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Invariant measures for bipermutative cellular automata [article]

Marcus Pivato
2003 arXiv   pre-print
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
more » ... he 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'.
arXiv:math/0306211v1 fatcat:bh5zq6k7x5hgtma4cbvnxcuxda

Ergodic Theory of Cellular Automata [chapter]

Marcus Pivato
2012 Computational Complexity  
Proof: For (a), see (Hedlund, 1969, Theorem 5.9) or (Lind and Marcus, 1995, Corollary 8.1.20, p.271) .  ...  (b) The case M = Z is (Lind and Marcus, 1995, Corollary 8.1.20) (actually this holds for any sofic subshift); see also Fiorenzi (2000) .  ... 
doi:10.1007/978-1-4614-1800-9_62 fatcat:6chdeuo5n5b3dj434scnmeplwq

Linear cellular automata, asymptotic randomization, and entropy [article]

Marcus Pivato
2002 arXiv   pre-print
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.
arXiv:math/0210241v1 fatcat:75osk7mx2nchvefcypvgeo7ttm

Pyramidal Democracy

Marcus J. Pivato
2009 Journal of Deliberative Democracy  
Tier 4 Tier 3 Tier 2 Tier 1 t o T i e r 5 b ac k to T ie r 3 This essay is available in Journal of Public Deliberation: Pivato  ...  Pivato: Pyramidal Democracy See(Riker, 1982, Ch.7),(Austen-Smith and Banks, 1999, Ch.6), or(Mueller, 2003, §5.12.1) for a summary.17 See Nurmi and Uusi-Heikkilä (1985) , Wagner (1983 Wagner ( , 1984  ... 
doi:10.16997/jdd.82 fatcat:fx27pjgyijffdbjsdvnbjkb7u4

Embedding Bratteli-Vershik systems in cellular automata [article]

Marcus Pivato, Reem Yassawi
2007 arXiv   pre-print
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'
more » ... ap 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.
arXiv:0710.3608v1 fatcat:hs3qjnfhzrhodo26zjopj6rq6m

Statistical Utilitarianism [chapter]

Marcus Pivato
2016 The Political Economy of Social Choices  
to maximizing the expected value of the utilitarian SWF, and also equivalent to ex ante Pareto efficiency, as Schmitz and Troger observe in footnote 13 of their paper.3 SeeNitzan (2009, Ch.11-12) or Pivato  ... 
doi:10.1007/978-3-319-40118-8_8 fatcat:lf5udt53nfapvhhx5ab4ooyrde

Module Shifts and Measure Rigidity in Linear Cellular Automata [article]

Marcus Pivato
2007 arXiv   pre-print
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.
arXiv:0707.1408v1 fatcat:ceelgqmponer3nskdtxuplnkka

Positive expansiveness versus network dimension in symbolic dynamical systems [article]

Marcus Pivato
2009 arXiv   pre-print
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
more » ... itive 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.
arXiv:0907.2935v1 fatcat:43xdh4bkt5d6zn6mv3ak5r7ygy

Spectral domain boundaries in cellular automata [article]

Marcus Pivato
2007 arXiv   pre-print
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
more » ... 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.
arXiv:math/0507091v2 fatcat:cel6ehgjhjdfxg356shjjhv26a

The median rule in judgement aggregation

Klaus Nehring, Marcus Pivato
2021 Economic Theory  
There exists Q ∈ (0, 1) ∩ Q such that [−Q, Q] K ⊂ conv(X rk A ) (Nehring and Pivato 2011, Example 3.3) .  ...  (Or see Lemma D.7 from Nehring and Pivato (2019) .) Thus, it remains to show that F α satisfies Reinforcement.  ... 
doi:10.1007/s00199-021-01348-7 fatcat:lbji5megmvduvdbyaphzoeoifi

Social preference under twofold uncertainty

Philippe Mongin, Marcus Pivato
2019 Economic Theory  
See Mongin and Pivato (2016) for a review, and Fleurbaey and Mongin (2016) for a new defence of Harsanyi's position.  ...  See Mongin and Pivato (2015) . Part (a) follows from Theorem 1(c,d), and part (b) from Corollary 1(c,d).  ...  ., Karni's (2014) review of SEU theories. 6 For a recent extension of Anscombe and Aumann's theorem, see Mongin and Pivato (2015) .  ... 
doi:10.1007/s00199-019-01237-0 fatcat:lb5t25umkvgdxg63ovks4ksfuy

Building a Stationary Stochastic Process From a Finite-dimensional Marginal [article]

Marcus Pivato
2001 arXiv   pre-print
If A is a finite alphabet, Z^D is a D-dimensional lattice, U is a subset of Z^D, and mu_U is a probability measure on A^U that "looks like" the marginal projection of a stationary random field on A^(Z^D), then can we "extend" mu_U to such a field? Under what conditions can we make this extension ergodic, (quasi)periodic, or (weakly) mixing? After surveying classical work on this problem when D = 1, we provide some sufficient conditions and some necessary conditions for mu_U to be extendible for
more » ... D > 1, and show that, in general, the problem is not formally decidable.
arXiv:math/0108081v1 fatcat:2i4f3oghdzbzxgwffic5vlb4eq

Variable-population voting rules

Marcus Pivato
2013 Journal of Mathematical Economics  
groups are useful for representing infinite-horizon intertemporal preferences, non-probabilistic uncertainty, and preferences where some decision variables have lexicographical priority over others (Pivato  ... 
doi:10.1016/j.jmateco.2013.02.001 fatcat:rruhfqyn6nh6dh442ce3ly5wce

Voting rules as statistical estimators

Marcus Pivato
2012 Social Choice and Welfare  
Thus, a result of Pivato (2011) implies that there is some r > 0 and some function t : V−→R such that S(v, x) = r S(v, x) + t(v) for all v ∈ V and x ∈ X .  ... 
doi:10.1007/s00355-011-0619-1 fatcat:z5327izgpnhjnj3nkv7ul2lduq

Epistemic democracy with correlated voters

Marcus Pivato
2017 Journal of Mathematical Economics  
Pivato (2013a, Theorem 2) proved a generalization of Myerson's result which keeps neutrality but relaxes the Continuity condition, by allowing the score vectors to take "infinitesimal" values.  ...  Furthermore, in these contexts, the "nonasymptotic" part of the CJT can be refined: under certain conditions, the output of the voting rule is a maximum likelihood estimator of the correct answer (see Pivato  ... 
doi:10.1016/j.jmateco.2017.06.001 fatcat:wkawpnkx2jdgxi7xp3rsnuifee
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