Propositional Logics Complexity and the Sub-Formula Property

In 1979 Richard Statman proved, using proof-theory, that the purely implicational fragment of Intuitionistic Logic (M-imply) is PSPACE-complete. He showed a polynomially bounded translation from full Intuitionistic Propositional Logic into its implicational fragment. By the PSPACE-completeness of S4, proved by Ladner, and the Goedel translation from S4 into Intuitionistic Logic, the PSPACE- completeness of M-imply is drawn. The sub-formula principle for a deductive system for a logic L states that whenever F1,...,Fk proves A, there is a proof in which each formula occurrence is either a sub-formula of A or of some of Fi. In this work we extend Statman result and show that any propositional (possibly modal) structural logic satisfying a particular formulation of the sub-formula principle is in PSPACE. If the logic includes the minimal purely implicational logic then it is PSPACE-complete. As a consequence, EXPTIME-complete propositional logics, such as PDL and the common-knowledge epistemic logic with at least 2 agents satisfy this particular sub-formula principle, if and only if, PSPACE=EXPTIME. We also show how our technique can be used to prove that any finitely many-valued logic has the set of its tautologies in PSPACE.Comment: In Proceedings DCM 2014, arXiv:1504.0192

A dependent nominal type theory

Nominal abstract syntax is an approach to representing names and binding pioneered by Gabbay and Pitts. So far nominal techniques have mostly been studied using classical logic or model theory, not type theory. Nominal extensions to simple, dependent and ML-like polymorphic languages have been studied, but decidability and normalization results have only been established for simple nominal type theories. We present a LF-style dependent type theory extended with name-abstraction types, prove soundness and decidability of beta-eta-equivalence checking, discuss adequacy and canonical forms via an example, and discuss extensions such as dependently-typed recursion and induction principles

Mechanizing type environments in weak HOAS

We provide a paradigmatic case study, about the formalization of System F<:'s type language in the proof assistant Coq. Our approach relies on weak HOAS, for the sake of producing a readable and concise representation of the object language. Actually, we present and discuss two encoding strategies for typing environments which yield a remarkable influence on the whole formalization. Then, on the one hand we develop System F<:'s metatheory, on the other hand we address the equivalence of the two approaches internally to Coq

Behavioral equivalences for AbU: Verifying security and safety in distributed IoT systems

Attribute-based memory Updates ([Formula presented]in short) is an interaction mechanism recently introduced for adapting the Event-Condition-Action (ECA) programming paradigm to distributed reactive systems, such as autonomic and smart IoT device ensembles. In this model, an event (e.g., an input from a sensor, or a device state update) can trigger an ECA rule, whose execution can cause the state update of (possibly) many remote devices at once; the latter are selected “on the fly” by means of predicates over their state, without the need of a central coordinating entity. However, the combination of different [Formula presented]systems may yield unexpected interactions, e.g., when a new device is added to an existing secure system, potentially hindering the security of the whole ensemble of devices. This can be critical in the IoT, where smart devices are more and more pervasive in our daily life. In this paper, we consider the problem of ensuring security and safety requirements for [Formula presented]systems (and, in turn, for IoT devices). The first are a form of noninterference, as they correspond to avoid forbidden information flows (e.g., information flows violating confidentiality); while the second are a form of non-interaction, as they correspond to avoid unintended executions (e.g., leading to erroneous/unsafe states). In order to formally model these requirements, we introduce suitable behavioral equivalences for [Formula presented]. These equivalences are generalizations of hiding bisimilarity, i.e., a kind of weak bisimilarity where we can compare systems up to actions at different levels of security. Leveraging these behavioral equivalences, we propose (syntactic) sufficient conditions guaranteeing the requirements and, then, effective algorithms for statically verifying such conditions

A Natural Deduction Approach to Dynamic Logic

. Natural Deduction style presentations of program logics are useful in view of the implementation of such logics in interactive proof development environments, based on type theory, such as LEGO, Coq, etc. In fact, ND-style systems are the kind of systems which can take best advantage of the possibility of reasoning &quot;under assumptions&quot; o#ered by proof assistants generated by Logical Frameworks. In this paper we introduce and discuss sound and complete proof systems in Natural Deduction style for representing various &quot;truth&quot; consequence relations of Dynamic Logic. We discuss the design decisions which lead to adequate encodings of these logics in Coq. We derive in Dynamic Logic a set of rules representing a ND-style system for Hoare Logic