Substantive Bias and Natural Classes

Investigations of default logic have been so far mostly concerned with the notion of an extension of a default theory. It turns out, however, that default logic is much richer. Namely, there are other natural classes of objects that might be associated with default reasoning. We study two such classes of objects with emphasis on their relations with modal nonmonotonic formalisms. First, we introduce the concept of a weak extension and study its properties. It has long been suspected that there are close connections between default and autoepistemic logics. The notion of weak extension allows us to precisely describe the relationship between these two formalisms. In particular, we show that default logic with weak extensions is essentially equivalent to autoepistemic logic, that is, nonmonotonic logic KD45. In the paper we also study the notion of a set of formulas closed under a default theory. These objects are shown to correspond to stable theories and to modal logic S5. In particular, we show that skeptical reasoning with sets closed under default theories is closely related with provability in S5. As an application of our results we determine the complexity of reasoning with weak extensions and sets closed under default theories.

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UNDECIDABILITY OF THE THEORIES OF CLASSES OF STRUCTURES

Journal of Symbolic Logic ◽

10.1017/jsl.2014.54 ◽

2014 ◽

Vol 79 (4) ◽

pp. 1001-1019 ◽

Cited By ~ 2

Author(s):

ASHER M. KACH ◽

ANTONIO MONTALBÁN

Keyword(s):

Direct Product ◽

Disjoint Union ◽

Second Order ◽

Linear Orders ◽

Second Order Arithmetic ◽

Order Arithmetic ◽

Natural Classes ◽

Product Of Groups

AbstractMany classes of structures have natural functions and relations on them: concatenation of linear orders, direct product of groups, disjoint union of equivalence structures, and so on. Here, we study the (un)decidability of the theory of several natural classes of structures with appropriate functions and relations. For some of these classes of structures, the resulting theory is decidable; for some of these classes of structures, the resulting theory is bi-interpretable with second-order arithmetic.

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The blind choreographer: evolution of social norms and correlated equilibria

10.1101/435586 ◽

2018 ◽

Author(s):

Bryce Morsky ◽

Erol Akçay

Keyword(s):

Social Norms ◽

Social Life ◽

Evolutionary Dynamics ◽

Population Level ◽

Nash Bargaining ◽

Game Of Chicken ◽

Nash Bargaining Game ◽

Satisfactory Account ◽

Environmental Events ◽

Natural Classes

AbstractSocial norms regulate and coordinate most aspects of human social life, yet they emerge and change as a result of individual behaviours, beliefs, and expectations. A satisfactory account for the evolutionary dynamics of social norms therefore has to link individual beliefs and expectations to population-level dynamics, where individual norms change according to their consequences for individuals. Here we present a new model of evolutionary dynamics of social norms that encompasses this objective and addresses the emergence of social norms. In this model, a norm is a set of behavioural prescriptions and a set of environmental descriptions that describe the expected behaviours of those with whom the norm holder will interact. These pre-scriptions and descriptions are functions of exogenous environmental events. These events have no intrinsic meaning or effect on the payoffs to individuals, yet beliefs/- superstitions regarding them can effectuate coordination. Though a norm's prescriptions and descriptions are dependent upon one another, we show how they emerge from random accumulations of beliefs. We categorize the space of social norms into several natural classes and study the evolutionary competition between these classes of norms. We apply our model to the Game of Chicken and the Nash Bargaining Game. Further, we show how the space of norms and evolutionary stability is dependent upon the correlation structure of the environment, and under which such correlation structures social dilemmas can be ameliorated or exacerbated.

