Logical relations built on top of an operational semantics are one of the most successful proof methods in programming language semantics. In recent years, more and more expressive notions of operationally-based logical relations have been designed and applied to specific families of languages. However, a unifying abstract framework for operationally-based logical relations is still missing. We show how fibrations can provide a uniform treatment of operational logical relations, using as reference example a lambda-calculus with generic effects endowed with a novel, abstract operational semantics defined on a large class of categories. Moreover, this abstract perspective allows us to give a solid mathematical ground also to differential logical relations -- a recently introduced notion of higher-order distance between programs -- both pure and effectful, bringing them back to a common picture with traditional ones.
The author has recently introduced an abstract algebraic framework of analogical proportions within the general setting of universal algebra. This paper studies analogical proportions in the boolean domain consisting of two elements 0 and 1 within his framework. It turns out that our notion of boolean proportions coincides with two prominent models from the literature in different settings. This means that we can capture two separate modellings of boolean proportions within a single framework which is mathematically appealing and provides further evidence for the robustness and applicability of the general framework.
We study the extent to which it is possible to approximate the optimal value of a Unique Games instance in Fixed-Point Logic with Counting (FPC). Formally, we prove lower bounds against the accuracy of FPC-interpretations that map Unique Games instances (encoded as relational structures) to rational numbers giving the approximate fraction of constraints that can be satisfied. We prove two new FPC-inexpressibility results for Unique Games: the existence of a $(1/2, 1/3 + \delta)$-inapproximability gap, and inapproximability to within any constant factor. Previous recent work has established similar FPC-inapproximability results for a small handful of other problems. Our construction builds upon some of these ideas, but contains a novel technique. While most FPC-inexpressibility results are based on variants of the CFI-construction, ours is significantly different. We start with a graph of very large girth and label the edges with random affine vector spaces over $\mathbb{F}_2$ that determine the constraints in the two structures. Duplicator's strategy involves maintaining a partial isomorphism over a minimal tree that spans the pebbled vertices of the graph.
When translating a term calculus into a graphical formalism many inessential details are abstracted away. In the case of $\lambda$-calculus translated to proof-nets, these inessential details are captured by a notion of equivalence on $\lambda$-terms known as $\simeq_\sigma$-equivalence, in both the intuitionistic (due to Regnier) and classical (due to Laurent) cases. The purpose of this paper is to uncover a strong bisimulation behind $\simeq_\sigma$-equivalence, as formulated by Laurent for Parigot's $\lambda\mu$-calculus. This is achieved by introducing a relation $\simeq$, defined over a revised presentation of $\lambda\mu$-calculus we dub $\Lambda M$. More precisely, we first identify the reasons behind Laurent's $\simeq_\sigma$-equivalence on $\lambda\mu$-terms failing to be a strong bisimulation. Inspired by Laurent's \emph{Polarized Proof-Nets}, this leads us to distinguish multiplicative and exponential reduction steps on terms. Second, we enrich the syntax of $\lambda\mu$ to allow us to track the exponential operations. These technical ingredients pave the way towards a strong bisimulation for the classical case. We introduce a calculus $\Lambda M$ and a relation $\simeq$ that we show to be a strong bisimulation with respect to reduction in $\Lambda M$, ie. two $\simeq$-equivalent terms have the exact same reduction semantics, a result which fails for Regnier's $\simeq_\sigma$-equivalence in $\lambda$-calculus as well as for Laurent's […]
Quantum based systems are a relatively new research area for that different modelling languages including process calculi are currently under development. Encodings are often used to compare process calculi. Quality criteria are used then to rule out trivial or meaningless encodings. In this new context of quantum based systems, it is necessary to analyse the applicability of these quality criteria and to potentially extend or adapt them. As a first step, we test the suitability of classical criteria for encodings between quantum based languages and discuss new criteria. Concretely, we present an encoding, from a language inspired by CQP into a language inspired by qCCS. We show that this encoding satisfies compositionality, name invariance (for channel and qubit names), operational correspondence, divergence reflection, success sensitiveness, and that it preserves the size of quantum registers. Then we show that there is no encoding from qCCS into CQP that is compositional, operationally corresponding, and success sensitive.
