Probabilistic Feature Attention as an Alternative to Variables in Phonotactic Learning

Since Halle (1962), explicit algebraic variables (often called alpha notation) have been commonplace in phonological theory. However, Hayes and Wilson (2008) proposed a variable-free model of phonotactic learning, sparking a debate about whether such algebraic representations are necessary to capture human phonological acquisition. While past experimental work has found evidence that suggested a need for variables in models of phonology (Berent et al. 2012, Moreton 2012, Gallagher 2013), this paper presents a novel mechanism, Probabilistic Feature Attention (PFA), that allows a variable-free model of phonotactics to predict a number of these phenomena. Additionally, experimental results involving phonological generalization that cannot be explained by variables are captured by this novel approach. These results cast doubt on whether variables are necessary to capture human-like phonotactic learning and provide a useful alternative to such representations.

Minimization of Binary Decision Diagrams for Systems of Completely Defined Boolean Functions using Algebraic Representations of Cofactors

In design systems for digital VLSI (very large integrated circuits), the BDD is used for VLSI verification, as well as for technologically independent optimization, performed as the first stage in the synthesis of logic circuits in various technological bases. BDD is an acyclic graph defining a Boolean function or a system of Boolean functions. Each vertex of this graph corresponds to the complete or reduced Shannon expansion formula. Having constructed BDD representation for systems of Boolean functions, it is possible to perform additional logical optimization based on the proposed method of searching for algebraic representations of cofactors (subfunctions) of the same BDD level in the form of a disjunction or conjunction of other cofactors of this BDD level. The method allows to reduce the number of literals by replacing the Shannon expansion formulas with simpler formulas that are disjunctions or conjunctions of cofactors, and to reduce the number of literals in specifying a system of Boolean functions. The number of literals in algebraic multilevel representations of systems of fully defined Boolean functions is the main optimization criterion in the synthesis of combinational circuits from library logic gates.

On Polymorphic Sessions and Functions

This work exploits the logical foundation of session types to determine what kind of type discipline for the Λ-calculus can exactly capture, and is captured by, Λ-calculus behaviours. Leveraging the proof theoretic content of the soundness and completeness of sequent calculus and natural deduction presentations of linear logic, we develop the first mutually inverse and fully abstract processes-as-functions and functions-as-processes encodings between a polymorphic session π-calculus and a linear formulation of System F. We are then able to derive results of the session calculus from the theory of the Λ-calculus: (1) we obtain a characterisation of inductive and coinductive session types via their algebraic representations in System F; and (2) we extend our results to account for value and process passing, entailing strong normalisation.

TOPOLOGICAL NOETHERIANITY FOR ALGEBRAIC REPRESENTATIONS OF INFINITE RANK CLASSICAL GROUPS

Draisma recently proved that polynomial representations of GL∞ are topologically noetherian. We generalize this result to algebraic representations of infinite rank classical groups.

Seeing Stars: Sequences on U.S. Flag

Students analyze photographs of patterns and determine algebraic representations for the pattern growth.

Is This Vending Machine FUNCTIONing Correctly?

In this manuscript we describe a lesson that utilizes an applet we designed to help students develop a conceptual understanding of the concept of function. We describe how removing algebraic representations and focusing on a real world context can support students' development of these conceptual understandings of the function concept.