Axelsson, Roland (2010): Verification of NonRegular Program Properties. Dissertation, LMU München: Faculty of Mathematics, Computer Science and Statistics 

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Abstract
Most temporal logics which have been introduced and studied in the past decades can be embedded into the modal mucalculus. This is the case for e.g. PDL, CTL, CTL*, ECTL, LTL, etc. and entails that these logics cannot express nonregular program properties. In recent years, some novel approaches towards an increase in expressive power have been made: Fixpoint Logic with Chop enriches the mucalculus with a sequential composition operator and thereby allows to characterise contextfree processes. The Modal Iteration Calculus uses inflationary fixpoints to exceed the expressive power of the mucalculus. HigherOrder Fixpoint Logic (HFL) incorporates a simply typed lambdacalculus into a setting with extremal fixpoint operators and even exceeds the expressive power of Fixpoint Logic with Chop. But also PDL has been equipped with contextfree programs instead of regular ones. In terms of expressivity there is a natural demand for richer frameworks since program property specifications are simply not limited to the regular sphere. Expressivity however usually comes at the price of an increased computational complexity of logicrelated decision problems. For instance are the satisfiability problems for the above mentioned logics undecidable. We investigate in this work the model checking problem of three different logics which are capable of expressing nonregular program properties and aim at identifying fragments with feasible model checking complexity. Firstly, we develop a generic method for determining the complexity of model checking PDL over arbitrary classes of programs and show that the border to undecidability runs between PDL over indexed languages and PDL over contextsensitive languages. It is however still in PTIME for PDL over linear indexed languages and in EXPTIME for PDL over indexed languages. We present concrete algorithms which allow implementations of model checkers for these two fragments. We then introduce an extension of CTL in which the UNTIL and RELEASE operators are adorned with formal languages. These are interpreted over labeled paths and restrict the moments on such a path at which the operators are satisfied. The UNTILoperator is for instance satisfied if some path prefix forms a word in the language it is adorned with (besides the usual requirement that until that moment some property has to hold and at that very moment some other property must hold). Again, we determine the computational complexities of the model checking problems for varying classes of allowed languages in either operator. It turns out that either enabling contextsensitive languages in the UNTIL or contextfree languages in the RELEASE operator renders the model checking problem undecidable while it is EXPTIMEcomplete for indexed languages in the UNTIL and visibly pushdown languages in the RELEASE operator. PTIMEcompleteness is a result of allowing linear indexed languages in the UNTIL and deterministic contextfree languages in the RELEASE. We do also give concrete model checking algorithms for several interesting fragments of these logics. Finally, we turn our attention to the model checking problem of HFL which we have already studied in previous works. On finite state models it is kEXPTIMEcomplete for HFL(k), the fragment of HFL obtained by restricting functions in the lambdacalculus to order k. Novel in this work is however the generalisation (from the firstorder case to the case for functions of arbitrary order) of an idea to improve the best and average case behaviour of a model checking algorithm by using partial functions during the fixpoint iteration guided by the neededness of arguments. This is possible, because the semantics of a closed HFL formula is not a total function but the value of a function at some argument. Again, we give a concrete algorithm for such an improved model checker and argue that despite the very high model checking complexity this improvement is very useful in practice and gives feasible results for HFL with lower order fuctions, backed up by a statistical analysis of the number of needed arguments on a concrete example. Furthermore, we show how HFL can be used as a tool for the development of algorithms. Its high expressivity allows to encode a wide variety of problems as instances of model checking already in the firstorder fragment. The rather unintuitive  yet very succinct  problem encoding together with an analysis of the behaviour of the above sketched optimisation may give deep insights into the problem. We demonstrate this on the example of the universality problem for nondeterministic finite automata, where a slight variation of the optimised model checking algorithm yields one of the best known methods so far which was only discovered recently. We do also investigate typical modeltheoretic properties for each of these logics and compare them with respect to expressive power.
Item Type:  Thesis (Dissertation, LMU Munich) 

Keywords:  modelchecking, program verification, nonregular logic, pdl, ctl, hfl 
Subjects:  600 Natural sciences and mathematics > 510 Mathematics 600 Natural sciences and mathematics 
Faculties:  Faculty of Mathematics, Computer Science and Statistics 
Language:  English 
Date Accepted:  25. June 2010 
1. Referee:  Lange, Martin 
Persistent Identifier (URN):  urn:nbn:de:bvb:19116775 
MD5 Checksum of the PDFfile:  7500acce9b432b6443f1f02f1565f7cb 
Signature of the printed copy:  0001/UMC 18811 
ID Code:  11677 
Deposited On:  31. Aug 2010 12:55 
Last Modified:  16. Oct 2012 08:39 