Probabilistic Mu-Calculus: Decidability and Complete Axiomatization

We introduce a version of the probabilistic mu-calculus (PMC) built on top of a probabilistic modal logic that allows encoding n-ary inequational conditions on transition probabilities. PMC extends previously studied calculi and we prove that, despite its expressiveness, it enjoys a series of good meta-properties. Firstly, we prove the decidability of satisfiability checking by establishing the small model property. An algorithm for deciding the satisfiability problem is developed. As a second major result, we provide a complete axiomatization for the alternation-free fragment of PMC. The completeness proof is innovative in many aspects combining various techniques from topology and model theory

Alternation-free weighted mu-calculus : decidability and completeness

In this paper we introduce WMC, a weighted version of the alternation-free modal mu-calculus for weighted transition systems. WMC subsumes previously studied weighted extensions of CTL and resembles previously proposed time-extended versions of the modal mu-calculus. We develop, in addition, a symbolic semantics for WMC and demonstrate that the notion of satisfiability coincides with that of symbolic satisfiability. This central result allows us to prove two major meta-properties of WMC. The first is decidability of satisfiability for WMC. In contrast to the classical modal mu-calculus, WMC does not possess the finite model-property. Nevertheless, the finite model property holds for the symbolic semantics and decidability readily follows; and this contrasts to resembling logics for timed transitions systems for which satisfiability has been shown undecidable. As a second main contribution, we provide a complete axiomatization, which applies to both semantics. The completeness proof is non-standard, since the logic is non-compact, and it involves the notion of symbolic models

Adequacy and complete axiomatization for Timed Modal Logic

In this paper we develop the metatheory for Timed Modal Logic (TML), which is the modal logic used for the analysis of timed transition systems (TTSs). We solve a series of long-standing open problems related to TML. Firstly, we prove that TML enjoys the Hennessy-Milner property and solve one of the open questions in the field. Secondly, we prove that the set of validities are not recursively enumerable. Nevertheless, we develop a strongly-complete proof system for TML. Since the logic is not compact, the proof system contains infinitary rules, but only with countable sets of instances. Thus, we can involve topological results regarding Stone spaces, such as the Rasiowa-Sikorski lemma, to complete the proofs

decomposition of automata pdl and its extension

In this work we study the decomposition problem of automata PDL and one of its extension. We proved that automata PDL enjoys a good decomposition property under a very large class of process context, while this problem is more complicated for regular PDL, which has an exponential blow-up. We introduce proposition identifiers to automata PDL to obtain a language which has a good balance between expressiveness and ease of analysis. We prove that this extended specification language still has a good decomposition property for a large class of process contexts. After that we present a method to solve the weak bisimulation equations as an application of the extended language, by combining the decomposition property and the decision procedure proposed in [1].IAENG Society of Artificial Intelligence; IAENG Society of Bioinformatics; IAENG Society of Computer Science; IAENG Society of Data Mining; IAENG Society of Electrical EngineeringIn this work we study the decomposition problem of automata PDL and one of its extension. We proved that automata PDL enjoys a good decomposition property under a very large class of process context, while this problem is more complicated for regular PDL, which has an exponential blow-up. We introduce proposition identifiers to automata PDL to obtain a language which has a good balance between expressiveness and ease of analysis. We prove that this extended specification language still has a good decomposition property for a large class of process contexts. After that we present a method to solve the weak bisimulation equations as an application of the extended language, by combining the decomposition property and the decision procedure proposed in [1]

specification in pdl with recursion

By extending regular Propositional Dynamic Logic (PDL) with simple recursive propositions, we obtain a language which has enough expressiveness to allow interesting applications while still enjoying a relatively simple decision procedure. More specifically, it is strictly more expressive than the regular PDL and not more expressive than the single alternation fragment of the modal μ-calculus. We present a decision procedure for satisfiability of a large class of so called simple formulas. The decision procedure has a time complexity which is polynomial in the size of the programs and exponential in the number of the sub-formulas. We show a way to solve process equations of weak bisimulation as an application. © 2012 Springer-Verlag.By extending regular Propositional Dynamic Logic (PDL) with simple recursive propositions, we obtain a language which has enough expressiveness to allow interesting applications while still enjoying a relatively simple decision procedure. More specifically, it is strictly more expressive than the regular PDL and not more expressive than the single alternation fragment of the modal μ-calculus. We present a decision procedure for satisfiability of a large class of so called simple formulas. The decision procedure has a time complexity which is polynomial in the size of the programs and exponential in the number of the sub-formulas. We show a way to solve process equations of weak bisimulation as an application. © 2012 Springer-Verlag

WNetKAT:A Weighted SDN Programming and Verification Language

Programmability and verifiability lie at the heart of the software-defined networking paradigm. While OpenFlow and its match-action concept provide primitive operations to manipulate hardware configurations, over the last years, several more expressive network programming languages have been developed. This paper presents WNetKAT, the first network programming language accounting for the fact that networks are inherently weighted, and communications subject to capacity constraints (e.g., in terms of bandwidth) and costs (e.g., latency or monetary costs). WNetKAT is based on a syntactic and semantic extension of the NetKAT algebra. We demonstrate several relevant applications for WNetKAT, including cost- and capacity-aware reachability, as well as quality-of-service and fairness aspects. These applications do not only apply to classic, splittable and unsplittable (s; t)-flows, but also generalize to more complex network functions and service chains. For example, WNetKAT allows to model flows which need to traverse certain waypoint functions, which may change the traffic rate. This paper also shows the relation between the equivalence problem of WNetKAT and the equivalence problem of the weighted finite automata, which implies undecidability of the former. However, this paper also succeeds to prove the decidability of another useful problem, which is sufficient in many practical scnearios: whether an expression equals to 0. Moreover, we initiate the discussion of decidable subsets of the whole language

Concurrent weighted logic

We introduce Concurrent Weighted Logic (CWL), a multimodal logic for concurrent labeled weighted transition systems (LWSs). The synchronization of LWSs is de-scribed using dedicated functions that, in various concurrency paradigms, allow us to encode the compositionality of LWSs. To reflect these, CWL contains modal operators indexed with rational numbers to predicate over the numerical labels of LWSs as well as a binary modal operator that encodes properties concerning the (de-) composition of LWSs. We develop a Hilbert-style axiomatic system for CWL and we prove weak-and strong-completeness results for this logic. To complete these proofs we involve advanced topological techniques from Model Theory

Decidability and Expressiveness of Recursive Weighted Logic

Abstract. Labelled weighted transition systems (LWSs) are transition systems labelled with actions and real numbers. The numbers represent the costs of the corresponding actions in terms of resources. Recursive Weighted Logic (RWL) is a multimodal logic that expresses qualitative and quantitative properties of LWSs. It is endowed with simultaneous recursive equations, which specify the weakest properties satisfied by the recursive variables. We demonstrate that RWL is suffi-ciently expressive to characterize weighted-bisimilarity of LWSs. In addition, we prove that the logic is decidable, i.e., the satisfiability problem for RWL can be algorithmically solved