 xi2 ? ? ? xim ) = ?jm=1ij(n + 1)?j,
(9)
(10)
(11)
(12)
and ?(?) = ?(?) = 0. Thus ? represents xi1 xi2 ? ? ? xim as a reverse (n + 1)-adic number
and ? represents xi1 xi2 ? ? ? xim as a fractional number (0.xi1 xi2 ? ? ? xim )(n+1) (e.g. if n = 9,
then x1x2x3 is represented as ?(x1x2x3) = 32110 and ?(x1x2x3) = 0.12310, where subscript
10 denotes base 10). Note that ?w ? (? ? ?)?, ?(w) ? N and ?(w) ? [0, 1) ? Q. It is
not difficult to see that ?w1, w2 ? (? ? ?)?, (n + 1)|w1|?(w2) + ?(w1) = ?(w1w2) and
(n + 1)?|w1|?(w2) + ?(w1) = ?(w1w2).
Define ?00 : (? ? ?)? ? (? ? ?)? ? Q3?3 by
0 ?(u)?
(n + 1)?|v| ?(v)? .
0 1
It is easy to verify that ?00(u1, v1)?00(u2, v2) = ?00(u1u2, v1v2), i.e., ?00 is a homomorphism.
Let G00 = {?00(xi, g(xi)), ?00(xi, h(xi))|xi ? ?, 1 ? i ? n ? 2}, S00 = hG00i, ?00 = (1, 1, 0)T
and ? 00 = (0, 0, 1)T . Assume that there exist M1 = Gi1 Gi2 ? ? ? Git ? hG00i and M2 =
Gj1 Gj2 ? ? ? Gjt0 ? hG00i such that t 6= t0 or else at least one Gip 6= Gjp where 1 ? p ? t and
? = ?00T M1? 00 = ?00T M2? 00. We see that:
? = ?00T M1? 00 = ?(xi1 xi2 ? ? ? xit ) + ?(f1(xi1 )f2(xi2 ) ? ? ? ft(xit )),
? = ?00T M2? 00 = ?(xj1 xj2 ? ? ? xjt0 ) + ?(f10 (xj1 )f20 (xj2 ) ? ? ? ft00 (xjt0 )),
where each fi, fi0 ? {g, h}. Since ?(w) ? N and ?(w) ? (0, 1) ? Q, ?w ? (? ? ?)?,
injectivity of ? and ? implies that if ?00T M1? 00 = ?00T M2? 00, then t = t0 and ik = jk for
1 ? k ? t. Furthermore, if ?T M1? = ?T M2? , we have that ?(f1(xi1 )f2(xi2 ) ? ? ? ft(xit )) =
?(f10 (xi1 )f20 (xi2 ) ? ? ? ft0(xit )) and since at least one fp 6= fp0 for 1 ? p ? t by our above
assumption, then this corresponds to a correct solution to the MMPCP instance (h, g). On
the other hand, if there does not exist a solution to (h, g), then ?(f1(xi1 )f2(xi2 ) ? ? ? ft(xit )) 6=
?(f10 (xi1 )f20 (xi2 ) ? ? ? ft0(xit )), and injectivity of ? implies that ?00T M1? 00 6= ?00T M2? 00.
We now use our new technique to encode such matrices and vectors to a linearly ambiguous
four state PFA. We first define a mapping ?0 : (? ? ?)? ? (? ? ?)? ? N3?3 to make all
matrices be nonnegative integral:
?(n + 1)|u|+|v| 0 (n + 1)|v|?(u)?
?0(u, v) = (n + 1)|v|?00(u, v) = ? 0 1 (n + 1)|v|?(v)? ? N3?3
0 0 (n + 1)|v|
We next define the following morphism ? : (? ? ?)? ? (? ? ?)? ? Q4?4 to make all such
matrices be row stochastic:
?(u, v) = (n + 1)?k ??
?
?(n + 1)|u|+|v| 0 (n + 1)|v|?(u) ?1?
0 1 (n + 1)|v|?(v)
0 0 (n + 1)|v|
0 0 0 ?4
??32??? ,
where ?j ? N are chosen so that the row sum of each row of ?(u, v) is (n + 1)k for some k.
Any sufficiently large k can be used so long as each row has the same sum (n + 1)k and
thus ?(u, v) becomes row stochastic. We use the same k value for all matrices of G which we
define as G = {?(xi, g(xi)), ?(xi, h(xi))|xi ? ?, 1 ? i ? n ? 2}, so that S = hGi, and finally
? = (1, 1, 0, 0)T and ? = (0, 0, 1, 0)T are the initial and final state vectors respectively.
