The article looks briefly at Feferman’s most sweeping claims about categorical foundations, focuses on narrower points raised in Berkeley, and asks some questions about Feferman’s own foundations. Among many different senses of foundations, the one that mathematics needs in practice is a recognized body of truths adequate to organize definitions and proofs. Finding concise principles of this kind has been a huge achievement by mathematicians and logicians. We put ZFC and categorical foundations both into this context.

Feferman (1977) argues that category theory cannot stand on its own as a structuralist foundation for mathematics: he claims that, because the notions of operation and collection are both epistemically and logically prior, we require a background theory of operations and collections. Recently [2011], I have argued that in rationally reconstructing Hilbert’s organizational use of the axiomatic method, we can construct an algebraic version of category-theoretic structuralism. That is, in reply to Shapiro (2005), we can be structuralists all the way down; we do not have to appeal to some background theory to guarantee the truth of our axioms. In this paper, I again turn to Hilbert; I borrow his (Hilbert, 1900a) distinction between the genetic method and the axiomatic method to argue that even if the genetic method requires the notions of operation and collection, the axiomatic method does not. Even if the genetic method is in some sense epistemically or logically prior, the axiomatic method stands alone. Thus, if the claim that category theory can act as a structuralist foundation for mathematics arises from the organizational use of the axiomatic method, then it does not depend on the prior notions of operation or collection, and so we can be structuralists all the way up.

Racket is a large language that is built mostly within itself. Unlike the usual approach taken by non-Lisp languages, the self-hosting of Racket is not a matter of bootstrapping one implementation through a previous implementation, but instead a matter of building a tower of languages and libraries via macros. The upper layers of the tower include a class system, a component system, pedagogic variants of Scheme, a statically typed dialect of Scheme, and more. The demands of this language-construction effort require a macro system that is substantially more expressive than previous macro systems. In particular, while conventional Scheme macro systems handle stand-alone syntactic forms adequately, they provide weak support for macros that share information or macros that use existing syntactic forms in new contexts. This paper describes and models features of the Racket macro system, including support for general compile-time bindings, sub-form expansion and analysis, and environment management. The presentation assumes a basic familiarity with Lisp-style macros, and it takes for granted the need for macros that respect lexical scope. The model, however, strips away the pattern and template system that is normally associated with Scheme macros, isolating a core that is simpler, can support pattern and template forms themselves as macros, and generalizes naturally to Racket's other extensions.

Functional languages are suitable for transformational developments of programs. However, accumulative functions, or in particular tail-recursive functions, are known to be less suitable for manipulation. In this paper, we propose a program transformation named “IO swapping” that swaps call-time and return-time computations. It moves computations in accumulative parameters to results and thereby enables interesting transformations. We demonstrate effectiveness of IO swapping by several applications: deforestation, higher order removal, program inversion, and manipulation of circular programs.

The analysis of the dynamics of Biological Regulatory Networks (BRNs) requires innovative methods to cope with the state-space explosion. This paper settles an original approach for deciding reachability properties based on Process Hitting, which is a framework suitable for modelling dynamical complex systems. In particular, Process Hitting has been shown to be of interest in providing compact models of the dynamics of BRNs with discrete values. Process Hitting splits a finite number of processes into so-called sorts and describes the way each process is able to act upon (that is, to ‘hit’) another one (or itself) in order to ‘bounce’ it as another process of the same sort with further actions.

By using complementary abstract interpretations of the succession of actions in Process Hitting, we build a very efficient static analysis to over- and under-approximate reachability properties, which avoids the need to build the underlying states graph. The analysis is proved to have a low theoretical complexity, in particular when the number of processes per sorts is limited, while a very large number of sorts can be managed.

This makes such an approach very promising for the scalable analysis of abstract complex systems. We illustrate this through the analysis of a large BRN of 94 components. Our method replies quasi-instantaneously to reachability questions, while standard model-checking techniques regularly fail because of the combinatoric explosion of behaviours.

We introduce a novel formalism of label-structured modal transition systems that combines the classical may/must modalities on transitions with structured labels that represent quantitative aspects of the model. On the one hand, the specification formalism is general enough to include models like weighted modal transition systems and allows system developers to employ more complex label refinement than in previously studied theories. On the other hand, the formalism maintains the desirable properties required by any specification theory supporting compositional reasoning. In particular, we study modal and thorough refinement, determinisation, parallel composition, conjunction, quotient and logical characterisation of label-structured modal transition systems.

