Loop-checking and the uniform word problem for join-semilattices with an inflationary endomorphism

We solve in polynomial time two decision problems that occur in type checking when typings depend on universe level constraints

A Vernacular for Coherent Logic

We propose a simple, yet expressive proof representation from which proofs for different proof assistants can easily be generated. The representation uses only a few inference rules and is based on a frag- ment of first-order logic called coherent logic. Coherent logic has been recognized by a number of researchers as a suitable logic for many ev- eryday mathematical developments. The proposed proof representation is accompanied by a corresponding XML format and by a suite of XSL transformations for generating formal proofs for Isabelle/Isar and Coq, as well as proofs expressed in a natural language form (formatted in LATEX or in HTML). Also, our automated theorem prover for coherent logic exports proofs in the proposed XML format. All tools are publicly available, along with a set of sample theorems.Comment: CICM 2014 - Conferences on Intelligent Computer Mathematics (2014

Parametric polymorphism - universally

In the 1980s, John Reynolds postulated that a parametrically polymorphic function is an ad-hoc polymorphic function satisfying a uniformity principle. This allowed him to prove that his set-theoretic semantics has a relational lifting which satisfies the Identity Extension Lemma and the Abstraction Theorem. However, his definition (and subsequent variants) have only been given for specific models. In contrast, we give a model-independent axiomatic treatment by characterising Reynolds' definition via a universal property, and show that the above results follow from this universal property in the axiomatic setting

Computing the vertices of tropical polyhedra using directed hypergraphs

We establish a characterization of the vertices of a tropical polyhedron defined as the intersection of finitely many half-spaces. We show that a point is a vertex if, and only if, a directed hypergraph, constructed from the subdifferentials of the active constraints at this point, admits a unique strongly connected component that is maximal with respect to the reachability relation (all the other strongly connected components have access to it). This property can be checked in almost linear-time. This allows us to develop a tropical analogue of the classical double description method, which computes a minimal internal representation (in terms of vertices) of a polyhedron defined externally (by half-spaces or hyperplanes). We provide theoretical worst case complexity bounds and report extensive experimental tests performed using the library TPLib, showing that this method outperforms the other existing approaches.Comment: 29 pages (A4), 10 figures, 1 table; v2: Improved algorithm in section 5 (using directed hypergraphs), detailed appendix; v3: major revision of the article (adding tropical hyperplanes, alternative method by arrangements, etc); v4: minor revisio

Computational interpretations of analysis via products of selection functions

We show that the computational interpretation of full comprehension via two wellknown functional interpretations (dialectica and modified realizability) corresponds to two closely related infinite products of selection functions

The play's the thing

For very understandable reasons phenomenological approaches predominate in the field of sensory urbanism. This paper does not seek to add to that particular discourse. Rather it takes Rorty’s postmodernized Pragmatism as its starting point and develops a position on the role of multi-modal design representation in the design process as a means of admitting many voices and managing multidisciplinary collaboration. This paper will interrogate some of the concepts underpinning the Sensory Urbanism project to help define the scope of interest in multi-modal representations. It will then explore a range of techniques and approaches developed by artists and designers during the past fifty years or so and comment on how they might inform the question of multi-modal representation. In conclusion I will argue that we should develop a heterogeneous tool kit that adopts, adapts and re-invents existing methods because this will better serve our purposes during the exploratory phase(s) of any design project that deals with complexity

On completeness of logic programs

Program correctness (in imperative and functional programming) splits in logic programming into correctness and completeness. Completeness means that a program produces all the answers required by its specification. Little work has been devoted to reasoning about completeness. This paper presents a few sufficient conditions for completeness of definite programs. We also study preserving completeness under some cases of pruning of SLD-trees (e.g. due to using the cut). We treat logic programming as a declarative paradigm, abstracting from any operational semantics as far as possible. We argue that the proposed methods are simple enough to be applied, possibly at an informal level, in practical Prolog programming. We point out importance of approximate specifications.Comment: 20 page

Computational Interpretations of Classical Linear Logic

Abstract. We survey several computational interpretations of classical linear logic based on two-player one-move games. The moves of the games are higher-order functionals in the language of finite types. All interpretations discussed treat the exponential-free fragment of linear logic in a common way. They only differ in how much advantage one of the players has in the exponentials games. We dis-cuss how the several choices for the interpretation of the modalities correspond to various well-known functional interpretations of intuitionistic logic, including Gödel’s Dialectica interpretation and Kreisel’s modified realizability.