Assume that there exist M1 = Gi1 ? ? ? Git ? hGi and M2 = Gj1 ? ? ? Gjt0 ? hGi such that
t 6= t0 or else at least one Gip 6= Gjp for 1 ? p ? t and ? = ?T M1? = ?T M2? . We see that:
? = ?T M1? = (n + 1)?kt (?(xi1 xi2 ? ? ? xit ) + ?(f1(xi1 )f2(xi2 ) ? ? ? ft(xit ))) ,
? = ?T M2? = (n + 1)?kt0 ?(xj1 xj2 ? ? ? xjt0 ) + ?(f10 (xj1 )f20 (xj2 ) ? ? ? ft00 (xjt0 )) ,
where each fi, fi0 ? {g, h}. If t = t0, then the same argument as previously shows that ik = jk
for 1 ? k ? t. If t 6= t0, assume without loss of generality that t0 < t. In this case we see that:
(n + 1)?kt00 (?(xi1 ? ? ? xit ) + ?(f1(xi1 ) ? ? ? ft(xit ))) = ?(xj1 ? ? ? xjt0 ) + ?(f10 (xj1 ) ? ? ? ft00 (xjt0 )),
where t00 = t ? t0. This is a contradiction however since the number of nonzero digits (where
a digit is understood base (n + 1) here) in the left hand side of this expression is exactly 2t,
and the number of digits in the right expression is 2t0 < 2t. Note that the multiplication by
(n + 1)?kt00 does not alter the number of nonzero digits, it is only a right shift of all digits,
kt00 times. Thus, since the left and right sides have a different number of nonzero digits they
cannot be equal and thus t = t0 as required. J
4.2
Proof of Theorem 3
Proof. We use a reduction from the equal subset sum problem, defined thus: given a set
of positive integers S = {x1, x2, . . . , xk} ? N, do there exist two disjoint nonempty subsets
S1, S2 ? S such that P`?S1 ` = Pm?S2 m? This problem is known to be NP-complete [32].
Note that although there is a requirement that the sets S1 and S2 be disjoint, this is not
crucial so long as S1 6= S2 (since if some element xj is in both S1, S2, then the equality also
holds when xj is removed from both sets). We may therefore require that S1 =6 S2, with
both nonempty such that the sum of elements of each set is identical. We define the set of
matrices M = {Ai, Bi|1 ? i ? k} ? Q3?3 in the following way:
Note that Ai and Bi are thus row stochastic. Let u = (1, 0, 0)T be the initial probability
distribution, v = (0, 1, 0)T be the final state vector and let P = (u, {Ai, Bi}, v) be our PFA.
Define letter monotonic language L = (a1|b1)(a2|b2) ? ? ? (ak|bk) ? a1?b1?a2?b2? ? ? ? a?kb?k and define
a morphism ? : {ai, bi|1 ? i ? k}? ? {Ai, Bi|1 ? i ? k}? in the natural way (e.g. the
morphism induced by ?(ai) = Ai and ?(bi) = Bi). Now, for a word w = w1w2 ? ? ? wk ? L,
note that wj ? {aj, bj} for 1 ? j ? k. Define that v(ai) = xi and v(bi) = 0. In this case, we
see that (due to the structure of Ai and Bi)
uT ?(w1w2 ? ? ? wk)v =
1 k
X v(w`)
Pjk=1(xj + 1) `=1
Note of course that the factor Pjk=11(xj+1) is the same for any w ? L.
Assume then that there exists two words ?, ? ? L with ? 6= ? such that uT ?(?)v =
uT ?(?)v (i.e. assume that P is not free). Then P`k=1 v(?`) = Pik?S1 xi = Pik?S2 xi =
P`k=1 v(?`), where S1 = {xi; |?|ai > 0} and S2 = {xi; |?|ai > 0}. This is true if and only if
the instance S of the equal subset sum problem has a solution as required (note that only
the empty set has a sum of zero which has unique representation b1 ? ? ? bk). Since Ai and Bi
are upper-triangular, with initial state 1 and final state 2, then P is linearly ambiguous. J
5
Conclusion
There are a variety of open problems remaining. For example, does Theorem 1 still hold for
quadratic ambiguity, when taken alongside the other constraints (letter monotonic language
and commutative matrices). Another direction is to improve the complexity lower bound of
Theorem 3 to show it is either PSPACE-hard, EXPSPACE-hard or undecidable, under the
same constraints as in the theorem statement.
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L. Babai , R. Beals , J-Y. Cai , G. Ivanyos, and E. M. Luks . Multiplicative equations over commuting matrices . In Proc. of the Seventh Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 96 , 1996 .
L. Babai and S. Moran . Arthur-Merlin games: a randomized proof system, and a hierarchy of complexity classes . Journal of Computer and System Sciences , 36 : 254 - 276 , 1988 .
P. C. Bell , S. Chen , and L. M. Jackson . Scalar ambiguity and freeness in matrix semigroups over bounded languages . In Language and Automata Theory and Applications , volume LNCS 9618 , pages 493 - 505 , 2016 .
P. C. Bell , V. Halava , and M. Hirvensalo . Decision Problems for Probabilistic Finite Automata on Bounded Languages . Fundamenta Informaticae , 123 ( 1 ): 1 - 14 , 2012 .
