We settle the equivalence between the problem of hitting a polyhedral set by the orbit of a linear map and the intersection of a regular language and a language of permutations of binary words (the permutation filter realizability problem). The decidability of the both problems is presently unknown and the first one is a straightforward generalization of the famous Skolem problem and the nonnegativity problem in the theory of linear recurrent sequences. To show a `borderline' status of the permutation filter realizability problem with respect to computability we present some decidable and undecidable problems closely related to it.
Let Φ be a linear map of a vector space V into itself and let x ∈ V be a vector fin V . The iterations of Φ applied to x define an orbit Orb Φ x, i.e. the set
In the present paper we discuss algorithmic issues related to orbits. We assume that V is a rational coordinate space, so Φ and x are represented by their (rational) components.
An orbit description problem consists in finding specific relations which either hold for all vectors in the orbit or are violated by at least one vector in the orbit. Here we limit ourself to a simple case where the relations are formed from Boolean combinations of linear equalities and inequalities (see the exact definition below in Section 2).
Note that the most important case of the orbit problem is the chamber hitting problem. In this particular case we check whether an orbit intersects a closed polyhedron (i.e. a solution set for a finite system of nonstrict linear inequalities).
The orbit description problems are related to some problems on linear recurrent sequences.
A linear recurrent sequence (LRS) x n of a degree d is defined by:
x n = d i=1 a i x n-i when n > d,
x n = b n when 1 ≤ n ≤ d, where a i , b j are constants.
(
The famous Skolem problem is perhaps the most known algorithmic problem on linear recurrences.
The Skolem problem. Let {x n } be a specified LRS with integer coefficients. Whether x k = 0 for some k?
The decidability of the Skolem problem is presently an open question, but it is known that it is decidable for the degrees ≤ 5 (the cases d = 3, 4 are worked out by N. Vereshchagin [17], and the case d = 5 is solved by V. Halava et al. [7]).
In the opposite direction, the best hardness result on the Skolem problem is NPhardness [3].
LRS is called nonnegative if all its elements are nonnegative. One more important algorithmic question about LRS is called the nonnegativity problem and consists in checking nonnegativity of a LRS. This problem is at least as hard as the Skolem problem. Being a bit more formal it means that the Skolem problem is Turing reducible to the nonnegativity problem. This statement certainly belongs to the public mind, but see the proof of the statement in Section 4.
The nonnegativity problem is decidable for LRS of the degree ≤ 3 (V. Laohakosol, P. Tangsupphathawat [9]).
In this paper we often use some standard constructions from the theory of algorithms and the complexity theory, e.g., Turing reducibility, m-reducibility, polynomial reducibility. In particular, Turing reducibility of a Problem I to Problem II means that while solving Problem I the reduction may ask an oracle, who can give answers to Problem II (see, further [13,6].)
The relations between the orbit description problems and LRS are described in Section 2.
These results are certainly not new but we include them for completeness sake and in order to introduce necessary notation and terminology.
Let briefly sketch the contents of the paper. We recall basic properties of LRS in Section 1. LRS are closely related to regular languages. In particular, any LRS can be represented as a difference of the generating functions of a pair of regular languages (see, e.g., [11,Cor. 8.2]. We will use a modification of this result (see Theorem 3 from Section 4). (All necessary facts about regular languages could be found in [8,2].)
The main result of the present paper consists in an algorithmic equivalence between the chamber hitting problem and checking a particular property of the regular languages. Namely, the property involved consists in checking whether a regular language contains at least one word from a special set of words. We call this set a permutation filter. Speaking informally, an arbitrary word from the permutation filter gives a permutation of all binary words of a fixed length n. See a formal statement in Section 3 below.
The permutation filter is denoted by P B and the corresponding problem of checking this property of regular languages is called P B -realizability problem (see, Section 3).
The proof of an algorithmic equivalence of a problem of P B -realizability and the chamber hitting problem (Theorem 2 from Section 3) proceeds in several steps.
At first, we describe in Section 4 a polynomial reducibility of the chamber hitting problem to the P B -realizability problem. The reducibility uses the aforementioned Theorem 3.
As a consequence of this result we prove (see, theorem 4) NP-hardness of a P Brealizability problem. This result may be of independent interest.
To construct a reduction in the opposite direction, i.e. a reduction of the P Brealizability problem to the chamber hitting problem, we settle some technical difficulties. It turns out that a natural construction described in Subsection 5.1 gives a reduction of the P B -realizability problem to the problem of hitting a translate of an integral polyhedral cone represented by generators. To reduce this problem to the problem of hitting a rational cone we use some additio
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