Busy Beaver Scores and Alphabet Size

We investigate the Busy Beaver Game introduced by Rado (1962) generalized to non-binary alphabets. Harland (2016) conjectured that activity (number of steps) and productivity (number of non-blank symbols) of candidate machines grow as the alphabet size increases. We prove this conjecture for any alphabet size under the condition that the number of states is sufficiently large. For the measure activity we show that increasing the alphabet size from two to three allows an increase. By a classical construction it is even possible to obtain a two-state machine increasing activity and productivity of any machine if we allow an alphabet size depending on the number of states of the original machine. We also show that an increase of the alphabet by a factor of three admits an increase of activity

Termination of Triangular Integer Loops is Decidable

We consider the problem whether termination of affine integer loops is decidable. Since Tiwari conjectured decidability in 2004, only special cases have been solved. We complement this work by proving decidability for the case that the update matrix is triangular.Comment: Full version (with proofs) of a paper published in the Proceedings of the 31st International Conference on Computer Aided Verification (CAV '19), New York, NY, USA, Lecture Notes in Computer Science, Springer-Verlag, 201

Optimal Design of Robust Combinatorial Mechanisms for Substitutable Goods

In this paper we consider multidimensional mechanism design problem for selling discrete substitutable items to a group of buyers. Previous work on this problem mostly focus on stochastic description of valuations used by the seller. However, in certain applications, no prior information regarding buyers' preferences is known. To address this issue, we consider uncertain valuations and formulate the problem in a robust optimization framework: the objective is to minimize the maximum regret. For a special case of revenue-maximizing pricing problem we present a solution method based on mixed-integer linear programming formulation

Complexity of Bradley-Manna-Sipma Lexicographic Ranking Functions

In this paper we turn the spotlight on a class of lexicographic ranking functions introduced by Bradley, Manna and Sipma in a seminal CAV 2005 paper, and establish for the first time the complexity of some problems involving the inference of such functions for linear-constraint loops (without precondition). We show that finding such a function, if one exists, can be done in polynomial time in a way which is sound and complete when the variables range over the rationals (or reals). We show that when variables range over the integers, the problem is harder -- deciding the existence of a ranking function is coNP-complete. Next, we study the problem of minimizing the number of components in the ranking function (a.k.a. the dimension). This number is interesting in contexts like computing iteration bounds and loop parallelization. Surprisingly, and unlike the situation for some other classes of lexicographic ranking functions, we find that even deciding whether a two-component ranking function exists is harder than the unrestricted problem: NP-complete over the rationals and $\Sigma^P_2$-complete over the integers.Comment: Technical report for a corresponding CAV'15 pape

A lower bound on CNF encodings of the at-most-one constraint

Constraint "at most one" is a basic cardinality constraint which requires that at most one of its $n$ boolean inputs is set to $1$. This constraint is widely used when translating a problem into a conjunctive normal form (CNF) and we investigate its CNF encodings suitable for this purpose. An encoding differs from a CNF representation of a function in that it can use auxiliary variables. We are especially interested in propagation complete encodings which have the property that unit propagation is strong enough to enforce consistency on input variables. We show a lower bound on the number of clauses in any propagation complete encoding of the "at most one" constraint. The lower bound almost matches the size of the best known encodings. We also study an important case of 2-CNF encodings where we show a slightly better lower bound. The lower bound holds also for a related "exactly one" constraint.Comment: 38 pages, version 3 is significantly reorganized in order to improve readabilit

