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Marcus Pivato

Publications and source records attributed to Marcus Pivato.

At least 19 recordsLinked to original sources
arXiv2024

Coarse Descriptions and Cautious Preferences

We consider a model where an agent is must choose between alternatives that each provide only an imprecise description of the world (e.g. linguistic expressions). The set of alternatives is closed under logical conjunction and disjunction, but not necessarily negation. (Formally: it is a distributive lattice, but not necessarily a Boolean algebra). In our main result, each alternative is identified with a subset of an (endogenously defined) state space, and two axioms characterize maximin decision making. This means: from the agent's preferences over alternatives, we derive a preference order on the endogenous state space, such that alternatives are ranked in terms of their worst outcomes.

econ.TH↗
arXiv2021

Measure and integration on Boolean algebras of regular open subsets in a topological space

The regular open subsets of a topological space form a Boolean algebra, where the `join' of two regular open sets is the interior of the closure of their union. A `credence' is a finitely additive probability measure on this Boolean algebra, or on one of its subalgebras. We develop a theory of integration for such credences. We then explain the relationship between credences, residual charges, and Borel probability measures. We show that a credence can be represented by a normal Borel measure, augmented with a `liminal structure', which specifies how two or more regular open sets share the probability mass of their common boundary. In particular, a credence on a locally compact Hausdorff space can be represented by a normal Borel measure and a liminal structure on the Stone-Čech compactification of that space. We also show how credences can be represented by Borel measures on the Stone space of the underlying Boolean algebra of regular open sets. Finally, we show that these constructions are functorial.

math.GN↗
arXiv2015

Dynamic and stochastic systems as a framework for metaphysics and the philosophy of science

Scientists often think of the world (or some part of it) as a dynamical system, a stochastic process, or a generalization of such a system. Prominent examples of systems are (i) the system of planets orbiting the sun or any other classical mechanical system, (ii) a hydrogen atom or any other quantum-mechanical system, and (iii) the earth's atmosphere or any other statistical mechanical system. We introduce a simple and general framework for describing such systems and show how it can be used to examine some familiar philosophical questions, including the following: how can we define nomological possibility, necessity, determinism, and indeterminism; what are symmetries and laws; what regularities must a system display to make scientific inference possible; is there any metaphysical basis for invoking principles of parsimony such as Occam's Razor when we make such inferences; and what is the role of space and time in a system? Our framework is intended to serve as a toolbox for the formal analysis of systems that is applicable in several areas of philosophy.

physics.hist-ph↗
arXiv2009

Positive expansiveness versus network dimension in symbolic dynamical systems

A `symbolic dynamical system' is a continuous transformation F:X-->X of a closed perfect subset X of A^V, where A is a finite set and V is countable. (Examples include subshifts, odometers, cellular automata, and automaton networks.) The function F induces a directed graph structure on V, whose geometry reveals information about the dynamical system (X,F). The `dimension' dim(V) is an exponent describing the growth rate of balls in the digraph as a function of their radius. We show: if X has positive entropy and dim(V)>1, and the system (A^V,X,F) satisfies minimal symmetry and mixing conditions, then (X,F) cannot be positively expansive; this generalizes a well-known result of Shereshevsky about multidimensional cellular automata. We also construct a counterexample to a version of this result without the symmetry condition. Finally, we show that network dimension is invariant under topological conjugacies which are Holder-continuous.

math.DS↗
arXiv2007

Embedding Bratteli-Vershik systems in cellular automata

Many dynamical systems can be naturally represented as `Bratteli-Vershik' (or `adic') systems, which provide an appealing combinatorial description of their dynamics. If an adic system X satisfies two technical conditions (`focus' and `bounded width') then we show how to represent X using a two-dimensional subshift of finite type Y; each `row' in a Y-admissible configuration corresponds to an infinite path in the Bratteli diagram of X, and the vertical shift on Y corresponds to the `successor' map of X. Any Y-admissible configuration can then be recoded as the spacetime diagram of a one-dimensional cellular automaton F; in this way X is `embedded' in F (i.e. X is conjugate to a subsystem of F). With this technique, we can embed many odometers, Toeplitz systems, and constant-length substitution systems in one-dimensional cellular automata.

