Electronic Notes in Theoretical Computer Science最新文献

We describe a type theory or metalanguage for constructing and reasoning about higher-order programs with global and local state, and its categorical model. This provides an encapsulation primitive for abstracting global state and making it local to an object, so that it is passed only between its invocations. Our calculus and its semantics extend the interpretation of lambda-terms in a Cartesian closed category with a monoidal action on a category of evaluation contexts — the sequoid — which is dual to the action of the function type. This gives an interpretation of a new type constructor which allows the representation of both global state — via “state-passing-style” interpretation which uses it to represent output states — and local state, via encapsulation, which corresponds to the unique map into a final coalgebra for the sequoid. This provides the equational theory of our calculus with a coinduction rule for proving equivalence between objects with local state. We show that this theory is sound and complete with respect to the categorical semantics by constructing a term model and we show that it is consistent by giving a concrete example based on a category of games and strategies previously used to interpret general references.

It is commonly understood that Countable Choice holds constructively due to the underlying computational nature of constructivism. However, in this paper we demonstrate that invoking different notions of computation result in radically different behaviors regarding Countable Choice. In particular, we illustrate that, although deterministic computation guarantees Countable Choice, non-deterministic computation can negate Countable Choice. We then further show that using stateful computation can restore Countable Choice even in the presence of non-determinism. This finding suggests that much of the modern discourse of constructivism assumes a deterministic underlying computational system, despite non-determinism being a fundamental aspect of modern-day computation.

This article studies (multilayer perceptron) neural networks with an emphasis on the transformations involved — both forward and backward — in order to develop a semantic/logical perspective that is in line with standard program semantics. The common two-pass neural network training algorithms make this viewpoint particularly fitting. In the forward direction, neural networks act as state transformers, using Kleisli composition for the multiset monad — for the linear parts of network layers. In the reverse direction, however, neural networks change losses of outputs to losses of inputs, thereby acting like a (real-valued) predicate transformer. In this way, backpropagation is functorial by construction, as shown in other works recently. We illustrate this perspective by training a simple instance of a neural network.

Coinductive reasoning in terms of bisimulations is in practice routinely supported by carefully crafted up-to techniques that can greatly simplify proofs. However, designing and proving such bisimulation enhancements sound can be challenging, especially when striving for modularity. In this article, we present a theory of up-to techniques that builds on the notion of companion introduced by Pous and that extends our previous work which allows for powerful up-to techniques defined in terms of diacritical progress of relations. The theory of diacritical companion that we put forward works in any complete lattice and makes it possible to modularly prove soundness of up-to techniques which rely on the distinction between passive and active progresses, such as up to context in λ-calculi with control operators and extensionality.

We construct finitary set-truncated higher inductive types (HITs) from quotients and the propositional truncation. For that, we first define signatures as a modification of the schema by Basold et al., and we show they give rise to univalent categories of algebras in both sets and setoids. To interpret HITs, we use the well-known method of initial algebra semantics. The desired algebra is obtained by lifting the quotient adjunction to the level of algebras and adapting Dybjer's and Moeneclaey's interpretation of HITs in setoids. From this construction, we conclude that the equality types of HITs are freely generated and that HITs are unique. The results are formalized in the UniMath library.

The category of presheaves on a (small) category is a suitable semantic universe to study behaviour of various dynamical systems. In particular, presheaves can be used to record the executions of a system and their morphisms correspond to simulation maps for various kinds of state-based systems. In this paper, we introduce a notion of bisimulation maps between presheaves (or executions) to capture well known behavioural equivalences in an abstract way. We demonstrate the versatility of this framework by working out the characterisations for standard bisimulation, ∀-fair bisimulation, and branching bisimulation.

We examine some recent methods introduced to extend Ehrhard and Regnier's result on Taylor expansion: infinite linear combinations of approximants of a lambda-term can be normalized while keeping all coefficients finite. The methods considered allow to extend this result to non-uniform calculi; we show that when focusing on precise reduction strategies, such as Call-By-Value, Call-By-Need, PCF or variants of Call-By-Push-Value, the extension of Ehrhard and Regnier's finiteness result can hold or not, depending on the structure of the original calculus.

In particular, we introduce a resource calculus for Call-By-Need, and show that the finiteness result about its Taylor expansion can be derived from our Call-By-Value considerations. We also introduce a resource calculus for a presentation of PCF with an explicit fixpoint construction, and show how it interferes with the finiteness result. We examine then Ehrhard and Guerrieri's Bang Calculus which enjoys some Call-By-Push-Value features in a slightly different presentation.

Given an n-vertex graph G = (V,E) with m edges, a labeling f of V ∪ E that uses all the labels in the set {1,2,...,n + m} is edge-magic if there is an integer k such that f(u) + f(v) + f(uv) = k for every edge uv ∈ E. Furthermore, if the labels in {1,2,...,n} are given to the vertices, then f is called super edge-magic. Kotzig [On magic valuations of trichromatic graphs, Reports of the CRM, 1971] started the investigation of super edge-magic labelings of forests. Following this line of research, we prove that some forests of stars admit a super edge-magic labeling and that some forests of caterpillars admit an edge-magic labeling.

A (proper) k-coloring of a graph G = (V,E) is a function c : V (G) → {1,...,k} such that c(u) ≠ c(v) for every uv ∈ E(G). Given a graph G and a subgraph H of G, a q-backbone k-coloring of (G,H) is a k-coloring c of G such that q ≤ |c(u) − c(v)| for every edge uv ∈ E(H). The q-backbone chromatic number of (G,H), denoted by BBCq(G,H), is the minimum integer k for which there exists a q-backbone k-coloring of (G,H). Similarly, a circular q-backbone k-coloring of (G,H) is a function c: V (G) → {1,...,k} such that, for every edge uv ∈ E(G), we have |c(u)−c(v)| ≥ 1 and, for every edge uv ∈ E(H), we have k−q ≥ |c(u)−c(v)| ≥ q. The circular q-backbone chromatic number of (G,H), denoted by CBCq(G,H), is the smallest integer k such that there exists such coloring c.

In this work, we first prove that if G is a 3-chromatic graph and F is a galaxy, then CBCq(G,F) ≤ 2q + 2. Then, we prove that CBC3(G,M) ≤ 7 and CBCq(G,M) ≤ 2q, for every q ≥ 4, whenever M is a matching of a planar graph G. Moreover, we argue that both bounds are tight. Such bounds partially answer open questions in the literature. We also prove that one can compute BBC2(G,M) in polynomial time, whenever G is an outerplanar graph with a matching backbone M. Finally, we show a mistake in a proof that BBC2(G,M) ≤ Δ(G)+1, for any matching M of an arbitrary graph G [Miškuf et al., 2010] and we present how to fix it.

Gallai (1966) conjectured that the edge set of every graph G on n vertices can be covered by at most ⌈n/2⌉ edge-disjoint paths. Such a covering by edge-disjoint paths is called a path decomposition, and the size of a path decomposition with a minimum number of elements is called the path number of G. Peroche (1984) proved that the problem of computing the path number is NP-Complete; and Constantinou and Ellinas (2018) proved that it is polynomial for a family of complete bipartite graphs. In this paper we present an Integer Linear Programming model for computing the path number of a graph. This allowed us to verify Gallai's Conjecture for a large collection of graphs. As a result, following a work of Heinrich, Natale and Streicher on cycle decompositions (2017), we verify Gallai's Conjecture for graphs with at most 11 vertices; for bipartite graphs with at most 16 vertices; and for regular graphs with at most 14 vertices.