In the k-means problem with penalties, we are given a data set $${\cal D} \subseteq \mathbb{R}^\ell $$ of n points where each point $$j \in {\cal D}$$ is associated with a penalty cost pj and an integer k. The goal is to choose a set $${\rm{C}}S \subseteq {{\cal R}^\ell }$$ with |CS| ≤ k and a penalized subset $${{\cal D}_p} \subseteq {\cal D}$$ to minimize the sum of the total squared distance from the points in D / Dp to CS and the total penalty cost of points in Dp, namely $$\sum\nolimits_{j \in {\cal D}\backslash {{\cal D}_p}} {d^2}(j,{\rm{C}}S) + \sum\nolimits_{j \in {{\cal D}_p}} {p_j}$$. We employ the primal-dual technique to give a pseudo-polynomial time algorithm with an approximation ratio of (6.357+ε) for the k-means problem with penalties, improving the previous best approximation ratio 19.849+∊ for this problem given by Feng et al. in Proceedings of FAW (2019).

This article gives an overview of automatic amortized resource analysis (AARA), a technique for inferring symbolic resource bounds for programs at compile time. AARA has been introduced by Hofmann and Jost in 2003 as a type system for deriving linear worst-case bounds on the heap-space consumption of first-order functional programs with eager evaluation strategy. Since then AARA has been the subject of dozens of research articles, which extended the analysis to different resource metrics, other evaluation strategies, non-linear bounds, and additional language features. All these works preserved the defining characteristics of the original paper: local inference rules, which reduce bound inference to numeric (usually linear) optimization; a soundness proof with respect to an operational cost semantics; and the support of amortized analysis with the potential method.

Labeled graphs are particularly well adapted to represent objects in the context of topology-based geometric modeling. Thus, graph transformation theory is used to implement modeling operations and check their consistency. This article defines a class of graph transformation rules dedicated to embedding computations. Objects are here defined as a particular subclass of labeled graphs in which arc labels encode their topological structure (i.e., cell subdivision: vertex, edge, face) and node labels encode their embedding (i.e., relevant data: vertex positions, face colors, volume density). Object consistency is defined by labeling constraints which must be preserved by modeling operations that modify topology and/or embedding. Dedicated graph transformation variables allow us to access the existing embedding from the underlying topological structure (e.g., collecting all the points of a face) in order to compute the new embedding using user-provided functions (e.g., compute the barycenter of several points). To ensure the safety of the defined operations, we provide syntactic conditions on rules that preserve the object consistency constraints.

One of the most important results in domain theory is the Hofmann-Mislove Theorem, which reveals a very distinct characterization for the sober spaces via open filters. In this paper, we extend this result to the d-spaces and well-filtered spaces. We do this by introducing the notions of Hofmann-Mislove-system (HM-system for short) and $\Psi$ -well-filtered space, which provide a new unified approach to sober spaces, well-filtered spaces, and d-spaces. In addition, a characterization for $\Psi$ -well-filtered spaces is provided via $\Psi$ -sets. We also discuss the relationship between $\Psi$ -well-filtered spaces and H-sober spaces considered by Xu. We show that the category of complete $\Psi$ -well-filtered spaces is a full reflective subcategory of the category of $T_0$ spaces with continuous mappings. For each HM-system $\Psi$ that has a designated property, we show that a $T_0$ space X is $\Psi$ -well-filtered if and only if its Smyth power space $P_s(X)$ is $\Psi$ -well-filtered.

Convergent rewriting systems on algebraic structures give methods to solve decision problems, to prove coherence results, and to compute homological invariants. These methods are based on higher-dimensional extensions of the critical branching lemma that proves local confluence from confluence of the critical branchings. The analysis of local confluence of rewriting systems on algebraic structures, such as groups or linear algebras, is complicated because of the underlying algebraic axioms. This article introduces the structure of algebraic polygraph modulo that formalizes the interaction between the rules of an algebraic rewriting system and the inherent algebraic axioms, and we show a critical branching lemma for algebraic polygraphs. We deduce a critical branching lemma for rewriting systems on algebraic models whose axioms are specified by convergent modulo rewriting systems. We illustrate our constructions for string, linear, and group rewriting systems.

