Symbol Extension Research Articles

Tuning similarity-based fuzzy logic programs

We have recently designed a symbolic extension of FASILL (acronym of “Fuzzy Aggregators and Similarity Into a Logic Language”), where some truth degrees, similarity annotations and fuzzy connectives can be left unknown, so that the user can easily see the impact of their possible values at execution time. By extending our previous results in the development of tuning techniques not dealing yet with similarity relations, in this work we automatically tune FASILL programs by appropriately substituting the symbolic constants appearing on their rules and similarity relations with the concrete values that best satisfy the user's preferences. Firstly, we have formally proved two theoretical results with different levels of generality/practicability for tuning programs in a safe and effective way. Regarding efficiency, we have drastically reduced the exponential complexity of the tuning algorithms by splitting the initial set of symbolic constants in disjoint sets and using thresholding techniques. These effects have been evidenced by several experiments and benchmarks developed with the online tool we provide to verify in practice the high performance of the improved system.

Symbolic dynamics for pointwise hyperbolic systems on open regions

Abstract Under certain conditions, we construct a countable Markov partition for pointwise hyperbolic diffeomorphism $f:M\rightarrow M$ on an open invariant subset $O\subset M$ , which allows the Lyapunov exponents to be zero. From this partition, we define a symbolic extension that is finite-to-one and onto a subset of O that carries the same finite f-invariant measures as O. Our method relies upon shadowing theory of a recurrent-pointwise-pseudo-orbit that we introduce. As a canonical application, we estimate the number of closed orbits for f.

A novel SLM scheme for peak-to-average power ratio reduction in WOLA-OFDM system

This paper addresses high peak-to-average power ratio (PAPR) issue in the recently invented weighted overlap and add-orthogonal frequency division multiplexing (WOLA-OFDM) system. The selective mapping (SLM), one of the most widely used PAPR reduction schemes in OFDM-based multicarrier transmission strategies, can be used to reduce the PAPR of WOLA-OFDM signals. However, the overlapping of symbol extensions in the WOLA-OFDM transmitter considerably deteriorates the performance of SLM. To overcome this problem, we have developed an efficient SLM scheme named dual symbol optimization-SLM (DSO-SLM) for the WOLA-OFDM system. In this new scheme, instead of reducing the PAPR of adjacent symbols separately as in conventional SLM, each symbol is evaluated together with its overlapping predecessor symbol while searching for the optimal transmission signal with minimized PAPR. With the usage of DSO procedure, it becomes possible to enhance the performance of conventional SLM in the WOLA-OFDM system.

An Interference Alignment Design with Higher Multiplexing Gain by the Reduced Channel Extension

For a K-user Single Input Single Output fading interference channel, Cadambe and Jafar proposed a scheme that achieves K=2 multiplexing gain asymptotically employing interference alignment technique. This scheme is improved by Choi et al. in which for reduced channel extension, higher multiplexing gain is achievable. But the existing designs require an infinite number of channel extension to achieve the upperbound. Our proposed design attains higher multiplexing gain that is arbitrarily close to the upper-bound by using a lot lower number of channel extension in comparison with the other existing interference alignment schemes. On the other hand, due to the great symbol extension reduction of the channel, the computational complexity to achieve K=2 degrees of freedom (desired upper-bound) is also reduced significantly. Moreover, it can be shown that the proposed design achieves higher interference alignment quality. For a limited channel extension, the proposed scheme leads to a better allocation of the signal space at each receiver to the desired and interfered signal. This results to a higher quality of alignment.

Strongly Isomorphic Symbolic Extensions for Expansive Topological Flows

Abstract In this paper, we prove that finite-dimensional topological flows without fixed points and having a countable number of periodic orbits, have the small flow boundary property. This enables us to answer positively a question of Bowen and Walters from 1972: Any expansive topological flow has a strongly isomorphic symbolic flow extension, that is, an extension by a suspension flow over a subshift. Previously Burguet had shown this is true if the flow is assumed to be $C^{2}$-smooth.

