Undecidability of Two-dimensional Robot Games

Robot game is a two-player vector addition game played on the integer lattice $\mathbb{Z}^n$. Both players have sets of vectors and in each turn the vector chosen by a player is added to the current configuration vector of the game. One of the players, called Eve, tries to play the game from the initial configuration to the origin while the other player, Adam, tries to avoid the origin. The problem is to decide whether or not Eve has a winning strategy. In this paper we prove undecidability of the robot game in dimension two answering the question formulated by Doyen and Rabinovich in 2011 and closing the gap between undecidable and decidable cases

Pienten kokonaislukumatriisien kuolevuusongelman ratkeavuudesta

Tämä tutkielma käsittelee kokonaislukumatriisien kuolevuusongelman ratkeavuutta. Kuolevuusongelmassa kysytään onko annettujen matriisien jokin tulo nollamatriisi. Ongelma todistettiin ratkeamattomaksi 3 × 3 matriiseille vuonna 1970, mutta 2 × 2 matriiseille ongelma on kiinnostuksesta huolimatta edelleenkin avoin. Tutkielman pääpainona on 2 × 2 matriisien kuolevuusongelman kahden erikoistapauksen ratkeavaksi osoittaminen. Ensimmäisessä erikoistapauksessa rajoitutaan matriiseihin, joiden determinantti on 0 tai ±1. Toisessa erikoistapauksessa tarkastellaan kahden matriisin kuolevuutta. Lisäksi tarkastellaan yleisesti, miten matriisijoukon koko vaikuttaa kuolevuusongelman ratkeavuuteen.Siirretty Doriast

On the Identity and Group Problems for Complex Heisenberg Matrices

We study the Identity Problem, the problem of determining if a finitely generated semigroup of matrices contains the identity matrix; see Problem 3 (Chapter 10.3) in ``Unsolved Problems in Mathematical Systems and Control Theory'' by Blondel and Megretski (2004). This fundamental problem is known to be undecidable for $\mathbb{Z}^{4 \times 4}$ and decidable for $\mathbb{Z}^{2 \times 2}$. The Identity Problem has been recently shown to be in polynomial time by Dong for the Heisenberg group over complex numbers in any fixed dimension with the use of Lie algebra and the Baker-Campbell-Hausdorff formula. We develop alternative proof techniques for the problem making a step forward towards more general problems such as the Membership Problem. We extend our techniques to show that the fundamental problem of determining if a given set of Heisenberg matrices generates a group, can also be decided in polynomial time

Monadic Decomposability of Regular Relations

Monadic decomposibility - the ability to determine whether a formula in a given logical theory can be decomposed into a boolean combination of monadic formulas - is a powerful tool for devising a decision procedure for a given logical theory. In this paper, we revisit a classical decision problem in automata theory: given a regular (a.k.a. synchronized rational) relation, determine whether it is recognizable, i.e., it has a monadic decomposition (that is, a representation as a boolean combination of cartesian products of regular languages). Regular relations are expressive formalisms which, using an appropriate string encoding, can capture relations definable in Presburger Arithmetic. In fact, their expressive power coincide with relations definable in a universal automatic structure; equivalently, those definable by finite set interpretations in WS1S (Weak Second Order Theory of One Successor). Determining whether a regular relation admits a recognizable relation was known to be decidable (and in exponential time for binary relations), but its precise complexity still hitherto remains open. Our main contribution is to fully settle the complexity of this decision problem by developing new techniques employing infinite Ramsey theory. The complexity for DFA (resp. NFA) representations of regular relations is shown to be NLOGSPACE-complete (resp. PSPACE-complete)

On the Identity Problem for the Special Linear Group and the Heisenberg Group

We study the identity problem for matrices, i.e., whether the identity matrix is in a semigroup generated by a given set of generators. In particular we consider the identity problem for the special linear group following recent NP-completeness result for SL(2,Z) and the undecidability for SL(4,Z) generated by 48 matrices. First we show that there is no embedding from pairs of words into 3 × 3 integer matrices with determinant one, i.e., into SL(3,Z) extending previously known result that there is no embedding into C^2×2. Apart from theoretical importance of the result it can be seen as a strong evidence that the computational problems in SL(3, Z) are decidable. The result excludes the most natural possibility of encoding the Post correspondence problem into SL(3,Z), where the matrix products extended by the right multiplication correspond to the Turing machine simulation. Then we show that the identity problem is decidable in polynomial time for an important subgroup of SL(3,Z), the Heisenberg group H(3,Z). Furthermore, we extend the decidability result for H(n,Q) in any dimension n. Finally we are tightening the gap on decidability question for this long standing open problem by improving the undecidability result for the identity problem in SL(4, Z) substantially reducing the bound on the size of the generator set from 48 to 8 by developing a novel reduction technique

Integer Weighted Automata on Infinite Words

In this paper we combine two classical generalisations of finite automata (weighted automata and automata on infinite words) into a model of integer weighted automata on infinite words and study the universality and the emptiness problems under zero weight acceptance. We show that the universality problem is undecidable for three-state automata by a direct reduction from the infinite Post correspondence problem. We also consider other more general acceptance conditions as well as their complements with respect to the universality and the emptiness problems. Additionally, we build a universal integer weighted automaton with fixed transitions. This automaton has an additional integer input that allows it to simulate any semi-Thue system. </jats:p

On the Identity and Group Problems for Complex Heisenberg Matrices

We study the Identity Problem, the problem of determining if a finitely generated semigroup of matrices contains the identity matrix; see Problem 3 (Chapter 10.3) in “Unsolved Problems in Mathematical Systems and Control Theory” by Blondel and Megretski (2004). This fundamental problem is known to be undecidable for Z4 × 4 and decidable for Z2 × 2. The Identity Problem has been recently shown to be in polynomial time by Dong for the Heisenberg group over complex numbers in any fixed dimension with the use of Lie algebra and the Baker-Campbell-Hausdorff formula. We develop alternative proof techniques for the problem making a step forward towards more general problems such as the Membership Problem. We extend our techniques to show that the fundamental problem of determining if a given set of Heisenberg matrices generates a group, can also be decided in polynomial time