Electronic Notes in Theoretical Computer Science最新文献

In flowchart languages, predicates play an interesting double role. In the textual representation, they are often presented as conditions, i.e., expressions which are easily combined with other conditions (often via Boolean combinators) to form new conditions, though they only play a supporting role in aiding branching statements choose a branch to follow. On the other hand, in the graphical representation they are typically presented as decisions, intrinsically capable of directing control flow yet mostly oblivious to Boolean combination.

While categorical treatments of flowchart languages are abundant, none of them provide a treatment of this dual nature of predicates. In the present paper, we argue that extensive restriction categories are precisely categories that capture such a condition/decision duality, by means of morphisms which, coincidentally, are also called decisions. Further, we show that having these categorical decisions amounts to having an internal logic: Analogous to how subobjects of an object in a topos form a Heyting algebra, we show that decisions on an object in an extensive restriction category form a De Morgan quasilattice, the algebraic structure associated with the (three-valued) weak Kleene logic $K_3^w$. Full classical propositional logic can be recovered by restricting to total decisions, yielding extensive categories in the usual sense, and confirming (from a different direction) a result from effectus theory that predicates on objects in extensive categories form Boolean algebras.

As an application, since (categorical) decisions are partial isomorphisms, this approach provides naturally reversible models of classical propositional logic and weak Kleene logic.

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.

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.

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.

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.

In this paper we present approximation preserving reductions from the Budgeted and Generalized Maximum Coverage Problems to the Knapsack Problem with Conflict Graphs. The reductions are used to yield Polynomial Time Approximation Schemes for special classes of instances of these problems. Using these approximation schemes, the existence of pseudo-polynomial algorithms are proven and, in more particular cases, these algorithms are shown to have polynomial time complexity. Moreover, the characteristics of the instances that admit these algorithms are analyzed.

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.

Graph searches and the corresponding search trees can exhibit important structural properties and are used in various graph algorithms. The problem of deciding whether a given spanning tree of a graph is a search tree of a particular search on this graph was introduced by Hagerup and Nowak in 1985, and independently by Korach and Ostfeld in 1989 where the authors showed that this problem is efficiently solvable for DFS trees. A linear time algorithm for BFS trees was obtained by Manber in 1990. In this paper we prove that the search tree problem is also in P for LDFS, in contrast to LBFS, MCS, and MNS, where we show NP-completeness. We complement our results by providing linear time algorithms for these searches on split graphs.