Three Applications to Rational Relations of the High Undecidability of the Infinite Post Correspondence Problem in a Regular omega-Language

It was noticed by Harel in [Har86] that "one can define $\Sigma_1^1$-complete versions of the well-known Post Correspondence Problem". We first give a complete proof of this result, showing that the infinite Post Correspondence Problem in a regular $\omega$-language is $\Sigma_1^1$-complete, hence located beyond the arithmetical hierarchy and highly undecidable. We infer from this result that it is $\Pi_1^1$-complete to determine whether two given infinitary rational relations are disjoint. Then we prove that there is an amazing gap between two decision problems about $\omega$-rational functions realized by finite state B\"uchi transducers. Indeed Prieur proved in [Pri01, Pri02] that it is decidable whether a given $\omega$-rational function is continuous, while we show here that it is $\Sigma_1^1$-complete to determine whether a given $\omega$-rational function has at least one point of continuity. Next we prove that it is $\Pi_1^1$-complete to determine whether the continuity set of a given $\omega$-rational function is $\omega$-regular. This gives the exact complexity of two problems which were shown to be undecidable in [CFS08].Comment: To appear in: Special Issue: Frontier Between Decidability and Undecidability and Related Problems, International Journal of Foundations of Computer Scienc

A new proof for the decidability of D0L ultimate periodicity

We give a new proof for the decidability of the D0L ultimate periodicity problem based on the decidability of p-periodicity of morphic words adapted to the approach of Harju and Linna.Comment: In Proceedings WORDS 2011, arXiv:1108.341

Buyback Problem - Approximate matroid intersection with cancellation costs

In the buyback problem, an algorithm observes a sequence of bids and must decide whether to accept each bid at the moment it arrives, subject to some constraints on the set of accepted bids. Decisions to reject bids are irrevocable, whereas decisions to accept bids may be canceled at a cost that is a fixed fraction of the bid value. Previous to our work, deterministic and randomized algorithms were known when the constraint is a matroid constraint. We extend this and give a deterministic algorithm for the case when the constraint is an intersection of $k$ matroid constraints. We further prove a matching lower bound on the competitive ratio for this problem and extend our results to arbitrary downward closed set systems. This problem has applications to banner advertisement, semi-streaming, routing, load balancing and other problems where preemption or cancellation of previous allocations is allowed

The Identity Correspondence Problem and its Applications

In this paper we study several closely related fundamental problems for words and matrices. First, we introduce the Identity Correspondence Problem (ICP): whether a finite set of pairs of words (over a group alphabet) can generate an identity pair by a sequence of concatenations. We prove that ICP is undecidable by a reduction of Post's Correspondence Problem via several new encoding techniques. In the second part of the paper we use ICP to answer a long standing open problem concerning matrix semigroups: "Is it decidable for a finitely generated semigroup S of square integral matrices whether or not the identity matrix belongs to S?". We show that the problem is undecidable starting from dimension four even when the number of matrices in the generator is 48. From this fact, we can immediately derive that the fundamental problem of whether a finite set of matrices generates a group is also undecidable. We also answer several question for matrices over different number fields. Apart from the application to matrix problems, we believe that the Identity Correspondence Problem will also be useful in identifying new areas of undecidable problems in abstract algebra, computational questions in logic and combinatorics on words.Comment: We have made some proofs clearer and fixed an important typo from the published journal version of this article, see footnote 3 on page 1

Weighted Automata on Infinite Words in the Context of Attacker-Defender Games

The paper is devoted to several infinite-state Attacker–Defender games with reachability objectives. We prove the undecidability of checking for the existence of a winning strategy in several low-dimensional mathematical games including vector reachability games, word games and braid games. To prove these results, we consider a model of weighted automata operating on infinite words and prove that the universality problem is undecidable for this new class of weighted automata. We show that the universality problem is undecidable by using a non-standard encoding of the infinite Post correspondence problem

Enumeration and Decidable Properties of Automatic Sequences

We show that various aspects of k-automatic sequences -- such as having an unbordered factor of length n -- are both decidable and effectively enumerable. As a consequence it follows that many related sequences are either k-automatic or k-regular. These include many sequences previously studied in the literature, such as the recurrence function, the appearance function, and the repetitivity index. We also give some new characterizations of the class of k-regular sequences. Many results extend to other sequences defined in terms of Pisot numeration systems