The logic of information flows (LIF) has recently been proposed as a general framework in the field of knowledge representation. In this framework, tasks of procedural nature can still be modeled in a declarative, logic-based fashion. In this paper, we focus on the task of query processing under limited access patterns, a well-studied problem in the database literature. We show that LIF is well-suited for modeling this task. Toward this goal, we introduce a variant of LIF called "forward" LIF (FLIF), in a first-order setting. FLIF takes a novel graph-navigational approach; it is an XPath-like language that nevertheless turns out to be equivalent to the "executable" fragment of first-order logic defined by Nash and Ludäscher. One can also classify the variables in FLIF expressions as inputs and outputs. Expressions where inputs and outputs are disjoint, referred to as io-disjoint FLIF expressions, allow a particularly transparent translation into algebraic query plans that respect the access limitations. Finally, we show that general FLIF expressions can always be put into io-disjoint form.
We investigate the impact of non-regular path expressions on the decidability of satisfiability checking and querying in description logics extending ALC. Our primary objects of interest are ALCreg and ALCvpl, the extensions of with path expressions employing, respectively, regular and visibly-pushdown languages. The first one, ALCreg, is a notational variant of the well-known Propositional Dynamic Logic of Fischer and Ladner. The second one, ALCvpl, was introduced and investigated by Loding and Serre in 2007. The logic ALCvpl generalises many known decidable non-regular extensions of ALCreg. We provide a series of undecidability results. First, we show that decidability of the concept satisfiability problem for ALCvpl is lost upon adding the seemingly innocent Self operator. Second, we establish undecidability for the concept satisfiability problem for ALCvpl extended with nominals. Interestingly, our undecidability proof relies only on one single non-regular (visibly-pushdown) language, namely on r#s# := { r^n s^n | n in N } for fixed role names r and s. Finally, in contrast to the classical database setting, we establish undecidability of query entailment for queries involving non-regular atoms from r#s#, already in the case of ALC-TBoxes.
In the literature on Kleene algebra, a number of variants have been proposed which impose additional structure specified by a theory, such as Kleene algebra with tests (KAT) and the recent Kleene algebra with observations (KAO), or make specific assumptions about certain constants, as for instance in NetKAT. Many of these variants fit within the unifying perspective offered by Kleene algebra with hypotheses, which comes with a canonical language model constructed from a given set of hypotheses. For the case of KAT, this model corresponds to the familiar interpretation of expressions as languages of guarded strings. A relevant question therefore is whether Kleene algebra together with a given set of hypotheses is complete with respect to its canonical language model. In this paper, we revisit, combine and extend existing results on this question to obtain tools for proving completeness in a modular way. We showcase these tools by giving new and modular proofs of completeness for KAT, KAO and NetKAT, and we prove completeness for new variants of KAT: KAT extended with a constant for the full relation, KAT extended with a converse operation, and a version of KAT where the collection of tests only forms a distributive lattice.
The pebbling comonad, introduced by Abramsky, Dawar and Wang, provides a categorical interpretation for the k-pebble games from finite model theory. The coKleisli category of the pebbling comonad specifies equivalences under different fragments and extensions of infinitary k-variable logic. Moreover, the coalgebras over this pebbling comonad characterise treewidth and correspond to tree decompositions. In this paper we introduce the pebble-relation comonad, which characterises pathwidth and whose coalgebras correspond to path decompositions. We further show that the existence of a coKleisli morphism in this comonad is equivalent to truth preservation in the restricted conjunction fragment of k-variable infinitary logic. We do this using Dalmau's pebble-relation game and an equivalent all-in-one pebble game. We then provide a similar treatment to the corresponding coKleisli isomorphisms via a bijective version of the all-in-one pebble game. Finally, we show as a consequence a new Lovász-type theorem relating pathwidth to the restricted conjunction fragment of k-variable infinitary logic with counting quantifiers.
