Petar Maksimovic

First steps towards probabilistic justification logic

In this article, we introduce the probabilistic justification logic PJ, a logic in which we can reason about the probability of justification statements. We present its syntax and semantics, and establish a strong completeness theorem. Moreover, we investigate the relationship between PJ and the logic of uncertain justifications.

LF P : a logical framework with external predicates

The LFP Framework is an extension of the Harper-Honsell-Plotkins Edinburgh Logical Framework LF with external predicates. This is accomplished by defining lock type constructors, which are a sort of diamond-modality constructors, releasing their argument under the condition that a possibly external predicate is satisfied on an appropriate typed judgement. Lock types are defined using the standard pattern of constructive type theory, i.e. via introduction, elimination, and equality rules. Using LFP, one can factor out the complexity of encoding specific features of logical systems which would otherwise be awkwardly encoded in LF, e.g. side-conditions in the application of rules in Modal Logics, and substructural rules, as in non-commutative Linear Logic. The idea of LFP is that these conditions need only to be specified, while their verification can be delegated to an external proof engine, in the style of the Poincaré Principle. We investigate and characterize the metatheoretical properties of the calculus underpinning LFP: strong normalization, confluence, and subject reduction. This latter property holds under the assumption that the predicates are well-behaved, i.e. closed under weakening, permutation, substitution, and reduction in the arguments.

A logic with conditional probability operators

This paper presents a sound and strongly complete axiomatization of the reasoning about linear combinations of conditional probabilities, including compar- ative statements. The developed logic is decidable, with a PSPACE containment for the decision procedure.

An open logical framework

The LFP Framework is an extension of the Harper-Honsell-Plotkins Edinburgh Logical Framework LF with external predicates, hence the name Open Logical Framework. This is accomplished by defining lock type constructors, which are a sort of Star-modality constructors, releasing their argument under the condition that a possibly external predicate is satisfied on an appropriate typed judgement. Lock types are defined using the standard pattern of constructive type theory, i.e. via introduction, elimination, and equality rules. Using LFP, one can factor out the complexity of encoding specific features of logical systems which would otherwise be awkwardly encoded in LF, e.g. side-conditions in the application of rules in Modal Logics, and sub-structural rules, as in non-commutative Linear Logic. The idea of LFP is that these conditions need only to be specified, while their verification can be delegated to an external proof engine, in the style of the Poincare Principle or Deduction Modulo. Indeed such paradigms can be adequately formalized in LFP. We investigate and characterize the meta-theoretical properties of the calculus underpinning LFP : strong normalization, confluence, and subject reduction. This latter property holds under the assumption that the predicates are well-behaved, i.e. closed under weakening, permutation, substitution, and reduction in the arguments. Moreover, we provide a canonical presentation of LFP, based on a suitable extension of the notion of βη-long normal form, allowing for smooth formulations of adequacy statements.

A backward-compatible protocol for inter-routing over heterogeneous overlay networks

Overlay networks are logical networks running on the highest level of the OSI stack: they are applicative networks used by millions of users everyday. In many scenarios, it would be desirable for peers belonging to overlays running different protocols to communicate with each other and exchange certain information. However, due to differences in their respective protocols, this communication is often difficult or even impossible to be achieved efficiently, even if the overlays are sharing common objectives and functionalities. In this paper, we address this problem by presenting a new overlay protocol, called OGP (Overlay Gateway Protocol), allowing different existing networks to route messages between each other in a backward-compatible fashion, by making use of specialized peers joined together into a super-overlay. Experimental results on a large scale Grid5000 infrastructure show that having only a small number of nodes running the OGP protocol is sufficient for achieving efficient routing between heterogeneous overlay networks.

An extension and cooperation mechanism for heterogeneous overlay networks

In real-world peer-to-peer applications, the scalability of data lookup is heavily affected by network artifacts. A common solution to improve scalability, robustness and security is to increase the local properties of nodes, by clustering them together. This paper presents a framework which allows for the development of distributed applications on top of interconnected overlay network. Here, message routing between overlays is accomplished by using co-located nodes, i.e. nodes belonging to more than one overlay network at the same time. These co-located nodes serve as distributed gateways, enabling the routing of requests across overlays, while keeping overlay maintenance operations local. The protocol has been evaluated via simulations and client deployment, showing that the ability, of reaching the totality of the overlays in a federated configuration can be preserved even with the simplest routing, proving the feasibility of federated overlay configurations.

HOCore in Coq

We consider a recent publication on higher-order process calculi [12] and describe how its main results have been formalized in the Coq proof assistant. We highlight a number of important technical issues that we have uncovered in the original publication. We believe that these issues are not unique to the paper under consideration and require particular care to be avoided.

Analyzing the exhaustiveness of the synapse protocol

The Synapse protocol is a scalable protocol designed for information retrieval over inter-connected heterogeneous overlay networks. In this paper, we give a formal description of Synapse using the Abstract State Machines framework. The formal description pertains to Synapse actions that manipulate distributed keys. Based on this formal description, we present results concerning the expected exhaustiveness for a number of scenarios and systems maintained by the Synapse protocol, and provide comparisons to the results of the corresponding simulations and experiments. We show that the predicted theoretical results match the obtained experimental results, and give recommendations on the design of systems using Synapse.

Gluing together Proof Environments: Canonical extensions of LF Type Theories featuring Locks

We present two extensions of the LF Constructive Type Theory featuring monadic locks. A lock is a monadic type construct that captures the effect of an external call to an oracle. Such calls are the basic tool for gluing together diverse Type Theories and proof development environments. The oracle can either be invoked in order to check that a constraint holds or to provide a suitable witness. The systems are presented in the canonical style developed by the CMU School. The first system, CLLF_P , is the canonical version of the system LLF_P , presented earlier by the authors. The second system, CLLF_P? , features the possibility of invoking the oracle to obtain a witness satisfying a given constraint. We discuss encodings of Fitch-Prawitz Set theory, call-by-value λ-calculi, and systems of Light Linear Logic. Finally, we show how to use Fitch-Prawitz Set Theory to define a type system that types precisely the strongly normalizing terms.