On insertion-deletion systems over relational words

We introduce a new notion of a relational word as a finite totally ordered set of positions endowed with three binary relations that describe which positions are labeled by equal data, by unequal data and those having an undefined relation between their labels. We define the operations of insertion and deletion on relational words generalizing corresponding operations on strings. We prove that the transitive and reflexive closure of these operations has a decidable membership problem for the case of short insertion-deletion rules (of size two/three and three/two). At the same time, we show that in the general case such systems can produce a coding of any recursively enumerable language leading to undecidabilty of reachability questions.Comment: 24 pages, 8 figure

Successful Curriculum Change in Health Management and Leadership Studies for the Specialist Training Programs in Medicine in Finland

In Finland, the specialization programs in Medicine and Dentistry can be undertaken at all five university medical faculties in 50 specialization programs and in five programs for Dentistry. The specialist training requires 5 or 6 years (300–360 ECTS credits) of medical practice including 9 months of service in primary health care centers, theoretical substance specific education, management studies, and passing a national written exam. The renovation of the national curriculum for the specialization programs was implemented, first in 2008 and officially in August 2009, when theoretical multi-professional social, health management and leadership studies (10–30 ECTS credits) were added to the curriculum. According to European Credit Transfer and Accumulation System (ECTS), 1 ECTS credit (henceforth, simply “ECTS”) means 27–30 h of academic work1 National guidelines for the multi-professional leadership training include the basics of organizational management and leadership, the social and healthcare system, human resources (HR) management, leadership interaction and organizational communication, healthcare economy, legislation (HR) and data management. Each medical faculty has implemented management studies autonomously but according to national guidelines. This paper will describe how the compulsory management studies (10 ECTS) have been executed at the Universities of Tampere and Turku. In Tampere, the 10 ECTS management studies follow a flexible design of six academic modules. Versatile modern teaching methods such as technology-assisted and student orientated learning are used. Advanced supplementary management studies (20 ECTS) are also available. In Turku, the 10 ECTS studies consist of academic lectures, portfolio and project work. Attendees select contact studies (4–6 ECTS) from yearly available 20 ECTS and proceed at their own pace. Portfolio and project comprise 2–5 ECTS each. The renovation of medical specializing physicians' management and leadership education has been a successful reform. It has been observed that positive attitudes and interest toward management overall are increasing among younger doctors. In addition, management and leadership education will presumably facilitate medical doctors' work as managers also. Continuous development of medical doctors' management and leadership education for physicians and dentists is needed while the changing and complex healthcare environment requires both professional and leadership expertise

Efficient Enumeration of Non-Equivalent Squares in Partial Words with Few Holes

International audienceA partial word is a word with holes (also called don't cares: special symbols which match any symbol). A p-square is a partial word matching at least one standard square without holes (called a full square). Two p-squares are called equivalent if they match the same sets of full squares. Denote by psquares(T) the number of non-equivalent p-squares which are subwords of a partial word T. Let PSQUARES k (n) be the maximum value of psquares(T) over all partial words of length n with k holes. We show asympthotically tight bounds: c1 · min(nk 2 , n 2) ≤ PSQUARES k (n) ≤ c2 · min(nk 2 , n 2) for some constants c1, c2 > 0. We also present an algorithm that computes psquares(T) in O(nk 3) time for a partial word T of length n with k holes. In particular, our algorithm runs in linear time for k = O(1) and its time complexity near-matches the maximum number of non-equivalent p-squares

Reachability Problems in Nondeterministic Polynomial Maps on the Integers

We study the reachability problems in various nondeterministic polynomial maps in Zn. We prove that the reachability problem for very simple three-dimensional affine maps (with independent variables) is undecidable and is PSPACE-hard for two-dimensional quadratic maps. Then we show that the complexity of the reachability problem for maps without functions of the form ±x + b is lower. In this case the reachability problem is PSPACE-complete in general, and NP-hard for any fixed dimension. Finally we extend the model by considering maps as language acceptors and prove that the universality problem is undecidable for two-dimensional affine maps