The ZX-calculus is a powerful framework for reasoning in quantum computing. It provides in particular a compact representation of matrices of interests. A peculiar property of the ZX-calculus is the absence of a formal sum allowing the linear combinations of arbitrary ZX-diagrams. The universality of the formalism guarantees however that for any two ZX-diagrams, the sum of their interpretations can be represented by a ZX-diagram. We introduce a general, inductive definition of the addition of ZX-diagrams, relying on the construction of controlled diagrams. Based on this addition technique, we provide an inductive differentiation of ZX-diagrams. Indeed, given a ZX-diagram with variables in the description of its angles, one can differentiate the diagram according to one of these variables. Differentiation is ubiquitous in quantum mechanics and quantum computing (e.g. for solving optimization problems). Technically, differentiation of ZX-diagrams is strongly related to summation as witnessed by the product rules. We also introduce an alternative, non inductive, differentiation technique rather based on the isolation of the variables. Finally, we apply our results to deduce a diagram for an Ising Hamiltonian.
The Skolem problem is a long-standing open problem in linear dynamical systems: can a linear recurrence sequence (LRS) ever reach 0 from a given initial configuration? Similarly, the positivity problem asks whether the LRS stays positive from an initial configuration. Deciding Skolem (or positivity) has been open for half a century: the best known decidability results are for LRS with special properties (e.g., low order recurrences). But these problems are easier for "uninitialized" variants, where the initial configuration is not fixed but can vary arbitrarily: checking if there is an initial configuration from which the LRS stays positive can be decided in polynomial time (Tiwari in 2004, Braverman in 2006). In this paper, we consider problems that lie between the initialized and uninitialized variants. More precisely, we ask if 0 (resp. negative numbers) can be avoided from every initial configuration in a neighborhood of a given initial configuration. This can be considered as a robust variant of the Skolem (resp. positivity) problem. We show that these problems lie at the frontier of decidability: if the neighbourhood is given as part of the input, then robust Skolem and robust positivity are Diophantine hard, i.e., solving either would entail major breakthroughs in Diophantine approximations, as happens for (non-robust) positivity. However, if one asks whether such a neighbourhood exists, then the problems turn out to be decidable with PSPACE complexity. Our […]
We propose a relaxation to the definition of well-structured transition systems (\WSTS) while retaining the decidability of boundedness and non-termination. In this class, the well-quasi-ordered (wqo) condition is relaxed such that it is applicable only between states that are reachable one from another. Furthermore, the monotony condition is relaxed in the same way. While this retains the decidability of non-termination and boundedness, it appears that the coverability problem is undecidable. To this end, we define a new notion of monotony, called cover-monotony, which is strictly more general than the usual monotony and still allows us to decide a restricted form of the coverability problem.
We describe a general approach to deriving linear-time logics for a wide variety of state-based, quantitative systems, by modelling the latter as coalgebras whose type incorporates both branching and linear behaviour. Concretely, we define logics whose syntax is determined by the type of linear behaviour, and whose domain of truth values is determined by the type of branching behaviour, and we provide two semantics for them: a step-wise semantics akin to that of standard coalgebraic logics, and a path-based semantics akin to that of standard linear-time logics. The former semantics is useful for model checking, whereas the latter is the more natural semantics, as it measures the extent with which qualitative properties hold along computation paths from a given state. Our main result is the equivalence of the two semantics. We also provide a semantic characterisation of a notion of logical distance induced by these logics. Instances of our logics support reasoning about the possibility, likelihood or minimal cost of exhibiting a given linear-time property.
In this paper, we prove measurability of event for which a general continuous-time stochastic process satisfies continuous-time Metric Temporal Logic (MTL) formula. Continuous-time MTL can define temporal constrains for physical system in natural way. Then there are several researches that deal with probability of continuous MTL semantics for stochastic processes. However, proving measurability for such events is by no means an obvious task, even though it is essential. The difficulty comes from the semantics of "until operator", which is defined by logical sum of uncountably many propositions. Given the difficulty involved in proving the measurability of such an event using classical measure-theoretic methods, we employ a theorem from stochastic analysis. This theorem is utilized to prove the measurability of hitting times for stochastic processes, and it stands as a profound result within the theory of capacity. Next, we provide an example that illustrates the failure of probability approximation when discretizing the continuous semantics of MTL formulas with respect to time. Additionally, we prove that the probability of the discretized semantics converges to that of the continuous semantics when we impose restrictions on diamond operators to prevent nesting.