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The Natural Classes of Two-Handed Signs

Proceedings of the Annual Meeting of the Berkeley Linguistics Society ◽

10.3765/bls.v24i1.1245 ◽

1998 ◽

Vol 24 (1) ◽

pp. 256

Author(s):

Janine Toole ◽

Linda Uyechi

Keyword(s):

Annual Meeting ◽

General Session ◽

Natural Classes

Proceedings of the Twenty-Fourth Annual Meeting of the Berkeley Linguistics Society: General Session and Parasession on Phonetics and Phonological Universals (1998)

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Exact bounds for tail probabilities of martingales with bounded differences

Lietuvos matematikos rinkinys ◽

10.15388/lmr.2009.73 ◽

2009 ◽

Vol 50 ◽

Author(s):

Dainius Dzindzalieta

Keyword(s):

Random Walks ◽

General Principle ◽

Isoperimetric Problem ◽

Maximal Inequality ◽

Tail Probabilities ◽

Bounds For Tail Probabilities ◽

Maximal Inequalities ◽

Exact Bounds ◽

Natural Classes

We consider random walks, say Wn = {0, M1, . . ., Mn} of length n starting at 0 and based on a martingale sequence Mk = X1 + ··· + Xk with differences Xm. Assuming |Xk| \leq 1 we solve the isoperimetric problem Bn(x) = supP\{Wn visits an interval [x,∞)\}, (1) where sup is taken over all possible Wn. We describe random walks which maximize the probability in (1). We also extend the results to super-martingales.For martingales our results can be interpreted as a maximalinequalitiesP\{max 1\leq k\leq n Mk \geq x\} \leq Bn(x).The maximal inequality is optimal since the equality is achieved by martingales related to the maximizing random walks. To prove the result we introduce a general principle – maximal inequalities for (natural classes of) martingales are equivalent to (seemingly weaker) inequalities for tail probabilities, in our caseBn(x) = supP{Mn \geq x}.Our methods are similar in spirit to a method used in [1], where a solution of an isoperimetric problem (1), for integer x is provided and to the method used in [4], where the isoperimetric problem of type (1) for conditionally symmetric bounded martingales was solved for all x ∈ R.

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Degrees of classes of RE sets

Journal of Symbolic Logic ◽

10.1017/s0022481200051252 ◽

1976 ◽

Vol 41 (3) ◽

pp. 695-696 ◽

Cited By ~ 7

Author(s):

J. R. Shoenfield

Keyword(s):

Order Theory ◽

Turing Degree ◽

Natural Numbers ◽

First Order ◽

First Order Theory ◽

Simple Sets ◽

Natural Classes

In [3], Martin computed the degrees of certain classes of RE sets. To state the results succinctly, some notation is useful.If A is a set (of natural numbers), dg(A) is the (Turing) degree of A. If A is a class of sets, dg(A) = {dg(A): A ∈ A). Let M be the class of maximal sets, HHS the class of hyperhypersimple sets, and DS the class of dense simple sets. Martin showed that dg(M), dg(HHS), and dg(DS) are all equal to the set H of RE degrees a such that a′ = 0″.Let M* be the class of coinfinite RE sets having no superset in M; and define HHS* and DS* similarly. Martin showed that dg(DS*) = H. In [2], Lachlan showed (among other things) that dg(M*)⊆K, where K is the set of RE degrees a such that a″ > 0″. We will show that K ⊆ dg (HHS*). Since maximal sets are hyperhypersimple, this gives dg(M*) = dg (HHS*) = K.These results suggest a problem. In each case in which dg(A) has been calculated, the set of nonzero degrees in dg(A) is either H or K or the empty set or the set of all nonzero RE degrees. Is this always the case for natural classes A? Natural here might mean that A is invariant under all automorphisms of the lattice of RE sets; or that A is definable in the first-order theory of that lattice; or anything else which seems reasonable.

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The Path to Partition

Impossible Persons ◽

10.7551/mitpress/9780262034739.003.0002 ◽

2016 ◽

Author(s):

Daniel Harbour

Keyword(s):

Status Quo ◽

Skewed Distribution ◽

Historical Factors ◽

Morphological Theory ◽

Natural Classes

Morphological theory generally takes syncretism to be a key source of evidence about features and natural classes. The current chapter argues against this status quo, showing that there are no categorical constraints on possible versus impossible syncretisms. Partitions, by contrast, offer a means of marshalling the data productively. Defined as the total set of distinctions made by all paradigms within a given grammatical domain, partitions transform person in a categorical problem of the attested versus the unattested. Focusing narrowly on three-person pronominal systems, the chapter shows that the skewed distribution of partitions is not attributable to functional or historical factors and demands a deeper explanation.

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