8 research outputs found
Finite-State Abstractions for Probabilistic Computation Tree Logic
Probabilistic Computation Tree Logic (PCTL) is the established temporal
logic for probabilistic verification of discrete-time Markov chains. Probabilistic
model checking is a technique that verifies or refutes whether a property
specified in this logic holds in a Markov chain. But Markov chains are often
infinite or too large for this technique to apply. A standard solution to
this problem is to convert the Markov chain to an abstract model and to
model check that abstract model. The problem this thesis therefore studies
is whether or when such finite abstractions of Markov chains for model
checking PCTL exist.
This thesis makes the following contributions. We identify a sizeable fragment
of PCTL for which 3-valued Markov chains can serve as finite abstractions;
this fragment is maximal for those abstractions and subsumes many
practically relevant specifications including, e.g., reachability. We also develop
game-theoretic foundations for the semantics of PCTL over Markov
chains by capturing the standard PCTL semantics via a two-player games.
These games, finally, inspire a notion of p-automata, which accept entire
Markov chains. We show that p-automata subsume PCTL and Markov
chains; that their languages of Markov chains have pleasant closure properties;
and that the complexity of deciding acceptance matches that of probabilistic
model checking for p-automata representing PCTL formulae. In addition,
we offer a simulation between p-automata that under-approximates
language containment. These results then allow us to show that p-automata
comprise a solution to the problem studied in this thesis
Model-Checking Over Multi-Valued Logics
Classical logic cannot be used to effectively reason about systems with uncertainty (lack of essential information) or inconsistency (contradictory information often occurring when information is gathered from multiple sources). In this paper we propose the use of quasi-boolean multi-valued logics for reasoning about such systems. We also give semantics to a multi-valued extension of CTL, describe an implementation of a symbolic multi-valued CTL model-checker called chek, and analyze its correctness and running time
Data Structures for Symbolic Multi-Valued Model-Checking
Multi-valued logics can be effectively used to reason about incomplete and/or inconsistent systems, e.g. during early software requirements or as the systems evolve. In our earlier work we identified a useful family of multi-valued logics: those specified over finite distributive lattices where negation preserves involution, i.e., �¦������ � for every element � of the logic. Model-checking over this family of logics allows not only to extend the domain of applicability of automated reasoning to new problems, but also to speed up some classical verification problems. Symbolic model-checking over multi-valued logics can be cast in terms of operations over multivalued sets: sets whose membership functions are multi-valued. In this paper we propose and empirically evaluate several choices for implementing multi-valued sets with decision diagrams. In particular, we describe two major approaches: (1) representing the multi-valued membership function canonically, using MDDs or ADDs; (2) representing multi-valued sets as a collection of classical sets, using a vector of either MBTDDs or BDDs. The naive implementation of (2) includes having a classical set for each value of the logic. We exploit a result of lattice theory to reduce the number of such sets that need to be represented. The major contribution of this paper is the evaluation of the different implementations of multivalued sets, done via a series of experiments and using several case studies.