Symbolic Extensions of Amenable Group Actions and the Comparison Property

In topological dynamics, the Symbolic Extension Entropy Theorem (SEET) (Boyle and Downarowicz, 2004) describes the possibility of a lossless digitalization of a dynamical system by extending it to a subshift on finitely many symbols. The theorem gives a precise estimate on the entropy of such a symbolic extension (and hence on the necessary number of symbols). Unlike in the measure-theoretic case, where Kolmogorov–Sinai entropy serves as an estimate in an analogous problem, in the topological setup the task reaches beyond the classical theory of measure-theoretic and topological entropy. Necessary are tools from an extended theory of entropy, the theory of entropy structures developed in Downarowicz (2005). The main goal of this paper is to prove the analog of the SEET for actions of (discrete infinite) countable amenable groups: Let a countable amenable group G G act by homeomorphisms on a compact metric space X X and let M G ( X ) \mathcal {M}_{G}(X) denote the simplex of all G G -invariant Borel probability measures on X X . A function E A {E}_{\mathsf {A}} on M G ( X ) \mathcal {M}_{G}(X) equals the extension entropy function h π h^\pi of a symbolic extension π : ( Y , G ) → ( X , G ) \pi :(Y,G)\to (X,G) , where h π ( μ ) = sup { h ν ( Y , G ) : ν ∈ π − 1 ( μ ) } h^\pi (\mu )=\sup \{h_\nu (Y,G): \nu \in \pi ^{-1}(\mu )\} ( μ ∈ M G ( X ) \mu \in \mathcal {M}_{G}(X) ), if and only if E A {E}_{\mathsf {A}} is a finite affine superenvelope of the entropy structure of ( X , G ) (X,G) . Of course, the statement is preceded by the presentation of the concepts of an entropy structure and its superenvelopes, adapted from the case of Z \mathbb {Z} -actions. In full generality we are able to prove a slightly weaker version of SEET, in which symbolic extensions are replaced by quasi-symbolic extensions, i.e., extensions in form of a joining of a subshift with a zero-entropy tiling system. The notion of a tiling system is a subject of several earlier works (e.g. Downarowicz and Huczek (2018) and Downarowicz, Huczek, and Zhang (2019)) and in this paper we review and complement the theory developed there. The full version of the SEET (with genuine symbolic extensions) is proved for groups which are either residually finite or enjoy the so-called comparison property. In order to describe the range of our theorem more clearly, we devote a large portion of the paper to studying the comparison property. Our most important result in this aspect is showing that all subexponential groups have the comparison property (and thus satisfy the SEET). To summarize, the heart of the paper is the presentation of the following four major topics and the interplay between them: Symbolic extensions, Entropy structures, Tiling systems (and their encodability), The comparison property.

Use of quasi-SMILES to build models based on quantitative results from experiments with nanomaterials.

Quasi-SMILES deviate from traditional SMILES (simplified molecular input-line entry system) by the extension of additional symbols that encode for conditions of an experiment. Descriptors calculated with SMILES are useful for the development of quantitative structure-property/activity relationships (QSPRs/QSARs), while descriptors calculated with quasi-SMILES can be useful for the development of quantitative models of experimental results obtained under different conditions. Here, this approach has been applied for the development of generalized models using aquatic nanotoxicity data (i.e., related to fish and daphnia). The statistical quality of the above models (pLC50) is quite good with a determination coefficient for the external validation set ranging from 0.62 to 0.71 and RMSE ranging from 0.58 to 0.60. The principle of the approach includes splitting the experimental data into three random distributions defining training, calibration, and validation sets.

The Symbolic Extension Theory in Topological Dynamics

In this survey we will present the symbolic extension theory in topological dynamics, which was built over the past twenty years.

Zero-dimensional and symbolic extensions of topological flows

<p style='text-indent:20px;'>A zero-dimensional (resp. symbolic) flow is a suspension flow over a zero-dimensional system (resp. a subshift). We show that any topological flow admits a principal extension by a zero-dimensional flow. Following [<xref ref-type="bibr" rid="b6">6</xref>] we deduce that any topological flow admits an extension by a symbolic flow if and only if its time-<inline-formula><tex-math id="M1">\begin{document}$ t $\end{document}</tex-math></inline-formula> map admits an extension by a subshift for any <inline-formula><tex-math id="M2">\begin{document}$ t\neq 0 $\end{document}</tex-math></inline-formula>. Moreover the existence of such an extension is preserved under orbit equivalence for regular topological flows, but this property does not hold for singular flows. Finally we investigate symbolic extensions for singular suspension flows. In particular, the suspension flow over the full shift on <inline-formula><tex-math id="M3">\begin{document}$ \{0,1\}^{\mathbb Z} $\end{document}</tex-math></inline-formula> with a roof function <inline-formula><tex-math id="M4">\begin{document}$ f $\end{document}</tex-math></inline-formula> vanishing at the zero sequence <inline-formula><tex-math id="M5">\begin{document}$ 0^\infty $\end{document}</tex-math></inline-formula> admits a principal symbolic extension or not depending on the smoothness of <inline-formula><tex-math id="M6">\begin{document}$ f $\end{document}</tex-math></inline-formula> at <inline-formula><tex-math id="M7">\begin{document}$ 0^\infty $\end{document}</tex-math></inline-formula>.</p>

Symbolic territorialities in a terreiro de candomblé: the morphology of a sacred space

