First-class patterns

Pure pattern calculus supports pattern-matching functions in which patterns are first-class citizens that can be passed as parameters, evaluated and returned as results. This new expressive power supports two new forms of polymorphism. Path polymorphism allows recursive functions to traverse arbitrary data structures. Pattern polymorphism allows patterns to be treated as parameters which may be collected from various sources or generated from training data. A general framework for pattern calculi is developed. It supports a proof of confluence that is parameterised by the nature of the matching algorithm, suitable for the pure pattern calculus and all other known pattern calculi. © 2009 Copyright Cambridge University Press

Normalisation for Dynamic Pattern Calculi

The Pure Pattern Calculus (PPC) extends the lambda-calculus, as well as the family of algebraic pattern calculi, with first-class patterns; that is, patterns can be passed as arguments, evaluated and returned as results. The notion of matching failure of the PPC not only provides a mechanism to define functions by pattern matching on cases but also supplies PPC with parallel-or-like, non-sequential behaviour. Therefore, devising normalising strategies for PPC to obtain well-behaved implementations turns out to be challenging. This paper focuses on normalising reduction strategies for PPC. We define a (multistep) strategy and show that it is normalising. The strategy generalises the leftmost-outermost strategy for lambda-calculus and is strictly finer than parallel-outermost. The normalisation proof is based on the notion of necessary set of redexes, a generalisation of the notion of needed redex encompassing non-sequential reduction systems

Non-linear Pattern Matching with Backtracking for Non-free Data Types

Non-free data types are data types whose data have no canonical forms. For example, multisets are non-free data types because the multiset $\{a,b,b\}$ has two other equivalent but literally different forms $\{b,a,b\}$ and $\{b,b,a\}$. Pattern matching is known to provide a handy tool set to treat such data types. Although many studies on pattern matching and implementations for practical programming languages have been proposed so far, we observe that none of these studies satisfy all the criteria of practical pattern matching, which are as follows: i) efficiency of the backtracking algorithm for non-linear patterns, ii) extensibility of matching process, and iii) polymorphism in patterns. This paper aims to design a new pattern-matching-oriented programming language that satisfies all the above three criteria. The proposed language features clean Scheme-like syntax and efficient and extensible pattern matching semantics. This programming language is especially useful for the processing of complex non-free data types that not only include multisets and sets but also graphs and symbolic mathematical expressions. We discuss the importance of our criteria of practical pattern matching and how our language design naturally arises from the criteria. The proposed language has been already implemented and open-sourced as the Egison programming language

A standardisation proof for algebraic pattern calculi

This work gives some insights and results on standardisation for call-by-name pattern calculi. More precisely, we define standard reductions for a pattern calculus with constructor-based data terms and patterns. This notion is based on reduction steps that are needed to match an argument with respect to a given pattern. We prove the Standardisation Theorem by using the technique developed by Takahashi and Crary for lambda-calculus. The proof is based on the fact that any development can be specified as a sequence of head steps followed by internal reductions, i.e. reductions in which no head steps are involved.Comment: In Proceedings HOR 2010, arXiv:1102.346

Reconciling positional and nominal binding

We define an extension of the simply-typed lambda calculus where two different binding mechanisms, by position and by name, nicely coexist. In the former, as in standard lambda calculus, the matching between parameter and argument is done on a positional basis, hence alpha-equivalence holds, whereas in the latter it is done on a nominal basis. The two mechanisms also respectively correspond to static binding, where the existence and type compatibility of the argument are checked at compile-time, and dynamic binding, where they are checked at run-time.Comment: In Proceedings ITRS 2012, arXiv:1307.784

Attractor Modulation and Proliferation in 1+∞\infty Dimensional Neural Networks

We extend a recently introduced class of exactly solvable models for recurrent neural networks with competition between 1D nearest neighbour and infinite range information processing. We increase the potential for further frustration and competition in these models, as well as their biological relevance, by adding next-nearest neighbour couplings, and we allow for modulation of the attractors so that we can interpolate continuously between situations with different numbers of stored patterns. Our models are solved by combining mean field and random field techniques. They exhibit increasingly complex phase diagrams with novel phases, separated by multiple first- and second order transitions (dynamical and thermodynamic ones), and, upon modulating the attractor strengths, non-trivial scenarios of phase diagram deformation. Our predictions are in excellent agreement with numerical simulations.Comment: 16 pages, 15 postscript figures, Late