Axiomatizing Omega and Omega-op Powers of Words

In 1978, Courcelle asked for a complete set of axioms and rules for the equational theory of (discrete regular) words equipped with the operations of product, omega power and omega-op power. In this paper we find a simple set of equations and prove they are complete. Moreover, we show that the equational theory is decidable in polynomial time

Chromatic Number in Time O(2.4023^n) Using Maximal Independent Sets

In this paper we improve an algorithm by Eppstein (2001) for finding the chromatic number of a graph. We modify the algorithm slightly, and by using a bound on the number of maximal independent sets of size  k from our recent paper (2003), we prove that the running time is O(2.4023^n). Eppstein's algorithm runs in time O(2.4150^n). The space usage for both algorithms is O(2^n)

Checking Consistency of Pedigree Information is NP-complete (Preliminary Report)

Consistency checking is a fundamental computational problem in genetics. Given a pedigree and information on the genotypes of some of the individuals in it, the aim of consistency checking is to determine whether these data are consistent with the classic Mendelian laws of inheritance. This problem arose originally from the geneticists' need to filter their input data from erroneous information, and is well motivated from both a biological and a sociological viewpoint. This paper shows that consistency checking is NP-complete, even in the presence of three alleles. Several other results on the computational complexity of problems from genetics that are related to consistency checking are also offered. In particular, it is shown that checking the consistency of pedigrees over two alleles can be done in polynomial time

Extended Temporal Logic on Finite Words and Wreath Product of Monoids with Distinguished Generators

We associate a modal operator with each language belonging to a given class of regular languages and use the (reverse) wreath product of monoids with distinguished generators to characterize the expressive power of the resulting logic

Using IDDs for Packet Filtering

Firewalls are one of the key technologies used to control the traffic going in and out of a network. A central feature of the firewall is the packet filter. In this paper, we propose a complete framework for packet classification. Through two applications we demonstrate that both performance and security can be improved. We show that a traditional ordered rule set can always be expressed as a first-order logic formula on integer variables. Moreover, we emphasize that, with such specification, the packet filtering problem is known to be constant time. We propose to represent the first-order logic formula as Interval Decision Diagrams. This structure has several advantages. First, the algorithm for removing redundancy and unnecessary tests is very simple. Secondly, it allows us to handle integer variables which makes it efficient on a generic CPUs. And, finally, we introduce an extension of IDDs called Multi-Terminal Interval Decision Diagrams in order to deal with any number of policies. In matter of efficiency, we evaluate the performance our framework through a prototype toolkit composed by a compiler and a packet filter. The results of the experiments shows that this method is efficient in terms of CPU usage and has a low storage requirements. Finally, we outline a tool, called Network Access Verifier. This tool demonstrates how the IDD representation can be used for verifying access properties of a network. In total, potentially improving the security of a network

CCS with Hennessy's merge has no finite-equational axiomatization

Abstract This paper confirms a conjecture of Bergstra and Klop¿s from 1984 by establishing that the process algebra obtained by adding an auxiliary operator proposed by Hennessy in 1981 to the recursion free fragment of Milner¿s Calculus of Communicationg Systems is not finitely based modulo bisimulation equivalence. Thus Hennessy¿s merge cannot replace the left merge and communication merge operators proposed by Bergstra and Klop, at least if a finite axiomatization of parallel composition is desired. 2000 MATHEMATICS SUBJECT CLASSIFICATION: 08A70, 03B45, 03C05, 68Q10, 68Q45, 68Q55, 68Q70. CR SUBJECT CLASSIFICATION (1991): D.3.1, F.1.1, F.1.2, F.3.2, F.3.4, F.4.1. KEYWORDS AND PHRASES: Concurrency, process algebra, CCS, bisimulation, Hennessy¿s merge, left merge, communication merge, parallel composition, equational logic, complete axiomatizations, non-finitely based algebras

Greibach Normal Form in Algebraically Complete Semirings

We give inequational and equational axioms for semirings with a fixed-point operator and formally develop a fragment of the theory of context-free languages. In particular, we show that Greibach's normal form theorem depends only on a few equational properties of least pre-fixed-points in semirings, and elimination of chain- and deletion rules depend on their inequational properties (and the idempotency of addition). It follows that these normal form theorems also hold in non-continuous semirings having enough fixed-points

Computer Aided Verification

This open access two-volume set LNCS 13371 and 13372 constitutes the refereed proceedings of the 34rd International Conference on Computer Aided Verification, CAV 2022, which was held in Haifa, Israel, in August 2022. The 40 full papers presented together with 9 tool papers and 2 case studies were carefully reviewed and selected from 209 submissions. The papers were organized in the following topical sections: Part I: Invited papers; formal methods for probabilistic programs; formal methods for neural networks; software Verification and model checking; hyperproperties and security; formal methods for hardware, cyber-physical, and hybrid systems. Part II: Probabilistic techniques; automata and logic; deductive verification and decision procedures; machine learning; synthesis and concurrency. This is an open access book