Minsky machines and algorithmic problems

This is a survey of using Minsky machines to study algorithmic problems in semigroups, groups and other algebraic systems.Comment: 19 page

On the time complexity of 2-tag systems and small universal Turing machines

We show that 2-tag systems efficiently simulate Turing machines. As a corollary we find that the small universal Turing machines of Rogozhin, Minsky and others simulate Turing machines in polynomial time. This is an exponential improvement on the previously known simulation time overhead and improves a forty year old result in the area of small universal Turing machines.Comment: Slightly expanded and updated from conference versio

Undecidability of Multiplicative Subexponential Logic

Subexponential logic is a variant of linear logic with a family of exponential connectives--called subexponentials--that are indexed and arranged in a pre-order. Each subexponential has or lacks associated structural properties of weakening and contraction. We show that classical propositional multiplicative linear logic extended with one unrestricted and two incomparable linear subexponentials can encode the halting problem for two register Minsky machines, and is hence undecidable.Comment: In Proceedings LINEARITY 2014, arXiv:1502.0441

Dense-choice Counter Machines revisited

This paper clarifies the picture about Dense-choice Counter Machines, which have been less studied than (discrete) Counter Machines. We revisit the definition of "Dense Counter Machines" so that it now extends (discrete) Counter Machines, and we provide new undecidability and decidability results. Using the first-order additive mixed theory of reals and integers, we give a logical characterization of the sets of configurations reachable by reversal-bounded Dense-choice Counter Machines

Temporal logic with predicate abstraction

A predicate linear temporal logic LTL_{\lambda,=} without quantifiers but with predicate abstraction mechanism and equality is considered. The models of LTL_{\lambda,=} can be naturally seen as the systems of pebbles (flexible constants) moving over the elements of some (possibly infinite) domain. This allows to use LTL_{\lambda,=} for the specification of dynamic systems using some resources, such as processes using memory locations, mobile agents occupying some sites, etc. On the other hand we show that LTL_{\lambda,=} is not recursively axiomatizable and, therefore, fully automated verification of LTL_{\lambda,=} specifications is not, in general, possible.Comment: 14 pages, 4 figure

О разрешимости проблем ограниченности для счетчиковых машин Минского

In the paper the decidability of boundedness problems for counter Minsky machines is investigated. It is proved, that for Minsky machines with two counters the boundedness is partial decidable, but for the total boundedness problem does not even exist a semidecision algorithm. On the other hand, for one-counter Minsky machines all these problems are polinomial (quantitatively of local machine states) decidable.Исследуется разрешимость проблем ограниченности для счетчиковых машин Минского. Доказывается, что для машин Минского с двумя счетчиками проблема ограниченности лишь частично разрешима, а проблема тотальной ограниченности не является даже частично разрешимой. Для односчет-чиковых машин Минского указанные проблемы разрешимы за время, полиномиально зависящее от общего количества локальных состояний счетчиковой машины

On the boundaries of solvability and unsolvability in tag systems. Theoretical and Experimental Results

Several older and more recent results on the boundaries of solvability and unsolvability in tag systems are surveyed. Emphasis will be put on the significance of computer experiments in research on very small tag systems