It is well established that equational algebraic theories and the monads they generate can be used to encode computational effects. An important insight of Power and Shkaravska is that comodels of an algebraic theory $\mathbb{T}$ – i.e., models in the opposite category $\mathcal{S}\mathrm{et}^{\mathrm{op}}$ – provide a suitable environment for evaluating the computational effects encoded by $\mathbb{T}$ . As already noted by Power and Shkaravska, taking comodels yields a functor from accessible monads to accessible comonads on $\mathcal{S}\mathrm{et}$ . In this paper, we show that this functor is part of an adjunction – the “costructure–cosemantics adjunction” of the title – and undertake a thorough investigation of its properties. We show that, on the one hand, the cosemantics functor takes its image in what we term the presheaf comonads induced by small categories; and that, on the other, costructure takes its image in the presheaf monads induced by small categories. In particular, the cosemantics comonad of an accessible monad will be induced by an explicitly-described category called its behaviour category that encodes the static and dynamic properties of the comodels. Similarly, the costructure monad of an accessible comonad will be induced by a behaviour category encoding static and dynamic properties of the comonad coalgebras. We tie these results together by showing that the costructure–cosemantics adjunction is idempotent, with fixpoints to either side given precisely by the presheaf monads and comonads. Along the way, we illustrate the value of our results with numerous examples drawn from computation and mathematics.

Hoare Logic has a long tradition in formal verification and has been continuously developed and used to verify a broad class of programs, including sequential, object-oriented, and concurrent programs. Here we focus on partial and total correctness assertions within the framework of Hoare logic and show that a comprehensive categorical analysis of its axiomatic semantics needs the languages of indexed and fibered category theory. We consider Hoare formulas with local, finite contexts, of program and logical variables. The structural features of Hoare assertions are presented in an indexed setting, while the logical features of deduction are modeled in the fibered one.

In this paper, we mainly study the function spaces related to H-sober spaces. For an irreducible subset system H and $T_{0}$ spaces X and Y, it is proved that the following three conditions are equivalent: (1) the Scott space $\Sigma \mathcal O(X)$ of the lattice of all open sets of X is H-sober; (2) for every H-sober space Y, the function space $\mathbb{C}(X, Y)$ of all continuous mappings from X to Y equipped with the Isbell topology is H-sober; (3) for every H-sober space Y, the Isbell topology on $\mathbb{C}(X, Y)$ has property S with respect to H. One immediate corollary is that for a $T_{0}$ space X, Y is a d-space (resp., well-filtered space) iff the function space $\mathbb{C}(X, Y)$ equipped with the Isbell topology is a d-space (resp., well-filtered space). It is shown that for any $T_0$ space X for which the Scott space $\Sigma \mathcal O(X)$ is non-sober, the function space $\mathbb{C}(X, \Sigma 2)$ equipped with the Isbell topology is not sober. The function spaces $\mathbb{C}(X, Y)$ equipped with the Scott topology, the compact-open topology and the pointwise convergence topology are also discussed. Our study also leads to a number of questions, whose answers will deepen our understanding of the function spaces related to H-sober spaces.

Classical definitions of weak higher-dimensional categories are given inductively, for example, a bicategory has a set of objects and hom categories, and a tricategory has a set of objects and hom bicategories. However, more recent definitions of weak n-categories for all natural numbers n, or of weak $\omega$ -categories, take more sophisticated approaches, and the nature of the ‘hom is often not immediate from the definitions’. In this paper, we focus on Leinster’s definition of weak $\omega$ -category based on an earlier definition by Batanin and construct, for each weak $\omega$ -category $\mathcal{A}$ , an underlying (weak $\omega$ -category)-enriched graph consisting of the same objects and for each pair of objects x and y, a hom weak $\omega$ -category $\mathcal{A}(x,y)$ . We also show that our construction is functorial with respect to weak $\omega$ -functors introduced by Garner.

