On Problems Dual to Unification: The String-Rewriting Case

In this paper, we investigate problems which are dual to the unification problem, namely the Fixed Point (FP) problem, Common Term (CT) problem and the Common Equation (CE) problem for string rewriting systems. Our main motivation is computing fixed points in systems, such as loop invariants in programming languages. We show that the fixed point (FP) problem is reducible to the common term problem. Our new results are: (i) the fixed point problem is undecidable for finite convergent string rewriting systems whereas it is decidable in polynomial time for finite, convergent and dwindling string rewriting systems, (ii) the common term problem is undecidable for the class of dwindling string rewriting systems, and (iii) for the class of finite, monadic and convergent systems, the common equation problem is decidable in polynomial time but for the class of dwindling string rewriting systems, common equation problem is undecidable.Comment: 28 pages, 6 figures, will be submitted for LMCS journal. arXiv admin note: substantial text overlap with arXiv:1706.0560

Reconfiguration in bounded bandwidth and treedepth

We show that several reconfiguration problems known to be PSPACE-complete remain so even when limited to graphs of bounded bandwidth. The essential step is noticing the similarity to very limited string rewriting systems, whose ability to directly simulate Turing Machines is classically known. This resolves a question posed open in [Bonsma P., 2012]. On the other hand, we show that a large class of reconfiguration problems becomes tractable on graphs of bounded treedepth, and that this result is in some sense tight.Comment: 14 page

A complete characterization of termination of 0p1q → 1r0s

We completely characterize termination of one-rule string rewriting systems of the form 0p1q → 1r0s for every choice of positive integers p, q, r, and s. For the simply terminating cases, we give a sharp estimate of the complexity of derivation lengths

Monoid presentations of groups by finite special string-rewriting systems

We show that the class of groups which have monoid presentations by means of finite special [λ]-confluent string-rewriting systems strictly contains the class of plain groups (the groups which are free products of a finitely generated free group and finitely many finite groups), and that any group which has an infinite cyclic central subgroup can be presented by such a string-rewriting system if and only if it is the direct product of an infinite cyclic group and a finite cyclic group

Infinite families of finite string rewriting systems and their confluence

International audienceWe introduce parameterized rewrite systems for describing infinite families of finite string rewrite systems depending upon non-negative integer pa- rameters, as well as ways to reason uniformly over these families. Unlike previous work, the vocabulary on which a rewrite system in the family is built depends it- self on the integer parameters. Rewriting makes use of a toolkit for parameterized words which allows to describe a rewrite step made independently by all systems in an infinite family by a single, effective parameterized rewrite step. The main result is a confluence test for all systems in a family at once, based on a critical pair lemma classically based on computing finitely many overlaps between left- hand sides of parameterized rules and then checking for their joinability (which decidability is not garanteed)

Weighted automata define a hierarchy of terminating string rewriting systems

The "matrix method" (Hofbauer and Waldmann 2006) proves termination of string rewriting via linear monotone interpretation into the domain of vectors over suitable semirings. Equivalently, such an interpretation is given by a weighted finite automaton. This is a general method that has as parameters the choice of the semiring and the dimension of the matrices (equivalently, the number of states of the automaton). We consider the semirings of nonnegative integers, rationals, algebraic numbers, and reals; with the standard operations and ordering. Monotone interpretations also allow to prove relative termination, which can be used for termination proofs that consist of several steps. The number of steps gives another hierarchy parameter. We formally define the hierarchy and we prove that it is infinite in both directions (dimension and steps)

On Protocell "Computation"

The EU FP6 Integrated Project PACE ('Programmable Artificial Cell Evolution') is investigating the creation, de novo, of chemical 'protocells'. These will be minimal 'wetware' chemical systems integrating molecular information carriers, primitive energy conversion (metabolism) and containment (membrane). Ultimately they should be capable of autonomous reproduction, and be 'programmable' to realise specific desired function. A key objective of PACE is to explore the application of such protocell technology to build novel nanoscale computational devices. In principle, such computation might be realised either at the level of an individual protocell or at the level of self-assembling, multi-cellular, aggregates. In the case of the individual protocell level, a form of 'molecular computation' may be possible in the manner of 'cell signalling networks' in modern cells. This might be particularly appropriate where a protocell is deployed to interface directly with molecular systems, such as in 'smart drug' applications. 'Programming' of molecular computation functionality might be realised by evolutionary techniques, i.e., applying selection to polulations of (reproducing) protocells. Reflexive string rewriting systems may provide an appropriate formal model of molecular computation. The behaviour of minimal reflexive string rewriting systems, incorporated in reproducing containers (protocells), is being explored in simulation. This is a basis for possible design of minimal protocell 'computers'

Towards 3-Dimensional Rewriting Theory

String rewriting systems have proved very useful to study monoids. In good cases, they give finite presentations of monoids, allowing computations on those and their manipulation by a computer. Even better, when the presentation is confluent and terminating, they provide one with a notion of canonical representative of the elements of the presented monoid. Polygraphs are a higher-dimensional generalization of this notion of presentation, from the setting of monoids to the much more general setting of n-categories. One of the main purposes of this article is to give a progressive introduction to the notion of higher-dimensional rewriting system provided by polygraphs, and describe its links with classical rewriting theory, string and term rewriting systems in particular. After introducing the general setting, we will be interested in proving local confluence for polygraphs presenting 2-categories and introduce a framework in which a finite 3-dimensional rewriting system admits a finite number of critical pairs