Ordered Sets in the Calculus of Data Structures

Our goal is to identify families of relations that are useful for reasoning about software. We describe such families using decidable quantifier-free classes of logical constraints with a rich set of operations. A key challenge is to define such classes of constraints in a modular way, by combining multiple decidable classes. Working with quantifierfree combinations of constraints makes the combination agenda more realistic and the resulting logics more likely to be tractable than in the presence of quantifiers. Our approach to combination is based on reducing decidable fragments to a common class, Boolean Algebra with Presburger Arithmetic (BAPA). This logic was introduced by Feferman and Vaught in 1959 and can express properties of uninterpreted sets of elements, with set algebra operations and equicardinality relation (consequently, it can also express Presburger arithmetic constraints on cardinalities of sets). Combination by reduction to BAPA allows us to obtain decidable quantifierfree combinations of decidable logics that share BAPA operations. We use the term Calculus of Data Structures to denote a family of decidable constraints that reduce to BAPA. This class includes, for example, combinations of formulas in BAPA, weak monadic second-order logic of k-successors, two-variable logic with counting, and term algebras with certain homomorphisms. The approach of reduction to BAPA generalizes the Nelson-Oppen combination that forms the foundation of constraint solvers used in software verification. BAPA is convenient as a target for reductions because it admits quantifier elimination and its quantifier-free fragment is NP-complete. We describe a new member of the Calculus of Data Structures: a quantifier-free fragment that supports 1) boolean algebra of finite and infinite sets of real numbers, 2) linear arithmetic over real numbers, 3) formulas that can restrict chosen set or element variables to range over integers (providing, among others, the power of mixed integer arithmetic and sets of integers), 4) the cardinality operators, stating whether a given set has a given finite cardinality or is infinite, 5) infimum and supremum operators on sets. Among the applications of this logic are reasoning about the externally observable behavior of data structures such as sorted lists and priority queues, and specifying witness functions for the BAPA synthesis problem. We describe an abstract reduction to BAPA for our logic, proving that the satisfiability of the logic is in NP and that it can be combined with the other fragments of the Calculus of Data Structures

Actor Network Procedures as Psi-calculi for Security Ceremonies

The actor network procedures of Pavlovic and Meadows are a recent graphical formalism developed for describing security ceremonies and for reasoning about their security properties. The present work studies the relations of the actor network procedures (ANP) to the recent psi-calculi framework. Psi-calculi is a parametric formalism where calculi like spi- or applied-pi are found as instances. Psi-calculi are operational and largely non-graphical, but have strong foundation based on the theory of nominal sets and process algebras. One purpose of the present work is to give a semantics to ANP through psi-calculi. Another aim was to give a graphical language for a psi-calculus instance for security ceremonies. At the same time, this work provides more insight into the details of the ANPs formalization and the graphical representation.Comment: In Proceedings GraMSec 2014, arXiv:1404.163

SMT Solving for Functional Programming over Infinite Structures

We develop a simple functional programming language aimed at manipulating infinite, but first-order definable structures, such as the countably infinite clique graph or the set of all intervals with rational endpoints. Internally, such sets are represented by logical formulas that define them, and an external satisfiability modulo theories (SMT) solver is regularly run by the interpreter to check their basic properties. The language is implemented as a Haskell module.Comment: In Proceedings MSFP 2016, arXiv:1604.0038

Relational type-checking for MELL proof-structures. Part 1: Multiplicatives

Relational semantics for linear logic is a form of non-idempotent intersection type system, from which several informations on the execution of a proof-structure can be recovered. An element of the relational interpretation of a proof-structure R with conclusion $\Gamma$ acts thus as a type (refining $\Gamma$) having R as an inhabitant. We are interested in the following type-checking question: given a proof-structure R, a list of formulae $\Gamma$, and a point x in the relational interpretation of $\Gamma$, is x in the interpretation of R? This question is decidable. We present here an algorithm that decides it in time linear in the size of R, if R is a proof-structure in the multiplicative fragment of linear logic. This algorithm can be extended to larger fragments of multiplicative-exponential linear logic containing $\lambda$-calculus

Mathematical Foundations for a Compositional Distributional Model of Meaning

We propose a mathematical framework for a unification of the distributional theory of meaning in terms of vector space models, and a compositional theory for grammatical types, for which we rely on the algebra of Pregroups, introduced by Lambek. This mathematical framework enables us to compute the meaning of a well-typed sentence from the meanings of its constituents. Concretely, the type reductions of Pregroups are `lifted' to morphisms in a category, a procedure that transforms meanings of constituents into a meaning of the (well-typed) whole. Importantly, meanings of whole sentences live in a single space, independent of the grammatical structure of the sentence. Hence the inner-product can be used to compare meanings of arbitrary sentences, as it is for comparing the meanings of words in the distributional model. The mathematical structure we employ admits a purely diagrammatic calculus which exposes how the information flows between the words in a sentence in order to make up the meaning of the whole sentence. A variation of our `categorical model' which involves constraining the scalars of the vector spaces to the semiring of Booleans results in a Montague-style Boolean-valued semantics.Comment: to appea

Logics for Unranked Trees: An Overview

Labeled unranked trees are used as a model of XML documents, and logical languages for them have been studied actively over the past several years. Such logics have different purposes: some are better suited for extracting data, some for expressing navigational properties, and some make it easy to relate complex properties of trees to the existence of tree automata for those properties. Furthermore, logics differ significantly in their model-checking properties, their automata models, and their behavior on ordered and unordered trees. In this paper we present a survey of logics for unranked trees

Intensional and Extensional Semantics of Bounded and Unbounded Nondeterminism

We give extensional and intensional characterizations of nondeterministic functional programs: as structure preserving functions between biorders, and as nondeterministic sequential algorithms on ordered concrete data structures which compute them. A fundamental result establishes that the extensional and intensional representations of non-deterministic programs are equivalent, by showing how to construct a unique sequential algorithm which computes a given monotone and stable function, and describing the conditions on sequential algorithms which correspond to continuity with respect to each order. We illustrate by defining may and must-testing denotational semantics for a sequential functional language with bounded and unbounded choice operators. We prove that these are computationally adequate, despite the non-continuity of the must-testing semantics of unbounded nondeterminism. In the bounded case, we prove that our continuous models are fully abstract with respect to may and must-testing by identifying a simple universal type, which may also form the basis for models of the untyped lambda-calculus. In the unbounded case we observe that our model contains computable functions which are not denoted by terms, by identifying a further "weak continuity" property of the definable elements, and use this to establish that it is not fully abstract