This work highlights the symbolic territorialities of a ‘terreiro de candomblé’ with fluid spatial dimensions, located in the city of Cajazeiras, Paraíba. The experiences are based on the ethnographic field research conducted between January and April. We outline the morphology of the sacred space represented by geosymbols and their meanings, which refer to the mythology of the African òrìṣà, rescued in the ‘terreiro’. We also describe, from the perspective of Costa’s micro-territoriality (2017), geosymbols outside the ‘terreiro’, eventually accessed by ritual occasion, revealing a symbolic extension of the territory that transgresses previously constructed determinations and constructs new identity spaces

Symbolic extensions for 3-dimensional diffeomorphisms

We prove that every $${{\cal C}^r}$$ diffeomorphism with r > 1 on a three-dimensional manifold admits symbolic extensions, i.e., topological extensions which are subshifts over a finite alphabet. This answers positively a conjecture of Downarowicz and Newhouse in dimension three.

Periodic points and measures for a class of skew-products

We consider the C1-open set \(\cal{V}\) of partially hyperbolic diffeomorphisms on the space \(\mathbb{T}^{2}\times\mathbb{T}^{2}\) whose non-wandering set is not stable, introduced by M. Shub in [57]. Firstly, we show that the non-wandering set of each diffeormorphism in \(\cal{V}\) is a limit of horseshoes in the sense of entropy. Afterwards, we establish the existence of a C2-open set \(\cal{U}\) of C2-diffeomorphisms in \(\cal{V}\) and of a C2-residual subset ℜ of \(\cal{U}\) such that any diffeomorphism in ℜ has equal topological and periodic entropies, is asymptotic per-expansive, has a sub-exponential growth rate of the periodic orbits and admits a principal strongly faithful symbolic extension with embedding. Besides, such a diffeomorphism has a unique probability measure with maximal entropy describing the distribution of periodic orbits. Under an additional assumption, we prove that the skew-products in \(\cal{U}\) preserve a unique ergodic SRB measure, which is physical, whose basin has full Lebesgue measure and which coincides with the measure with maximal entropy.

Symbolic Variable Elimination for Discrete and Continuous Graphical Models

Probabilistic reasoning in the real-world often requires inference incontinuous variable graphical models, yet there are few methods for exact, closed-form inference when joint distributions are non-Gaussian. To address this inferential deficit, we introduce SVE -- a symbolic extension of the well-known variable elimination algorithm to perform exact inference in an expressive class of mixed discrete and continuous variable graphical models whose conditional probability functions can be well-approximated as piecewise combinations of polynomials with bounded support. Using this representation, we show that we can compute all of the SVE operations exactly and in closed-form, which crucially includes definite integration w.r.t. multivariate piecewise polynomial functions. To aid in the efficient computation and compact representation of this solution, we use an extended algebraic decision diagram (XADD) data structure that supports all SVE operations. We provide illustrative results for SVE on probabilistic inference queries inspired by robotics localization and tracking applications that mix various continuous distributions; this represents the first time a general closed-form exact solution has been proposed for this expressive class of discrete/continuous graphical models.

Benefits of Local Cooperation in Sectorized Cellular Networks Under a Complexity Constraint

The paper presents upper and lower bounds on the per-sector degrees of freedom (DoF) of a sectorized hexagonal cellular model when neighboring base stations (BSs) can cooperate during at most Δ interaction rounds over rate-limited backhaul links. The lower bound is based on practically implementable beamforming and adapts the way BSs cooperate to the sectorization of the cells. It improves over the naive approach that ignores this sectorization in terms of the sum-rate, both at finite signal-to-noise ratio (SNR) and in the high-SNR limit. For moderate SNR, the new scheme improves also over an opportunistic cooperation strategy where each message is decoded based on the signals received at the three adjacent sectors with the best SNR. The upper bound is information-theoretic and holds for all possible coding schemes, including for example interference alignment with unlimited symbol extensions whose practical implementation currently seems out of reach. Lower and upper bounds show that the complexity constraint, imposed by limiting the number of interaction rounds Δ, indeed limits the largest achievable sum-rate and DoF. In particular, irrespective of the backhaul capacity μ, the per-sector DoF cannot exceed a threshold which depends on Δ.

On pro-p-extensions of number fields with restricted ramification over intermediate [formula omitted]-extensions

We consider pro-p-extensions of a number field in which the ramification and decomposition are restricted over an intermediate Zp-extension. For such a maximal pro-p-extension under a certain restriction condition, we obtain a Koch type presentation of the pro-p Galois group by generators and relations. As its applications, we also obtain an extension of triple cubic residue symbols, a new example of arithmetical mild pro-p group, and another proof of a criterion of finiteness of certain Iwasawa modules.