In this short note we study the semantics of two basic computational effects, exceptions and states, from a new point of view. In the handling of exceptions we dissociate the control from the elementary operation that recovers from the exception. In this way it becomes apparent that there is a duality, in the categorical sense, between exceptions and states.

Consider the random graph process where we start with an empty graph on n vertices and, at time t, are given an edge et chosen uniformly at random among the edges which have not appeared so far. A classical result in random graph theory asserts that w.h.p. the graph becomes Hamiltonian at time (1/2+o(1))n log n. On the contrary, if all the edges were directed randomly, then the graph would have a directed Hamilton cycle w.h.p. only at time (1+o(1))n log n. In this paper we further study the directed case, and ask whether it is essential to have twice as many edges compared to the undirected case. More precisely, we ask if, at time t, instead of a random direction one is allowed to choose the orientation of et, then whether or not it is possible to make the resulting directed graph Hamiltonian at time earlier than n log n. The main result of our paper answers this question in the strongest possible way, by asserting that one can orient the edges on-line so that w.h.p. the resulting graph has a directed Hamilton cycle exactly at the time at which the underlying graph is Hamiltonian.

Measuring lexical semantic relatedness is an important task in Natural Language Processing (NLP). It is often a prerequisite to many complex NLP tasks. Despite an extensive amount of work dedicated to this area of research, there is a lack of an up-to-date survey in the field. This paper aims to address this issue with a study that is focused on four perspectives: (i) a comparative analysis of background information resources that are essential for measuring lexical semantic relatedness; (ii) a review of the literature with a focus on recent methods that are not covered in previous surveys; (iii) discussion of the studies in the biomedical domain where novel methods have been introduced but inadequately communicated across the domain boundaries; and (iv) an evaluation of lexical semantic relatedness methods and a discussion of useful lessons for the development and application of such methods. In addition, we discuss a number of issues in this field and suggest future research directions. It is believed that this work will be a valuable reference to researchers of lexical semantic relatedness and substantially support the research activities in this field.

Let χ be the primitive Dirichlet character of conductor 49 defined by χ(3)=ζ for ζ a primitive 42nd root of unity. We explicitly compute the slopes of the U7 operator acting on the space of overconvergent modular forms on X1(49) with weight k and character χ7k−6 or χ8−7k, depending on the embedding of ℚ(ζ)into ℂ7. By applying results of Coleman and of Cohen and Oesterlé, we are then able to deduce the slopes of U7 acting on all classical Hecke newforms of the same weight and character.

We present an efficient algorithm for the condensation of homomorphism spaces. This provides an improvement over the known tensor condensation method which is essentially due to a better choice of bases. We explain the theory behind this approach and describe the implementation in detail. Finally, we give timings to compare with previous methods.

We construct and classify all groups given by triangular presentations associated to the smallest thick generalized quadrangle that act simply transitively on the vertices of hyperbolic triangular buildings of the smallest non-trivial thickness. Our classification yields 23 non-isomorphic torsion-free groups (which were obtained in an earlier work) and 168 non-isomorphic torsion groups acting on one of two possible buildings with the smallest thick generalized quadrangle as the link of each vertex. In analogy with the case, we find both torsion and torsion-free groups acting on the same building.

We study the equation a2−2b6=cp and its specialization a2−2=cp, where p is a prime, using the modular method. In particular, we use a ℚ-curve defined over for which the solution (a,b,c)=(±1,±1,−1) gives rise to a CM-form. This allows us to apply the modular method to resolve the equation a2−2b6=cp for p in certain congruence classes. For the specialization a2−2=cp, we use the multi-Frey technique of Siksek to obtain further refined results.

Over the past two decades, robots have been increasingly used in biomedical applications such as bone cutting. Traditional automated manufacturing processes are often unable to meet the safety and accuracy requirements for such applications, particularly for cutting inhomogeneous constitutions of bone. In this case, human–robot cooperation may prove to be an effective approach. In this paper, we demonstrate that a hybrid parallel manipulator under cooperative force control can achieve accurate bone cutting with sufficient safety guaranteed. First, a hybrid parallel manipulator was constructed to provide the required rigidity for bone cutting. Then a two-loop controller was designed to implement the human–robot cooperation in bone cutting. The position control loop of adaptive fuzzy control is responsible for achieving high-tracking performance by overcoming varying friction forces from the mechanism. The force control loop of the cooperative force control adjusts the feed rate of the cutter according to the bone slicing conditions and operator's supervisory commands. The experimental results show that the proposed controller can effectively achieve the required accuracy in bone cutting with required safety.