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# Model checking

In computer science, model checking refers to the following problem: Given a model of a system, test automatically whether this model meets a given specification. Typically, the systems one has in mind are hardware or software systems, and the specification contains safety requirements such as the absence of deadlocks and similar critical states that can cause the system to crash. Model checking is a technique for automatically verifying correctness properties of finite-state systems.

In order to solve such a problem algorithmically, both the model of the system and the specification are formulated in some precise mathematical language: To this end, it is formulated as a task in logic, namely to check whether a given structure satisfies a given logical formula. The concept is general and applies to all kinds of logics and suitable structures. A simple model-checking problem is verifying whether a given formula in the propositional logic is satisfied by a given structure.

## Overview

An important class of model checking methods have been developed for checking models of hardware and software designs where the specification is given by a temporal logic formula. Pioneering work in the model checking of temporal logic formulae was done by E. M. Clarke and E. A. Emerson and by J. P. Queille and J. Sifakis. Clarke, Emerson, and Sifakis shared the 2007 Turing Award for their work on model checking.

Model checking is most often applied to hardware designs. For software, because of undecidability (see computability theory) the approach cannot be fully algorithmic; typically it may fail to prove or disprove a given property.

The structure is usually given as a source code description in an industrial hardware description language or a special-purpose language. Such a program corresponds to a finite state machine (FSM), i.e., a directed graph consisting of nodes (or vertices) and edges. A set of atomic propositions is associated with each node, typically stating which memory elements are one. The nodes represent states of a system, the edges represent possible transitions which may alter the state, while the atomic propositions represent the basic properties that hold at a point of execution.

Formally, the problem can be stated as follows: given a desired property, expressed as a temporal logic formula p, and a structure M with initial state s, decide if $M,s \models p$. If M is finite, as it is in hardware, model checking reduces to a graph search.

## Algorithms

state space enumeration, symbolic state space enumeration, abstract interpretation, symbolic simulation, symbolic trajectory evaluation, symbolic execution

## Tools

Model checking tools face a combinatorial blow up of the state-space, commonly known as the state explosion problem, that must be addressed to solve most real-world problems. There are several approaches to combat this problem.

1. Symbolic algorithms avoid ever building the graph for the FSM; instead, they represent the graph implicitly using a formula in quantified propositional logic. The use of binary decision diagrams (BDDs) was made popular by the work of Ken McMillan.
2. Bounded model checking algorithms unroll the FSM for a fixed number of steps k and check whether a property violation can occur in k or fewer steps. This typically involves encoding the restricted model as an instance of SAT. The process can be repeated with larger and larger values of k until all possible violations have been ruled out (cf. Iterative deepening depth-first search).
3. Partial order reduction can be used (on explicitly represented graphs) to reduce the number of independent interleavings of concurrent processes that need to be considered. The basic idea is that if it does not matter, for the kind of things one intends to prove, whether A or B is executed first, then it is a waste of time to consider both the AB and the BA interleavings.
4. Abstraction attempts to prove properties on a system by first simplifying it. The simplified system usually does not satisfy exactly the same properties as the original one so that a process of refinement may be necessary. Generally, one requires the abstraction to be sound (the properties proved on the abstraction are true of the original system); however, most often, the abstraction is not complete (not all true properties of the original system are true of the abstraction). An example of abstraction is, on a program, to ignore the values of non boolean variables and to only consider boolean variables and the control flow of the program; such an abstraction, though it may appear coarse, may in fact be sufficient to prove e.g. properties of mutual exclusion.
5. Counter-example guided abstraction refinement (CEGAR) begins checking with a coarse (imprecise) abstraction and iteratively refines it. When a violation (counter-example) is found, the tool analyzes it for feasibility (i.e., is the violation genuine or the result of an incomplete abstraction?). If the violation is feasible, it is reported to the user; if it is not, the proof of infeasibility is used to refine the abstraction and checking begins again.

Model checking tools were initially developed to reason about the logical correctness of discrete state systems, but have since been extended to deal with real-time and limited forms of hybrid systems.

## Tools

For a categorized list of tools see here.

• BLAST model checker
• CADP (Construction and Analysis of Distributed Processes) a toolbox for the design of communication protocols and distributed systems
• CHESS model checker
• CHIC
• FDR2 a model checker for verifying real-time systems modeled and specified as CSP Processes
• ISP code level verifier for MPI programs
• Java Pathfinder - open source model checker for Java programs
• Markov Reward Model Checker (MRMC)
• mCRL2 Toolset, Boost Software License, Based on ACP
• MoonWalker - open source model checker for .NET programs
• NuSMV, a new symbolic model checker
• ompca, an interactive symbolic simulator with API control for C/C++ programs with OpenMP directives. The tool is built as an application of REDLIB.
• PAT - an enhanced simulator, model checker and refinement checker for concurrent and real-time systems
• Prism, a probabilistic symbolic model checker
• Rabbit
• REDLIB, library for the model-checking of communicating timed automatas with BDD-like diagrams. Applications include a TCTL model-checker with timed fairness quantifications, fair simulation checker, and interactive symbolic simulator for C/C++ programs with OpenMP directives. GUI for model editing and symbolic simulation are also available.
• SMART Model checker , Symbolic Model checking Analyzer for Reliability and Timing
• SPIN model checker a general tool for verifying the correctness of distributed software models in a rigorous and mostly automated fashion.
• TAPAs: tool for the analysis of process algebra.
• Vereofy, a software model checker for component-based systems for operational correctness
• μCRL, GPL, Based on ACP
• UPPAAL an integrated tool environment for modeling, validation and verification of real-time systems modeled as networks of timed automata
• Roméo an integrated tool environment for modeling, simulation and verification of real-time systems modeled as parametric, time and stopwatch Petri nets
• TLA+ model checker by Leslie Lamport
• AlPiNA, AlPiNA stands for Algebraic Petri Nets Analyzer and is a model checker for Algebraic Petri Nets.
Related techniques
History

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