The Complexity of Model Checking Higher-Order Fixpoint Logic

Higher-Order Fixpoint Logic (HFL) is a hybrid of the simply typed \lambda-calculus and the modal \lambda-calculus. This makes it a highly expressive temporal logic that is capable of expressing various interesting correctness properties of programs that are not expressible in the modal \lambda-calculus. This paper provides complexity results for its model checking problem. In particular we consider those fragments of HFL built by using only types of bounded order k and arity m. We establish k-fold exponential time completeness for model checking each such fragment. For the upper bound we use fixpoint elimination to obtain reachability games that are singly-exponential in the size of the formula and k-fold exponential in the size of the underlying transition system. These games can be solved in deterministic linear time. As a simple consequence, we obtain an exponential time upper bound on the expression complexity of each such fragment. The lower bound is established by a reduction from the word problem for alternating (k-1)-fold exponential space bounded Turing Machines. Since there are fixed machines of that type whose word problems are already hard with respect to k-fold exponential time, we obtain, as a corollary, k-fold exponential time completeness for the data complexity of our fragments of HFL, provided m exceeds 3. This also yields a hierarchy result in expressive power.Comment: 33 pages, 2 figures, to be published in Logical Methods in Computer Scienc

Verification of Non-Regular Program Properties

Most temporal logics which have been introduced and studied in the past decades can be embedded into the modal mu-calculus. This is the case for e.g. PDL, CTL, CTL*, ECTL, LTL, etc. and entails that these logics cannot express non-regular program properties. In recent years, some novel approaches towards an increase in expressive power have been made: Fixpoint Logic with Chop enriches the mu-calculus with a sequential composition operator and thereby allows to characterise context-free processes. The Modal Iteration Calculus uses inflationary fixpoints to exceed the expressive power of the mu-calculus. Higher-Order Fixpoint Logic (HFL) incorporates a simply typed lambda-calculus 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 context-free 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 logic-related 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 non-regular 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 context-sensitive 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 UNTIL-operator 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 context-sensitive languages in the UNTIL or context-free languages in the RELEASE- operator renders the model checking problem undecidable while it is EXPTIME-complete for indexed languages in the UNTIL and visibly pushdown languages in the RELEASE- operator. PTIME-completeness is a result of allowing linear indexed languages in the UNTIL and deterministic context-free 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 k-EXPTIME-complete for HFL(k), the fragment of HFL obtained by restricting functions in the lambda-calculus to order k. Novel in this work is however the generalisation (from the first-order 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 first-order 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 model-theoretic properties for each of these logics and compare them with respect to expressive power

Proposed protective mechanism of the pancreas in the rat

<p>Abstract</p> <p>Background</p> <p>Heparan sulphate is known to have various functions in the animal body, including surveillance of tissue integrity. Administered intraperitoneally, it induces a systemic inflammatory response syndrome and when given locally in the pancreas it initiates a protective inflammatory response. The aim of the present study was to investigate the underlying mechanisms behind cell recruitment following intra-ductal infusion of heparan sulphate.</p> <p>Methods</p> <p>Rats were subjected to intraductal-infusion of heparan sulphate, lipopolysaccharide and phosphate buffered saline into the pancreas. Pancreatic tissue was harvested 1, 3, 6, 9 or 48 hours after infusion and stained immunohistochemically for myeloperoxidase, ED-1, CINC-1 and MCP-1, as well as using eosin hematoxylin staining. Furthermore, MPO activity and MCP-1 and CINC-1 concentrations of tissue homogenates were measured. All differences were analyzed statistically using the Mann-Whitney U-test.</p> <p>Results</p> <p>During HS infusion, a rapid influx of macrophages/monocytes, as visualized as ED-1 positive cells, was seen reaching a maximum at 6 hours. After 48 hours, the same levels of ED-1 positive cells were noted in the pancreatic tissue, but with different location and morphology. Increased neutrophil numbers of heparan sulphate treated animals compared to control could be detected only 9 hours after infusion. The number of neutrophils was lower than the number of ED-1 positive cells. On the contrary, LPS infusion caused increased neutrophil numbers to a larger extent than heparan sulphate. Furthermore, this accumulation of neutrophils preceded the infiltration of ED-1 positive cells. Chemokine expression correlates very well to the cell infiltrate. MCP-1 was evident in the ductal cells of both groups early on. MCP-1 preceded monocyte infiltration in both groups, while the CINC-1 increase was only noticeable in the LPS group.</p> <p>Conclusions</p> <p>Our data suggest that heparan and LPS both induce host defense reactions, though by using different mechanisms of cell-recruitment. This implies that the etiology of pancreatic inflammation may influence how the subsequent events will develop.</p

Separation of Test-Free Propositional Dynamic Logics over Context-Free Languages

For a class L of languages let PDL[L] be an extension of Propositional Dynamic Logic which allows programs to be in a language of L rather than just to be regular. If L contains a non-regular language, PDL[L] can express non-regular properties, in contrast to pure PDL. For regular, visibly pushdown and deterministic context-free languages, the separation of the respective PDLs can be proven by automata-theoretic techniques. However, these techniques introduce non-determinism on the automata side. As non-determinism is also the difference between DCFL and CFL, these techniques seem to be inappropriate to separate PDL[DCFL] from PDL[CFL]. Nevertheless, this separation is shown but for programs without test operators.Comment: In Proceedings GandALF 2011, arXiv:1106.081

Reversible Induction of Phantom Auditory Sensations through Simulated Unilateral Hearing Loss

Tinnitus, a phantom auditory sensation, is associated with hearing loss in most cases, but it is unclear if hearing loss causes tinnitus. Phantom auditory sensations can be induced in normal hearing listeners when they experience severe auditory deprivation such as confinement in an anechoic chamber, which can be regarded as somewhat analogous to a profound bilateral hearing loss. As this condition is relatively uncommon among tinnitus patients, induction of phantom sounds by a lesser degree of auditory deprivation could advance our understanding of the mechanisms of tinnitus. In this study, we therefore investigated the reporting of phantom sounds after continuous use of an earplug. 18 healthy volunteers with normal hearing wore a silicone earplug continuously in one ear for 7 days. The attenuation provided by the earplugs simulated a mild high-frequency hearing loss, mean attenuation increased from <10 dB at 0.25 kHz to >30 dB at 3 and 4 kHz. 14 out of 18 participants reported phantom sounds during earplug use. 11 participants presented with stable phantom sounds on day 7 and underwent tinnitus spectrum characterization with the earplug still in place. The spectra showed that the phantom sounds were perceived predominantly as high-pitched, corresponding to the frequency range most affected by the earplug. In all cases, the auditory phantom disappeared when the earplug was removed, indicating a causal relation between auditory deprivation and phantom sounds. This relation matches the predictions of our computational model of tinnitus development, which proposes a possible mechanism by which a stabilization of neuronal activity through homeostatic plasticity in the central auditory system could lead to the development of a neuronal correlate of tinnitus when auditory nerve activity is reduced due to the earplug