Symbolic dynamics for non-uniformly hyperbolic systems

AbstractThis survey describes the recent advances in the construction of Markov partitions for non-uniformly hyperbolic systems. One important feature of this development comes from a finer theory of non-uniformly hyperbolic systems, which we also describe. The Markov partition defines a symbolic extension that is finite-to-one and onto a non-uniformly hyperbolic locus, and this provides dynamical and statistical consequences such as estimates on the number of closed orbits and properties of equilibrium measures. The class of systems includes diffeomorphisms, flows, and maps with singularities.

Against Allegory

More than ten years on from the 2008 financial crisis, two trends of global statism remain dominant: Beijing-led “exceptional neoliberalism” and the emerging “illiberal democracy” topped by Trumponomics, with racist populism looming at the back of both. Even though these persistent programs are remnants of the ideological, national, and economic wars of the previous century, the boundary separating them is permeable. Jiang Rong 姜 戎’s prizewinning novel, Wolf Totem 《狼图腾》, helps us see this porosity. Wolf Totem is the first “Chinese Cultural Revolution” (fictional) memoir written explicitly for Chinese nationals and yet goes on to engage the sensibility of readers from a Western historical and ideological context. This essay critically identifies certain acts of reading Wolf Totem and looks at the way these selected readings, all allegorical in their approach, step across the literary subject to build symbolic extensions that stretch thin the wolves for various purposes. Collectively, such acts of reading expose both an important quality of our historical moment and the ideological function of literary intellectuals within it. They show that our era is one of skepticism about the status quo, one in which certain antidemocratic drives commiserate over historical conflicts and strategize for an extended, ongoing, and relentless process of global dominance. The popular reception of Wolf Totem crystallizes the thrust and conduct of these seeming competing drives. In the final analysis, this essay follows through the symptoms of these drives to reveal a kind of energetics or “primitivist social ethos” alive in the unified way humanity makes extinct any life forms unsubscribed to global statisms in their Beijing or “illiberal democratic” forms.

Achievable Secure Degrees of Freedom of MIMO Interference Channel With a Cooperative Jammer

In this letter, the secure degrees of freedom (SDoF) of 2-user multiple-input multiple-output (MIMO) interference channel with an external N-antenna cooperative jammer (CJ) is considered. In this model, each transmitter intends to send confidential messages to the corresponding receiver and keep secret with unintended receiver. Different from the previous studies on interference channels, our model has an external cooperative jammer, which can improve the SDoF by transmitting interference signals. We derive the converse and give the corresponding achievable schemes to determine the exact SDoF, and study the influence of CJ on the SDoF of this system. We find that adding one CJ antenna has the same effect as adding one transmitting antenna, and has a better effect than adding one receiving antenna. The converse is given by using vector extensions of the role of a helper and secrecy penalty lemmas. For arbitrary values of N and M, we provide different achievable schemes for the corresponding regimes by making use of the symbol extension and spatial alignment techniques.

Courtesan Singers as Courtiers: Power, Political Pawns, and the Arrest of virtuosa Nina Barcarola

This article gives a close reading of the “avvisi di Roma”—unpublished archival documents reporting on daily life in the city—that record the arrest in 1645 of famous Roman courtesan singer Nina Barcarola. Organized by the political enemies of Nina's main protector, Cardinal Antonio Barberini, the arrest was orchestrated so as to compromise the public honor of both. The reports of the arrest reflect a growing elite interest in female vocal performance in Rome, and attest to a rise in the social value of courtesan singers. Examining details provided in these reports, the article explores various aspects of Nina's life and courtesan singing culture more generally: the public honor and social practices of courtesan singers; the positive effect of singing on courtesan honor; the types of gatherings hosted by Nina; and her politically satirical public performances. It also analyzes Nina's relationship to various areas of contemporary politics—social, state, familial, and gender. The reports reveal that, in the public sphere, Nina, like Barberini's male dependents, served as a symbolic extension of the cardinal. By introducing courtesan singers—a significant, marginalized population—into musicological discourse on seventeenth-century Rome, the article broadens our understanding of Roman singing culture in this period.

Finite-to-one zero-dimensional covers of dynamical systems

In this paper, we study the existence of finite-to-one zero-dimensional covers of dynamical systems. Kulesza showed that any homeomorphism $f:X \to X$ on an $n$-dimensional compactum $X$ with zero-dimensional set $P(f)$ of periodic points can be covered by a homeomorphism on a zero-dimensional compactum via an at most $(n + 1)^n$-to-one map. Moreover, Ikegami, Kato and Ueda showed that in the theorem of Kulesza, the condition of at most $(n + 1)^n$-to-one map can be strengthened to the condition of at most $2^n$-to-one map. In this paper, we will show that the theorem is also true for more general maps except for homeomorphisms. In fact we prove that the theorem is true for a class of maps containing two-sided zero-dimensional maps. For the special case, we give a theorem of symbolic extensions of positively expansive maps. Finally, we study some dynamical zero-dimensional decomposition theorems of spaces related to such maps.