On the computational complexity of the languages of general symbolic dynamical systems and beta-shifts

AbstractWe consider the computational complexity of languages of symbolic dynamical systems. In particular, we study complexity hierarchies and membership of the non-uniform class P/poly. We prove: 1.For every time-constructible, non-decreasing function t(n)=ω(n), there is a symbolic dynamical system with language decidable in deterministic time O(n2t(n)), but not in deterministic time o(t(n)).2.For every space-constructible, non-decreasing function s(n)=ω(n), there is a symbolic dynamical system with language decidable in deterministic space O(s(n)), but not in deterministic space o(s(n)).3.There are symbolic dynamical systems having hard and complete languages under ≤mlogs- and ≤mp-reduction for every complexity class above LOGSPACE in the backbone hierarchy (hence, P-complete, NP-complete, coNP-complete, PSPACE-complete, and EXPTIME-complete sets).4.There are decidable languages of symbolic dynamical systems in P/poly for every alphabet of size |Σ|≥1.5.There are decidable languages of symbolic dynamical systems not in P/poly iff the alphabet size is >1.For the particular class of symbolic dynamical systems known as β-shifts, we prove that: 1.For all real numbers β>1, the language of the β-shift is in P/poly.2.If there exists a real number β>1 such that the language of the β-shift is NP-hard under ≤Tp-reduction, then the polynomial hierarchy collapses to the second level. As NP-hardness under ≤mp-reduction implies hardness under ≤Tp-reduction, this result implies that it is unlikely that a proof of existence of an NP-hard language of a β-shift will be forthcoming.3.For every time-constructible, non-decreasing function t(n)≥n, there is a real number 1<β<2 such that the language of the β-shift is decidable in time O(n2t(logn+1)), but not in any proper time bound g(n) satisfying g(4n)=o(t(n)/16n).4.For every space-constructible, non-decreasing function s(n)=ω(n2), there is a real number 1<β<2 such that the language of the β-shift is decidable in space O(s(n)), but not in space g(n) where g is any function satisfying g(n2)=o(s(n)).5.There exists a real number 1<β<2 such that the language of the β-shift is recursive, but not context-sensitive

Term Rewriting Systems as Topological Dynamical Systems

Topological dynamics is, roughly, the study of phenomena related to iterations of continuous maps from a metric space to itself. We show how the rewrite relation in term rewriting gives rise to dynamical systems in two distinct, natural ways: (A) One in which any deterministic rewriting strategy induces a dynamical system on the set of finite and infinite terms endowed with the usual metric, and (B) one in which the unconstrained rewriting relation induces a dynamical system on sets of sets of terms, specifically the set of compact subsets of the set of finite and infinite terms endowed with the Hausdorff metric. For both approaches, we give sufficient criteria for the induced systems to be well-defined dynamical systems and for (A) we demonstrate how the classic topological invariant called topological entropy turns out to be much less useful in the setting of term rewriting systems than in symbolic dynamics

Decidability Problems for Self-induced Systems Generated by a Substitution

International audienceIn this talk we will survey several decidability and undecidability results on topological properties of self-affine or self-similar fractal tiles. Such tiles are obtained as fixed point of set equations governed by a graph. The study of their topological properties is known to be complex in general: we will illustrate this by undecidability results on tiles generated by multitape automata. In contrast, the class of self affine tiles called Rauzy fractals is particularly interesting. Such fractals provide geometrical representations of self-induced mathematical processes. They are associated to one-dimensional combinatorial substitutions (or iterated morphisms). They are somehow ubiquitous as self-replication processes appear naturally in several fields of mathematics. We will survey the main decidable topological properties of these specific Rauzy fractals and detail how the arithmetic properties yields by the combinatorial substitution underlying the fractal construction make these properties decidable. We will end up this talk by discussing new questions arising in relation with continued fraction algorithm and fractal tiles generated by S-adic expansion systems

Active imaginative listening—a neuromusical critique

The parallel study of music in science and creative practice can be traced back to the ancients; and paralleling the emergence of music neuroscience, creative musical practitioners have employed neurobiological phenomena extensively in music composition and performance. Several examples from the author’s work in this area, which began in the 1960s, are cited and briefly described. From this perspective, the author also explores questions pertinent to current agendas evident in music neuroscience and speculates on potentially potent future directions

Conversational Movement Dynamics and Nonverbal Indicators of Second Language Development: A Microgenetic Approach

This dissertation study extends on current understandings of gesture and embodied interaction with the eco-social environment in second language development (SLD) while introducing new aspects of movement analysis through dynamical modeling. To understand the role of embodiment during learning activities, a second language learning task has been selected. Dyads consisting of a non-native English-speaking student and a native English-speaking tutor were video recorded during writing consultations centered on class assignments provided by the student. Cross-recurrence quantification analysis was used to measure interactional movement synchrony between the members of each dyad. Results indicate that students with varied English proficiency levels synchronize movements with their tutors over brief, frequent periods of time. Synchronous movement pattern complexity is highly variable across and within the dyads. Additionally, co-speech gesture and gesture independent of speech were analyzed qualitatively to identify the role of gesture as related to SLD events. A range of movement types were used during developmental events by the students and tutors to interact with their partner. The results indicated that language development occurs within a movement rich context through negotiated interaction which depends on a combination of synchronized and synergistic movements. Synchronized movements exhibited complex, dynamical behaviors including variability, self-organization, and emergent properties. Synergistic movement emergence revealed how the dualistic presence of the self/other in each dyad creates a functioning intersubjective space. Overall, the dyads demonstrated that movement is a salient factor in the writing consultation activity

Dynamical Directions in Numeration

International audienceWe survey definitions and properties of numeration from a dynamical point of view. That is we focuse on numeration systems, their associated compactifications, and the dynamical systems that can be naturally defined on them. The exposition is unified by the notion of fibred numeration system. A lot of examples are discussed. Various numerations on natural, integral, real or complex numbers are presented with a special attention payed to beta-numeration and its generalisations, abstract numeration systems and shift radix systems. A section of applications ends the paper