On Multiphase-Linear Ranking Functions

Multiphase ranking functions ($\mathit{M{\Phi}RFs}$) were proposed as a means to prove the termination of a loop in which the computation progresses through a number of "phases", and the progress of each phase is described by a different linear ranking function. Our work provides new insights regarding such functions for loops described by a conjunction of linear constraints (single-path loops). We provide a complete polynomial-time solution to the problem of existence and of synthesis of $\mathit{M{\Phi}RF}$ of bounded depth (number of phases), when variables range over rational or real numbers; a complete solution for the (harder) case that variables are integer, with a matching lower-bound proof, showing that the problem is coNP-complete; and a new theorem which bounds the number of iterations for loops with $\mathit{M{\Phi}RFs}$. Surprisingly, the bound is linear, even when the variables involved change in non-linear way. We also consider a type of lexicographic ranking functions, $\mathit{LLRFs}$, more expressive than types of lexicographic functions for which complete solutions have been given so far. We prove that for the above type of loops, lexicographic functions can be reduced to $\mathit{M{\Phi}RFs}$, and thus the questions of complexity of detection and synthesis, and of resulting iteration bounds, are also answered for this class.Comment: typos correcte

On Hilbert-Schmidt operator formulation of noncommutative quantum mechanics

This work gives value to the importance of Hilbert-Schmidt operators in the formulation of a noncommutative quantum theory. A system of charged particle in a constant magnetic field is investigated in this framework

The Power of Non-Determinism in Higher-Order Implicit Complexity

We investigate the power of non-determinism in purely functional programming languages with higher-order types. Specifically, we consider cons-free programs of varying data orders, equipped with explicit non-deterministic choice. Cons-freeness roughly means that data constructors cannot occur in function bodies and all manipulation of storage space thus has to happen indirectly using the call stack. While cons-free programs have previously been used by several authors to characterise complexity classes, the work on non-deterministic programs has almost exclusively considered programs of data order 0. Previous work has shown that adding explicit non-determinism to cons-free programs taking data of order 0 does not increase expressivity; we prove that this - dramatically - is not the case for higher data orders: adding non-determinism to programs with data order at least 1 allows for a characterisation of the entire class of elementary-time decidable sets. Finally we show how, even with non-deterministic choice, the original hierarchy of characterisations is restored by imposing different restrictions.Comment: pre-edition version of a paper accepted for publication at ESOP'1

Universality, limits and predictability of gold-medal performances at the Olympic Games

Inspired by the Games held in ancient Greece, modern Olympics represent the world's largest pageant of athletic skill and competitive spirit. Performances of athletes at the Olympic Games mirror, since 1896, human potentialities in sports, and thus provide an optimal source of information for studying the evolution of sport achievements and predicting the limits that athletes can reach. Unfortunately, the models introduced so far for the description of athlete performances at the Olympics are either sophisticated or unrealistic, and more importantly, do not provide a unified theory for sport performances. Here, we address this issue by showing that relative performance improvements of medal winners at the Olympics are normally distributed, implying that the evolution of performance values can be described in good approximation as an exponential approach to an a priori unknown limiting performance value. This law holds for all specialties in athletics-including running, jumping, and throwing-and swimming. We present a self-consistent method, based on normality hypothesis testing, able to predict limiting performance values in all specialties. We further quantify the most likely years in which athletes will breach challenging performance walls in running, jumping, throwing, and swimming events, as well as the probability that new world records will be established at the next edition of the Olympic Games.Comment: 8 pages, 3 figures, 1 table. Supporting information files and data are available at filrad.homelinux.or

Who is the best player ever? A complex network analysis of the history of professional tennis

We consider all matches played by professional tennis players between 1968 and 2010, and, on the basis of this data set, construct a directed and weighted network of contacts. The resulting graph shows complex features, typical of many real networked systems studied in literature. We develop a diffusion algorithm and apply it to the tennis contact network in order to rank professional players. Jimmy Connors is identified as the best player of the history of tennis according to our ranking procedure. We perform a complete analysis by determining the best players on specific playing surfaces as well as the best ones in each of the years covered by the data set. The results of our technique are compared to those of two other well established methods. In general, we observe that our ranking method performs better: it has a higher predictive power and does not require the arbitrary introduction of external criteria for the correct assessment of the quality of players. The present work provides a novel evidence of the utility of tools and methods of network theory in real applications.Comment: 10 pages, 4 figures, 4 table