One of the most fundamental properties of a proof system is analyticity, expressing the fact that a proof of a given formula F only uses subformulas of F. In sequent calculus, this property is usually proved by showing that the $\mathsf{cut}$ rule is admissible, i.e., the introduction of the auxiliary lemma H in the reasoning “if H follows from G and F follows from H, then F follows from G” can be eliminated. The proof of cut admissibility is usually a tedious, error-prone process through several proof transformations, thus requiring the assistance of (semi-)automatic procedures. In a previous work by Miller and Pimentel, linear logic ( $\mathsf{LL}$ ) was used as a logical framework for establishing sufficient conditions for cut admissibility of object logical systems (OL). The OL’s inference rules are specified as an $\mathsf{LL}$ theory and an easy-to-verify criterion sufficed to establish the cut-admissibility theorem for the OL at hand. However, there are many logical systems that cannot be adequately encoded in $\mathsf{LL}$ , the most symptomatic cases being sequent systems for modal logics. In this paper, we use a linear-nested sequent ( $\mathsf{LNS}$ ) presentation of $\mathsf{MMLL}$ (a variant of LL with subexponentials), and show that it is possible to establish a cut-admissibility criterion for $\mathsf{LNS}$ systems for (classical or substructural) multimodal logics. We show that the same approach is suitable for handling the $\mathsf{LNS}$ system for intuitionistic logic.

User-defined higher-order rewrite rules are becoming a standard in proof assistants based on intuitionistic type theory. This raises the question of proving that they preserve the properties of beta-reductions for the corresponding type systems. In a series of papers, we develop techniques based on van Oostrom’s decreasing diagrams that reduce confluence proofs to the checking of various forms of critical pairs for higher-order rewrite rules extending beta-reduction on pure lambda-terms. As shown in a previous paper of the two middle authors, confluence of a terminating set of left-linear rewrite rules is obtained when their critical pairs are joinable, beta-rewrite steps being disallowed. The present paper concentrates on the case where arbitrary beta-rewrite steps are allowed for joining critical pairs. The rewrite relation used for analyzing confluence may rewrite arbitrarily many non-overlapping redexes in a single step. This relation gives rise to critical pairs that overlap both horizontally, as with parallel rewriting, but also vertically, forming chains of successive overlaps. Practical examples of use of this technique are analyzed.

This paper is meant to be a survey about implicit characterizations of complexity classes by fragments of higher-order programming languages, with a special focus on type systems and subsystems of linear logic. Particular emphasis will be put on Martin Hofmann’s contributions to the subject, which very much helped in shaping the field.

The paper works within the framework of punctual computability, which is focused on eliminating unbounded search from constructions in algebra and infinite combinatorics. We study punctual numberings, that is, uniform computations for families S of primitive recursive functions. The punctual reducibility between numberings is induced by primitive recursive functions. This approach gives rise to upper semilattices of degrees, which are called Rogers pr-semilattices. We show that any infinite, uniformly primitive recursive family S induces an infinite Rogers pr-semilattice R. We prove that the semilattice R does not have minimal elements, and every nontrivial interval inside R contains an infinite antichain. In addition, every non-greatest element from R is a part of an infinite antichain. We show that the $\Sigma_1$ -fragment of the theory Th(R) is decidable.

The proof theory of the constructive modal logic S4 (hereafter $\mathsf{CS4}$ ) has been settled since the beginning of this century by means of either standard natural deduction and sequent calculi or by the reconstruction of modal logic through hypothetical and categorical judgments à la Martin-Löf, an approach carried out by using a special kind of sequents, which keeps two separated contexts representing ordinary and enhanced hypotheses, intuitively interpreted as true and valid assumptions. These so-called dual-context sequents, originated in linear logic, are used to define a natural deduction system handling judgments of validity, truth, and possibility, resulting in a formalism equivalent to an axiomatic system for $\mathsf{CS4}$ . However, this proof-theoretical study of $\mathsf{CS4}$ lacks, to the best of our knowledge, its third fundamental constituent, namely a sequent calculus. In this paper, we define such a dual-context formalism, called ${\bf DG_{CS4}}$ , and provide detailed proofs of the admissibility for the ordinary cut rule as well as the elimination of a second cut rule, which manipulates enhanced hypotheses. Furthermore, we make available a formal verification of the equivalence of this proposal with the previously defined axiomatic and dual-context natural deduction systems for $\mathsf{CS4}$ , using the Coq proof-assistant.

