Probability and Measure

The subject of this chapter is the study of formal languages (mostly languages recognizable by finite automata) in the framework of mathematical logic.

/pdf/languages-automata-and-logic-418xw4tas7.pdf

Languages, automata, and logic

Extrinsic information transfer (EXIT) charts are a tool for predicting the convergence behavior of iterative processors for a variety of communication problems. A model is introduced that applies to decoding problems, including the iterative decoding of parallel concatenated (turbo) codes, serially concatenated codes, low-density parity-check (LDPC) codes, and repeat-accumulate (RA) codes. EXIT functions are defined using the model, and several properties of such functions are proved for erasure channels. One property expresses the area under an EXIT function in terms of a conditional entropy. A useful consequence of this result is that the design of capacity-approaching codes reduces to a curve-fitting problem for all the aforementioned codes. A second property relates the EXIT function of a code to its Helleseth-Klove-Levenshtein information functions, and thereby to the support weights of its subcodes. The relation is via a refinement of information functions called split information functions, and via a refinement of support weights called split support weights. Split information functions are used to prove a third property that relates the EXIT function of a linear code to the EXIT function of its dual.

Extrinsic information transfer functions: model and erasure channel properties

THE criticism on the passage quoted from p. 3 of the book by Profs. Harkness and Morley (NATURE, February 23, p. 347) turns on the fact that, in dealing with number divorced from measurement, the authors have used the phrase “an infinity of objects” without an explicit statement of its meaning. I am not sure that I understand the passage in their letter which refers to this point; but it seems to me to imply that the distinction between “finite” and “infinite” is one which does not require definition. This is not the only accepted view. It is not, for instance, the view taken in Herr Dedekind's book, “Was sind und was sollen die Zahlen.” As regards the opening sentences of Chapter xv., the authors have apparently misunderstood the point of my objection. With the usually received definition of convergence of an infinite product, Π(1-αn), if convergent, is different from zero. So far as the passage quoted goes, Π(1-αn) might be zero; and it is therefore not shown to be convergent, if the usual definition of convergence be assumed. As to the passage quoted from p. 232, I must express to the authors my regret for having overlooked the fact that the particular rearrangement, there made use of, has been fully justified in Chapter viii. Whether Log x is or is not, at the beginning of Chapter iv., defined by means of a string and a cone, will be obvious to any one who will read the whole passage (p. 46, line 16, to p. 47, line 9) leading up to the definition.

Theory of Functions

https://www.cs.unipr.it/~bagnara/Papers/PDF/SCP08.pdf

The Parma Polyhedra Library: Toward a complete set of numerical abstractions for the analysis and verification of hardware and software systems

https://deepblue.lib.umich.edu/bitstream/handle/2027.42/30017/0000385.pdf;sequence=1

Regular languages in NC 1

A new method for obtaining lower bounds on the computational complexity of logical theories is presented. It extends widely used techniques for proving the undecidability of theories by interpreting models of a theory already known to be undecidable. New inseparability results related to the well known inseparability result of Trakhtenbrot and Vaught are the foundation of the method. Their use yields hereditary lower bounds (i.e., bounds which apply uniformly to all subtheories of a theory). By means of interpretations lower bounds can be transferred from one theory to another. Complicated machine codings are replaced by much simpler definability considerations, viz., the kinds of binary relations definable with short formulas on large finite sets. Numerous examples are given, including new proofs of essentially all previously known lower bounds for theories, and lower bounds for various theories of finite trees, which turn out to be particularly useful.

A uniform method for proving lower bounds on the computational complexity of logical theories

This is a survey of logical results concerning random structures. A class of relational structures on which a (finitely additive) probability measure has been defined has a 0–1 law for a particular logic if every sentence of that logic has probability either 0 or 1. The measure may be an asymptotic probability on finite structures or generated on a class of infinite structures by assigning fixed probabilities to independently occurring properties. Conditions under which all sentences of a logic have a probability, and under which 0–1 laws occur, are examined. Also, the complexity of computing probabilities of sentences is considered.

Laws in Logic and Combinatorics

The Unified Modeling Language has become widely accepted as a standard in software development. Several tools have been produced to support UML model validation. However most of them support either static or dynamic model checking; and no tools support to check both static and dynamic aspects of a UML model. But a UML model should include the static and dynamic aspects of a software system. Furthermore, these UML tools translate a UML model into a validation language such as PROMELA. But they have some shortcomings: there is no proof of correctness (with respect to the UML semantics) for these tools. In order to overcome these shortcomings, we present a toolset which can validate both static and dynamic aspects of a model; and this toolset is based on the semantic model using Abstract State Machines. Since the toolset is derived from the semantic model, the toolset is correct with respect to the semantic model.

/pdf/a-toolset-for-supporting-uml-static-and-dynamic-model-3lkqabfio7.pdf

Kevin J. Compton

Papers

Regular languages in NC 1

A uniform method for proving lower bounds on the computational complexity of logical theories

A uniform method for proving lower bounds on the computational complexity of logical theories

Laws in Logic and Combinatorics

A toolset for supporting UML static and dynamic model checking