math.DS↗
arXiv2007

Module Shifts and Measure Rigidity in Linear Cellular Automata

Suppose R is a finite commutative ring of prime characteristic, A is a finite R-module, M:=Z^D x N^E, and F is an R-linear cellular automaton on A^M. If mu is an F-invariant measure which is multiply shift-mixing in a certain way, then we show that mu must be the Haar measure on a coset of some submodule shift of A^M. Under certain conditions, this means mu must be the uniform Bernoulli measure on A^M.

math.DS↗
arXiv2007

Spectral domain boundaries in cellular automata

Let L:=Z^D be a D-dimensional lattice. Let A^L be the Cantor space of L-indexed configurations in a finite alphabet A, with the natural L-action by shifts. A `cellular automaton' is a continuous, shift-commuting self-map F:A^L-->A^L. An `F-invariant subshift' is a closed, F-invariant and shift-invariant subset X of A^L. Suppose x is an element of A^L that is X-admissible everywhere except for some small region of L which we call a `defect'. Such defects are analogous to `domain boundaries' in a crystalline solid. It has been empirically observed that these defects persist under iteration of F, and often propagate like `particles' which coalesce or annihilate on contact. We use spectral theory to explain the persistence of some defects under F, and partly explain the outcomes of their collisions.

math.DS↗
arXiv2007

Algebraic invariants for crystallographic defects in cellular automata

Let L:= Z^D be the D-dimensional lattice and let A^L be the Cantor space of L-indexed configurations in some finite alphabet A, with the natural L-action by shifts. A `cellular automaton' is a continuous, shift-commuting self-map F of A^L, and an `F-invariant subshift' is a closed, F-invariant and shift-invariant subset X of A^L. Suppose x is a configuration in A^L that is X-admissible everywhere except for some small region we call a `defect'. It has been empirically observed that such defects persist under iteration of F, and often propagate like `particles' which coalesce or annihilate on contact. We construct algebraic invariants for these defects, which explain their persistence under F, and partly explain the outcomes of their collisions. Some invariants are based on the cocycles of multidimensional subshifts; others arise from the higher-dimensional (co)homology/homotopy groups for subshifts, obtained by generalizing the Conway-Lagarias tiling groups and the Geller-Propp fundamental group.

math.DS↗
arXiv2007

RealLife: the continuum limit of Larger Than Life cellular automata

Let A:={0,1}. A `cellular automaton' (CA) is a shift-commuting transformation of A^{Z^D} determined by a local rule. Likewise, a `Euclidean automaton' is a shift-commuting transformation of A^{R^D} determined by a local rule. `Larger than Life' (LtL) CA are long-range generalizations of J.H. Conway's Game of Life CA, proposed by K.M. Evans. We prove a conjecture of Evans: as their radius grows to infinity, LtL CA converge to a `continuum limit' Euclidean automaton, which we call `RealLife'. We also show that the `life forms' (fixed points, periodic orbits, and propagating structures) of LtL CA converge to life forms of RealLife. Finally we prove a number of existence results for fixed points of RealLife.

math.DS↗
arXiv2007

Defect Particle Kinematics in One-Dimensional Cellular Automata

Let A^Z be the Cantor space of bi-infinite sequences in a finite alphabet A, and let sigma be the shift map on A^Z. A `cellular automaton' is a continuous, sigma-commuting self-map Phi of A^Z, and a `Phi-invariant subshift' is a closed, (Phi,sigma)-invariant subset X of A^Z. Suppose x is a sequence in A^Z which is X-admissible everywhere except for some small region we call a `defect'. It has been empirically observed that such defects persist under iteration of Phi, and often propagate like `particles'. We characterize the motion of these particles, and show that it falls into several regimes, ranging from simple deterministic motion, to generalized random walks, to complex motion emulating Turing machines or pushdown automata. One consequence is that some questions about defect behaviour are formally undecidable.