In the context of the stream calculus, we present an Implicit Function Theorem (IFT) for polynomial systems, and discuss its relations with the classical IFT from calculus. In particular, we demonstrate the advantages of the stream IFT from a computational point of view, and provide a few example applications where its use turns out to be valuable.
Presheaf models of dependent type theory have been successfully applied to model HoTT, parametricity, and directed, guarded and nominal type theory. There has been considerable interest in internalizing aspects of these presheaf models, either to make the resulting language more expressive, or in order to carry out further reasoning internally, allowing greater abstraction and sometimes automated verification. While the constructions of presheaf models largely follow a common pattern, approaches towards internalization do not. Throughout the literature, various internal presheaf operators ($\surd$, $\Phi/\mathsf{extent}$, $\Psi/\mathsf{Gel}$, $\mathsf{Glue}$, $\mathsf{Weld}$, $\mathsf{mill}$, the strictness axiom and locally fresh names) can be found and little is known about their relative expressivenes. Moreover, some of these require that variables whose type is a shape (representable presheaf, e.g. an interval) be used affinely. We propose a novel type former, the transpension type, which is right adjoint to universal quantification over a shape. Its structure resembles a dependent version of the suspension type in HoTT. We give general typing rules and a presheaf semantics in terms of base category functors dubbed multipliers. Structural rules for shape variables and certain aspects of the transpension type depend on characteristics of the multiplier. We demonstrate how the transpension type and the strictness axiom can be combined to implement all and improve some of […]
We propose a simple imperative programming language, ERC, that features arbitrary real numbers as primitive data type, exactly. Equipped with a denotational semantics, ERC provides a formal programming language-theoretic foundation to the algorithmic processing of real numbers. In order to capture multi-valuedness, which is well-known to be essential to real number computation, we use a Plotkin powerdomain and make our programming language semantics computable and complete: all and only real functions computable in computable analysis can be realized in ERC. The base programming language supports real arithmetic as well as implicit limits; expansions support additional primitive operations (such as a user-defined exponential function). By restricting integers to Presburger arithmetic and real coercion to the `precision' embedding $\mathbb{Z}\ni p\mapsto 2^p\in\mathbb{R}$, we arrive at a first-order theory which we prove to be decidable and model-complete. Based on said logic as specification language for preconditions and postconditions, we extend Hoare logic to a sound (w.r.t. the denotational semantics) and expressive system for deriving correct total correctness specifications. Various examples demonstrate the practicality and convenience of our language and the extended Hoare logic.
$\text{TT}^{\Box}_{{\mathcal C}}$ is a generic family of effectful, extensional type theories with a forcing interpretation parameterized by modalities. This paper identifies a subclass of $\text{TT}^{\Box}_{{\mathcal C}}$ theories that internally realizes continuity principles through stateful computations, such as reference cells. The principle of continuity is a seminal property that holds for a number of intuitionistic theories such as System T. Roughly speaking, it states that functions on real numbers only need approximations of these numbers to compute. Generally, continuity principles have been justified using semantical arguments, but it is known that the modulus of continuity of functions can be computed using effectful computations such as exceptions or reference cells. In this paper, the modulus of continuity of the functionals on the Baire space is directly computed using the stateful computations enabled internally in the theory.
In this paper, we examine the relationship between the stability of the dynamical system $x^{\prime}=f(x)$ and the computability of its basins of attraction. We present a computable $C^{\infty}$ system $x^{\prime}=f(x)$ that possesses a computable and stable equilibrium point, yet whose basin of attraction is robustly non-computable in a neighborhood of $f$ in the sense that both the equilibrium point and the non-computability of its associated basin of attraction persist when $f$ is slightly perturbed. This indicates that local stability near a stable equilibrium point alone is insufficient to guarantee the computability of its basin of attraction. However, we also demonstrate that the basins of attraction associated with a structurally stable - globally stable (robust) - planar system defined on a compact set are computable. Our findings suggest that the global stability of a system and the compactness of the domain play a pivotal role in determining the computability of its basins of attraction.