We use a simplified version of the framework of resource monoids, introduced by Dal Lago and Hofmann, to interpret simply typed λ-calculus with constants zero and successor. We then use this model to prove a simple quantitative result about bounding the size of the normal form of λ-terms. While the bound itself is already known, this is to our knowledge the first semantic proof of this fact. Our use of resource monoids differs from the other instances found in the literature, in that it measures the size of λ-terms rather than time complexity.

The concept of a synchronizing word is a very important notion in the theory of finite automata. We consider the associated decision problem to decide if a given DFA possesses a synchronizing word of length at most k, where k is the standard parameter. We show that this problem DFA-SW is equivalent to the problem Monoid Factorization introduced by Cai, Chen, Downey, and Fellows. Apart from the known $\textsf{W}[2]$ -hardness results, we show that these problems belong to $\textsf{A}[2]$ , $\textsf{W}[\textsf{P}],$ and $\textsf{WNL}$ . This indicates that DFA-SW is not complete for any of these classes, and hence, we suggest a new parameterized complexity class $\textsf{W}[\textsf{Sync}]$ as a proper home for these (and more) problems. We present quite a number of problems that belong to $\textsf{W}[\textsf{Sync}]$ or are hard or complete for this new class.

We study the reduction in a $\lambda$ -calculus derived from Moggi’s computational one, which we call the computational core. The reduction relation consists of rules obtained by orienting three monadic laws. Such laws, in particular associativity and identity, introduce intricacies in the operational analysis. We investigate the central notions of returning a value versus having a normal form and address the question of normalizing strategies. Our analysis relies on factorization results.

We investigate how various forms of bisimulation can be characterised using the technology of logical relations. The approach taken is that each form of bisimulation corresponds to an algebraic structure derived from a transition system, and the general result is that a relation R between two transition systems on state spaces S and T is a bisimulation if and only if the derived algebraic structures are in the logical relation automatically generated from R. We show that this approach works for the original Park–Milner bisimulation and that it extends to weak bisimulation, and branching and semi-branching bisimulation. The paper concludes with a discussion of probabilistic bisimulation, where the situation is slightly more complex, partly owing to the need to encompass bisimulations that are not just relations.

Higher-dimensional rewriting systems are tools to analyse the structure of formally reducing terms to normal forms, as well as comparing the different reduction paths that lead to those normal forms. This higher structure can be captured by finding a homotopy basis for the rewriting system. We show that the basic notions of confluence and wellfoundedness are sufficient to recursively build such a homotopy basis, with a construction reminiscent of an argument by Craig C. Squier. We then go on to translate this construction to the setting of homotopy type theory, where managing equalities between paths is important in order to construct functions which are coherent with respect to higher dimensions. Eventually, we apply the result to approximate a series of open questions in homotopy type theory, such as the characterisation of the homotopy groups of the free group on a set and the pushout of 1-types. This paper expands on our previous conference contribution Coherence via Wellfoundedness by laying out the construction in the language of higher-dimensional rewriting.

Weakest precondition transformers are useful tools in program verification. One of their key properties is composability, that is, the weakest precondition predicate transformer (wppt for short) associated to program $f;\;g$ should be equal to the composition of the wppts associated to f and g. In this paper, we study the categorical structure behind wppts from a fibrational point of view. We characterize the wppts that satisfy composability as the ones constructed from the Cartesian lifting of a monad. We moreover show that Cartesian liftings of monads along lax slice categories bijectively correspond to Eilenberg–Moore monotone algebras. We then instantiate our techniques by deriving wppts for commonplace effects such as the maybe monad, the nonempty powerset monad, the counter monad, or the distribution monad. We also show how to combine them to derive the wppts appearing in the literature of verification of probabilistic programs.