math.DS↗
arXiv2006

Asymptotic Randomization of Sofic Shifts by Linear Cellular Automata

Let M=Z^D be a D-dimensional lattice, and let A be an abelian group. A^M is then a compact abelian group; a `linear cellular automaton' (LCA) is a topological group endomorphism Φ:A^M --> A^M that commutes with all shift maps. Suppose μis a probability measure on A^M whose support is a subshift of finite type or sofic shift. We provide sufficient conditions (on Φand μ) under which Φ`asymptotically randomizes' μ, meaning that wk*lim_{J\ni j --> oo} Φ^j μ= η, where ηis the Haar measure on A^M, and J has Cesaro density 1. In the case when Φ=1+σ, we provide a condition on μthat is both necessary and sufficient. We then use this to construct an example of a zero-entropy measure which is asymptotically randomized by 1+σ(all previously known examples had positive entropy).

math.DS↗
arXiv2005

Prevalence of Odometers in Cellular Automata

We consider a left permutive cellular automaton Phi, with no memory and positive anticipation, defined on the space of all doubly infinite sequences with entries from a finite alphabet. For each such automaton that is not one-to-one, there is a dense set of points X (which is large in another sense too) such that the Phi-orbit closure of each x in X is topologically conjugate to an odometer (the ``+1'' map on a projective limit of finite cyclic groups). We identify this odometer in several cases.

math.DS↗
arXiv2005

Cellular Automata vs. Quasisturmian Shifts

If L=Z^D and A is a finite set, then A^L is a compact space. A cellular automaton (CA) is a continuous transformation F:A^L--> A^L that commutes with all shift maps. A quasisturmian (QS) subshift is a shift-invariant subset obtained by mapping the trajectories of an irrational torus rotation through a partition of the torus. The image of a QS shift under a CA is again QS. We study the topological dynamical properties of CA restricted to QS shifts, and compare them to the properties of CA on the full shift A^L. We investigate injectivity, surjectivity, transitivity, expansiveness, rigidity, fixed/periodic points, and invariant measures. We also study `chopping': how iterating the CA fragments the partition generating the QS shift.

math.DS↗
arXiv2003

Invariant measures for bipermutative cellular automata

A `right-sided, nearest neighbour cellular automaton' (RNNCA) is a continuous transformation F:A^Z-->A^Z determined by a local rule f:A^{0,1}-->A so that, for any a in A^Z and any z in Z, F(a)_z = f(a_{z},a_{z+1}) . We say that F is `bipermutative' if, for any choice of a in A, the map g:A-->A defined by g(b) = f(a,b) is bijective, and also, for any choice of b in A, the map h:A-->A defined by h(a)=f(a,b) is bijective. We characterize the invariant measures of bipermutative RNNCA. First we introduce the equivalent notion of a `quasigroup CA', to expedite the construction of examples. Then we characterize F-invariant measures when A is a (nonabelian) group, and f(a,b) = a*b. Then we show that, if F is any bipermutative RNNCA, and mu is F-invariant, then F must be mu-almost everywhere K-to-1, for some constant K . We use this to characterize invariant measures when A^Z is a `group shift' and F is an `endomorphic CA'.

math.DS↗
arXiv2003

Limit Measures for Affine Cellular Automata, II

If M is a monoid (e.g. the lattice Z^D), and A is an abelian group, then A^M is a compact abelian group; a linear cellular automaton (LCA) is a continuous endomorphism F:A^M --> A^M that commutes with all shift maps. If F is diffusive, and mu is a harmonically mixing (HM) probability measure on A^M, then the sequence {F^N mu} (N=1,2,3,...) weak*-converges to the Haar measure on A^M, in density. Fully supported Markov measures on A^Z are HM, and nontrivial LCA on A^{Z^D} are diffusive when A=Z/p is a prime cyclic group. In the present work, we provide sufficient conditions for diffusion of LCA on A^{Z^D} when A=Z/n is any cyclic group or when A=[Z/(p^r)]^J (p prime). We show that any fully supported Markov random field on A^{Z^D} is HM (where A is any abelian group).

math.DS↗
arXiv2002

Linear cellular automata, asymptotic randomization, and entropy

If A=Z/2, then A^Z is a compact abelian group. A `linear cellular automaton' is a shift-commuting endomorphism F of A^Z. If P is a probability measure on A^Z, then F `asymptotically randomizes' P if F^j P converges to the Haar measure as j-->oo, for j in a subset of Cesaro density one. Via counterexamples, we show that nonzero entropy of P is neither necessary nor sufficient for asymptotic randomization.

math.DS↗