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

A Cyclic Proof System for HFL_?

A cyclic proof system allows us to perform inductive reasoning without explicit inductions. We propose a cyclic proof system for HFLN, which is a higher-order predicate logic with natural numbers and alternating fixed-points. Ours is the first cyclic proof system for a higher-order logic, to our knowledge. Due to the presence of higher-order predicates and alternating fixed-points, our cyclic proof system requires a more delicate global condition on cyclic proofs than the original system of Brotherston and Simpson. We prove the decidability of checking the global condition and soundness of this system, and also prove a restricted form of standard completeness for an infinitary variant of our cyclic proof system. A potential application of our cyclic proof system is semi-automated verification of higher-order programs, based on Kobayashi et al.'s recent work on reductions from program verification to HFLN validity checking.Comment: 27 page

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

Temporal Logic with Recursion

We introduce extensions of the standard temporal logics CTL and LTL with a recursion operator that takes propositional arguments. Unlike other proposals for modal fixpoint logics of high expressive power, we obtain logics that retain some of the appealing pragmatic advantages of CTL and LTL, yet have expressive power beyond that of the modal ?-calculus or MSO. We advocate these logics by showing how the recursion operator can be used to express interesting non-regular properties. We also study decidability and complexity issues of the standard decision problems

Coherent Imaging Spectroscopy of a Quantum Many-Body Spin System

Quantum simulators, in which well controlled quantum systems are used to reproduce the dynamics of less understood ones, have the potential to explore physics that is inaccessible to modeling with classical computers. However, checking the results of such simulations will also become classically intractable as system sizes increase. In this work, we introduce and implement a coherent imaging spectroscopic technique to validate a quantum simulation, much as magnetic resonance imaging exposes structure in condensed matter. We use this method to determine the energy levels and interaction strengths of a fully-connected quantum many-body system. Additionally, we directly measure the size of the critical energy gap near a quantum phase transition. We expect this general technique to become an important verification tool for quantum simulators once experiments advance beyond proof-of-principle demonstrations and exceed the resources of conventional computers

Model Checking Timed Recursive CTL

We introduce Timed Recursive CTL, a merger of two extensions of the well-known branching-time logic CTL: Timed CTL is interpreted over real-time systems like timed automata; Recursive CTL introduces a powerful recursion operator which takes the expressiveness of this logic CTL well beyond that of regular properties. The result is an expressive logic for real-time properties. We show that its model checking problem is decidable over timed automata, namely 2-EXPTIME-complete

The Early Bird Catches The Term: Combining Twitter and News Data For Event Detection and Situational Awareness

Twitter updates now represent an enormous stream of information originating from a wide variety of formal and informal sources, much of which is relevant to real-world events. In this paper we adapt existing bio-surveillance algorithms to detect localised spikes in Twitter activity corresponding to real events with a high level of confidence. We then develop a methodology to automatically summarise these events, both by providing the tweets which fully describe the event and by linking to highly relevant news articles. We apply our methods to outbreaks of illness and events strongly affecting sentiment. In both case studies we are able to detect events verifiable by third party sources and produce high quality summaries