On sets of numbers rationally represented in a rational base number system

In this work, it is proved that a set of numbers closed under addition and whose representations in a rational base numeration system is a rational language is not a finitely generated additive monoid. A key to the proof is the definition of a strong combinatorial property on languages : the bounded left iteration property. It is both an unnatural property in usual formal language theory (as it contradicts any kind of pumping lemma) and an ideal fit to the languages defined through rational base number systems

Bounded Counter Languages

We show that deterministic finite automata equipped with $k$ two-way heads are equivalent to deterministic machines with a single two-way input head and $k-1$ linearly bounded counters if the accepted language is strictly bounded, i.e., a subset of $a_1^*a_2^*... a_m^*$ for a fixed sequence of symbols $a_1, a_2,..., a_m$. Then we investigate linear speed-up for counter machines. Lower and upper time bounds for concrete recognition problems are shown, implying that in general linear speed-up does not hold for counter machines. For bounded languages we develop a technique for speeding up computations by any constant factor at the expense of adding a fixed number of counters

Real-Time Vector Automata

We study the computational power of real-time finite automata that have been augmented with a vector of dimension k, and programmed to multiply this vector at each step by an appropriately selected $k \times k$ matrix. Only one entry of the vector can be tested for equality to 1 at any time. Classes of languages recognized by deterministic, nondeterministic, and "blind" versions of these machines are studied and compared with each other, and the associated classes for multicounter automata, automata with multiplication, and generalized finite automata.Comment: 14 page

Silent Transitions in Automata with Storage

We consider the computational power of silent transitions in one-way automata with storage. Specifically, we ask which storage mechanisms admit a transformation of a given automaton into one that accepts the same language and reads at least one input symbol in each step. We study this question using the model of valence automata. Here, a finite automaton is equipped with a storage mechanism that is given by a monoid. This work presents generalizations of known results on silent transitions. For two classes of monoids, it provides characterizations of those monoids that allow the removal of \lambda-transitions. Both classes are defined by graph products of copies of the bicyclic monoid and the group of integers. The first class contains pushdown storages as well as the blind counters while the second class contains the blind and the partially blind counters.Comment: 32 pages, submitte

Integer Vector Addition Systems with States

This paper studies reachability, coverability and inclusion problems for Integer Vector Addition Systems with States (ZVASS) and extensions and restrictions thereof. A ZVASS comprises a finite-state controller with a finite number of counters ranging over the integers. Although it is folklore that reachability in ZVASS is NP-complete, it turns out that despite their naturalness, from a complexity point of view this class has received little attention in the literature. We fill this gap by providing an in-depth analysis of the computational complexity of the aforementioned decision problems. Most interestingly, it turns out that while the addition of reset operations to ordinary VASS leads to undecidability and Ackermann-hardness of reachability and coverability, respectively, they can be added to ZVASS while retaining NP-completness of both coverability and reachability.Comment: 17 pages, 2 figure

The spatial coverage of dairy cattle urine patches in an intensively grazed pasture system

Accurate field data on the paddock area affected by cow urine depositions are critical to the estimation and modelling of nitrogen (N) losses and N management in grazed pasture systems. A new technique using survey-grade global positioning system (GPS) technology was developed to precisely measure the paddock spatial area coverage, diversity and distribution of dairy cattle urine patches in grazed paddocks over time. A 4-year study was conducted on the Lincoln University Dairy Farm (LUDF), Canterbury, New Zealand, from 2003 to 2007. Twelve field plots, each 100m² in area, were established on typical grazing areas of the farm. All urine and dung deposits within the plots were visually identified, the pasture response area (radius) measured and position marked with survey-grade GPS. The plots were grazed as part of the normal grazing rotation of the farm and urine and dung deposits measured at 12-week intervals. The data were collated using spatial (GIS) software and an assessment of annual urine patch coverage and spatial distribution was made. Grazing intensities ranged from 17645 to 30295 cow grazing h/ha/yr. Mean annual areas of urine patches ranged from 0·34 to 0·40m² (4-year mean 0·37±0·009m²), with small but significant variation between years and seasons. Mean annual urine patch numbers were 6240±124 patches/ha/yr. The mean proportional area coverage for a single sampling event or season was 0·058 and the mean proportional annual urine patch coverage was 0·232±0·0071. There was a strong linear relationship between annual cow grazing h/ha and urine patch numbers/ha (R²=0·69) and also annual urine patch area coverage (R²=0·77). Within the stocking densities observed in this study, an annual increase of 10 000 cow grazing h/ha increased urine patch numbers by 1800 urine patches/ha/yr and annual urine patch area coverage by 0·07. This study presents new quantitative data on urine patch size, numbers and the spatial coverage of patches on a temporal basis

Remarks on blind and partially blind one-way multicounter machines

AbstractWe consider one-way nondeterministic machines which have counters allowed to hold positive or negative integers and which accept by final state with all counters zero. Such machines are called blind if their action depends on state and input alone and not on the counter configuration. They are partially blind if they block when any counter is negative (i.e., only nonnegative counter contents are permissible) but do not know whether or not any of the counters contain zero. Blind multicounter machines are equivalent in power to the reversal bounded multicounter machines of Baker and Book [1], and for both blind and reversal bounded multicounter machines, the quasirealtime family is as powerful as the full family. The family of languages accepted by blind multicounter machines is the least intersection closed semiAFL containing {anbn|n⩾0} and also the least intersection closed semiAFL containing the two-sided Dyck set on one letter. Blind multicounter machines are strictly less powerful than quasirealtime partially blind multicounter machines. Quasirealtime partially blind multicounter machines accept the family of computation state sequences or Petri net languages which is equal to the least intersection closed semiAFL containing the one-sided Dyck set on one letter but is not a principal semiAFL. For partially blind multicounter machines, as opposed to blind machines, linear time is more powerful than quasirealtime. Assuming that the reachability problem for vector addition systems is decidable [16], partially blind multicounter machines accept only recursive sets and do not accept even {anbn|n⩾0∗, and quasirealtime partially blind multicounter machines are less powerful than general quasirealtime multicounter machines

One way finite visit automata

AbstractA one-way preset Turing machine with base L is a nondeterministic on-line Turing machine with one working tape preset to a member of L. FINITEREVERSAL(L) (FINITEVISIT (L)) is the class of languages accepted by one-way preset Turing machines with bases in L which are limited to a finite number of reversals (visits). For any full semiAFL L, FINITEREVERSAL (L) is the closure of L under homomorphic replication or, equivalently, the closure of L under iteration of controls on linear context-free grammars while FINITEVISIT (L) is the result of applying controls from L to absolutely parallel grammars or, equivalently, the closure of L under deterministic two-way finite state transductions. If L is a full AFL with L ≠ FINITEVISIT(L), then FINITEREVERSAL(L) ≠ FINITEVISIT(L). In particular, for one-way checking automata, k + 1 reversals are more powerful than k reversals, k + 1 visits are more powerful than k visits, k visits and k + 1 reversals are incomparable and there are languages definable within 3 visits but no finite number of reversals. Finite visit one-way checking automaton languages can be accepted by nondeterministic multitape Turing machines in space log2 n. Results on finite visit checking automata provide another proof that not all context-free languages can be accepted by one-way nonerasing stack automata