A. Bertoni . The solution of problems relative to probabilistic automata in the frame of the formal language theory . GI Jahrestagung , pages 107 - 112 , 1974 .
A. Bertoni , G. Mauri, and M. Torelli . Some recursively unsolvable problems relating to isolated cutpoints in probabilistic automata . In Automata, Languages and Programming , volume 52 , pages 87 - 94 , 1977 .
V. Blondel and V. Canterini . Undecidable problems for probabilistic automata of fixed dimension . Theory of Computing Systems , 36 : 231 - 245 , 2003 .
Systems and Control Letters , Elsevier, 41 :2: 135 - 140 , 2000 .
V. Blondel and J. N. Tsitsiklis . A survey of computational complexity results in systems and control . Automatica , 36 : 1249 - 1274 , 2000 .
R. G. Bukharaev . Probabilistic automata . Journal of Mathematical Sciences , 13 ( 3 ): 359 - 386 , 1980 .
J. Buresh-Oppenheim , M. Clegg , R. Impagliazzo , and T. Pitassi . Homogenization and the polynomial calculus . Computational complexity , 11 ( 3-4 ): 91 - 108 , 2002 .
J. Cassaigne , J. Karhum?ki , and T. Harju . On the Decidability of the Freeness of Matrix Semigroups . International Journal of Algebra and Computation , 9 ( 3 -4): 295 - 305 , 1999 .
?. Charlier and J. Honkala. The freeness problem over matrix semigroups and bounded languages . Information and Computation , 237 : 243 - 256 , 2014 .
V. Chonev , J. Ouaknine , and J. Worrell . On the Complexity of the Orbit Problem . Journal of the ACM , 63 ( 3 ): 1 - 18 , 2016 .
A. Condon and R. J. Lipton . On the complexity of space bounded interactive proofs . In Proceedings of the 29th Annual Symposium on Foundations of Computer Science (FOCS) , pages 462 - 467 , 1989 .
L. Daviaud , M. Jurdzinski , R. Lazic , F. Mazowiecki , G. A. P?rez , and J. Worrell . When is containment decidable for probabilistic automata ? In International Colloquium on Automata, Languages, and Programming (ICALP) , pages 121 : 1 - 121 : 14 , 2018 .
N. Fijalkow , C. Riveros , and J. Worrell . Probabilistic automata of bounded ambiguity . In 28th International Conference on Concurrency Theory (CONCUR) , pages 19 : 1 - 19 : 14 , 2017 .
V. Halava , J. Kari , and Y. Matiyasevich . On post correspondence problem for letter monotonic languages . Theoretical Computer Science , 410 : 30 - 32 , 2009 .
M. Hirvensalo . Improved undecidability results on the emptiness problem of probabilistic and quantum cut-point languages . SOFSEM 2007: Theory and Practice of Computer Science, Lecture Notes in Computer Science , 4362 : 309 - 319 , 2007 .
R. A. Horn and C. R. Johnson . Topics in matrix analysis . Cambridge University Press, 1991 .
O. Ibarra and B. Ravikumar . On sparseness, ambiguity and other decision problems for acceptors and transducers . In Proc. STACS 1986 , volume 210 , pages 171 - 179 , 1986 .
R. Kannan and R. J. Lipton . Polynomial-time algorithm for the orbit problem . Journal of the ACM , 33 ( 4 ): 808 - 821 , 1986 .
Yu. Matiyasevich. Hilbert's Tenth Problem . MIT Press, 1993 .
Computer Speech & Language, 16 ( 1 ): 69 - 88 , 2002 .
T. Neary . Undecidability in Binary Tag Systems and the Post Correspondence Problem for Five Pairs of Words . In STACS15 , pages 649 - 661 , 2015 .
A. Paz . Introduction to Probabilistic Automata. Academic Press, 1971 .
M. O. Rabin . Probabilistic automata . Information and Control , 6 : 230 - 245 , 1963 .
C. Reutenauer . Properti?t?s arithm?tiques et topologiques de s?ries rationnelles en variables non commutatives . Th?se troisi?me cycle , Universit? Paris VI, 1977 .
A. Salomaa and M. Soittola . Automata-Theoretic Aspects of Formal Power Series . SpringerVerlag, 1978 .
P. Turakainen . Generalized automata and stochastic languages . Proceedings of the American Mathematical Society , 21 : 303 - 309 , 1969 .
A. Weber and H. Seidl . On the degree of ambiguity of finite automata . Theoretical Computer Science , 88 ( 2 ): 325 - 349 , 1991 .
H. J. Woeginger and Z. Yu . On the equal-subset-sum problem . Information Processing Letters , 42 ( 6 ): 299 - 302 , 1992 .
A. Yakaryilmaz and A. C. Say . Unbounded-error quantum computation with small space bounds . Information and Computation , 209 ( 6 ): 873 - 892 , 2011 .