L. Menasché Schechter

A Propositional Dynamic Logic for CCS Programs

This work presents a Propositional Dynamic Logic in which the programs are CCS terms (CCS-PDL). Its goal is to reason about properties of concurrent systems specified in CCS. CCS is a process algebra that models the concurrency and interaction between processes through individual acts of communication. At a first step, we consider only CCS processes without constants and give a complete axiomatization for this logic, which is very similar to *-free PDL. Then, we proceed to include CCS processes with constants. In this case, we impose some restrictions on the form of the recursive equations that can be built with those constants. We also give an axiomatization for this second logic and prove its completeness using a Fischer-Ladner construction. Unlike Concurrent PDL (with channels) [1,2], our logic has a simple Kripke semantics, a complete axiomatization and the finite model property.

Using modal logics to express and check global graph properties

Graphs are among the most frequently used structures in Computer Science. Some of the properties that must be checked in many applications are connectivity, acyclicity and the Eulerian and Hamiltonian properties. In this work, we analyze how we can express these four properties with modal logics. This involves two issues: whether each of the modal languages under consideration has enough expressive power to describe these properties and how complex (computationally) it is to use these logics to actually test whether a given graph has some desired property. First, we show that these properties are not definable in a basic modal logic or in any bisimulation-invariant extension of it, like the modal μ-calculus. We then show that it is possible to express some of the above properties in a basic hybrid logic. Unfortunately, the Hamiltonian and Eulerian properties still cannot be efficiently checked. In a second attempt, we propose an extension of CTL∗ with nominals and show that the Hamiltonian property can be more efficiently checked in this logic than in the previous one. In a third attempt, we extend the basic hybrid logic with the ↓ operator and show that we can check the Hamiltonian property with optimal (NP) complexity in this logic. Finally, we tackle the Eulerian property in two different ways. First, we develop a generic method to express edge-related properties in hybrid logics and use it to express the Eulerian property. Second, we express a necessary and sufficient condition for the Eulerian property to hold using a graded modal logic.

Algebraic solutions of holomorphic foliations: An algorithmic approach

We present two algorithms that can be used to check whether a given holomorphic foliation of the projective plane has an algebraic solution, and discuss the performance of their implementations in the computer algebra system Singular.

A Propositional Dynamic Logic for Concurrent Programs Based on the π-Calculus

This work presents a Propositional Dynamic Logic (@pDL) in which the programs are described in a language based on the @p-Calculus without replication. Our goal is to build a dynamic logic that is suitable for the description and verification of properties of communicating concurrent systems, in a similar way as PDL is used for the sequential case. We build a simple Kripke semantics for this logic, provide a complete axiomatization for it and show that it has the finite model property.

GGH may not be dead after all

In 1997, Goldreich, Goldwasser and Halevi presented the GGH cryptosystem, which is based on hard lattice problems. Only two years later, Nguyen pointed out major flaws on the scheme. From that point on, the system was considered officially dead. However, in 2012, Yoshino and Kunihiro proposed some improvements on the GGH cryptosystem, claiming to have fixed the flaws pointed out by Nguyen. In this paper, we make a thorough analysis of this tweaked GGH scheme, showing that, in practice, it behaves mostly in the same way as the original scheme. We also propose some modifications that can effectively make the new GGH different from the original one.

A Logic of Plausible Justifications

In this work, we combine the frameworks of Justification Logics and Logics of Plausibility-Based Beliefs to build a logic for Multi-Agent Systems where each agent can explicitly state his justification for believing in a given sentence. Our logic is a normal modal logic based on the standard Kripke semantics, where we provide a semantic definition for the evidence terms and define the notion of plausible evidence for an agent, based on plausibility relations in the model. This way, unlike traditional Justification Logics, justifications can be actually faulty and unreliable. In our logic, agents can disagree not only over whether a sentence is true or false, but also on whether some evidence is a valid justification for a sentence or not. After defining our logic and its semantics, we provide a strongly complete axiomatic system for it and show that it has the finite model property and is decidable. Thus, this logic seems to be a good first step for the development of a dynamic logic that can model the processes of argumentation and debate in multi-agent systems.

Modal Expressiveness of Graph Properties

Graphs are among the most frequently used structures in Computer Science. In this work, we analyze how we can express some important graph properties such as connectivity, acyclicity and the Eulerian and Hamiltonian properties in a modal logic. First, we show that these graph properties are not definable in a basic modal language. Second, we discuss an extension of the basic modal language with fix-point operators, the modal @m-calculus. Unfortunately, even with all its expressive power, the @m-calculus fails to express these properties. This happens because @m-calculus formulas are invariant under bisimulations. Third, we show that it is possible to express some of the above properties in a basic hybrid logic. Fourth, we propose an extension of CTL* with nominals, that we call hybrid-CTL*, and then show that it can express the Hamiltonian property in a better way than the basic hybrid logic. Finally, we introduce a promising way of expressing properties related to edges and use it to express the Eulerian property.

A logic of plausible justifications

In this work, we combine features from Justification Logics and Logics of Plausibility-Based Beliefs to build a logic for Multi-Agent Systems where each agent can explicitly state his justification for believing in a given sentence. Our logic is a normal modal logic based on the standard Kripke semantics, where we provide a semantic definition for the evidence terms and define the notion of plausible evidence for an agent, based on plausibility relations in the model. As we deal with beliefs, justifications can be faulty and unreliable. In our logic, agents can disagree not only over whether a sentence is true or false, but also on whether some evidence is a valid justification for a sentence or not. After defining our logic and its semantics, we provide a strongly complete axiomatic system for it, show that it has the finite model property, analyze the complexity of its Model-Checking Problem and show that its Satisfiability Problem has the same complexity as the one from basic modal logics. Thus, this logic seems to be a good first step for the development of a dynamic logic that can model the processes of argumentation and debate in Multi-Agent Systems.

Propositional dynamic logics for communicating concurrent programs with CCS's parallel operator

This work presents three increasingly expressive Dynamic Logics in which the programs are described in a language based on CCS. Our goal is to build dynamic logics that are suitable for the description and verification of properties of communicating concurrent systems, in a similar way as PDL is used for the sequential case. In order to accomplish that, CCS’s operators and constructions are added to a basic modal logic. Doing this, the semantics of CCS’s parallel operator allows us to build dynamic logics that support communicating and concurrent programs. We build a simple Kripke semantics for these logics, provide complete axiomatizations for them and show that they have the finite model property. This contrasts with other dynamic logics with parallel operators presented in the literature, such as Peleg’s Concurrent PDL with Channels, where either the parallel programs cannot communicate, or at least one of the properties mentioned above (simple Kripke semantics, complete axiomatization and finite model property) is missing.

Product of Graphs and Hybrid Logic

Left and right commutativity and the Church-Rosser and reverse Church-Rosser properties are necessary conditions for a graph (frame) to be a (non-trivial) product of two other graphs, but their conjunction is not a sufficient condition. This work presents a fifth property, called H-V intransitivity, that, when added to the four previous properties, results in a necessary and sufficient condition for a finite and connected graph to be a product. Then, we show that although the first four properties can be defined in a modal logic (the reverse Church-Rosser property requires a converse modality), H-V intransitivity is not modally definable. We also show that no necessary and sufficient condition for a graph to be a product can be modally definable. Finally, we present a formula in a hybrid language that defines H-V intransitivity.