Higher-Order Fixpoint Logic Research Articles

Higher-Order Property-Directed Reachability

The property-directed reachability (PDR) has been used as a successful method for automated verification of first-order transition systems. We propose a higher-order extension of PDR, called HoPDR, where higher-order recursive functions may be used to describe transition systems. We formalize HoPDR for the validity checking problem for conjunctive nu-HFL(Z), a higher-order fixpoint logic with integers and greatest fixpoint operators. The validity checking problem can also be viewed as a higher-order extension of the satisfiability problem for Constrained Horn Clauses (CHC), and safety property verification of higher-order programs can naturally be reduced to the validity checking problem. We have implemented a prototype verification tool based on HoPDR and confirmed its effectiveness. We also compare our HoPDR procedure with the PDR procedure for first-order systems and previous methods for fully automated higher-order program verification.

A Probabilistic Higher-order Fixpoint Logic

We introduce PHFL, a probabilistic extension of higher-order fixpoint logic, which can also be regarded as a higher-order extension of probabilistic temporal logics such as PCTL and the $\mu^p$-calculus. We show that PHFL is strictly more expressive than the $\mu^p$-calculus, and that the PHFL model-checking problem for finite Markov chains is undecidable even for the $\mu$-only, order-1 fragment of PHFL. Furthermore the full PHFL is far more expressive: we give a translation from Lubarsky's $\mu$-arithmetic to PHFL, which implies that PHFL model checking is $\Pi^1_1$-hard and $\Sigma^1_1$-hard. As a positive result, we characterize a decidable fragment of the PHFL model-checking problems using a novel type system.

An Overview of the HFL Model Checking Project

In this article, we give an overview of our project on higher-order program verification based on HFL (higher-order fixpoint logic) model checking. After a brief introduction to HFL, we explain how it can be applied to program verification, and summarize the current status of the project.

The Complexity of Model-Checking Tail-Recursive Higher-Order Fixpoint Logic

Higher-Order Fixpoint Logic (HFL) is a modal specification language whose expressive power reaches far beyond that of Monadic Second-Order Logic, achieved through an incorporation of a typed λ-calculus into the modal μ-calculus. Its model checking problem on finite transition systems is decidable, albeit of high complexity, namely k-EXPTIME-complete for formulas that use functions of type order at most k < 0. In this paper we present a fragment with a presumably easier model checking problem. We show that so-called tail-recursive formulas of type order k can be model checked in (k − 1)-EXPSPACE, and also give matching lower bounds. This yields generic results for the complexity of bisimulation-invariant non-regular properties, as these can typically be defined in HFL.

The Complexity of Model-Checking the Tail-Recursive Fragment of Higher-Order Modal Fixpoint Logic

Higher-Order Fixpoint Logic (HFL) is a modal specification language whose expressive power reaches far beyond that of Monadic Second-Order Logic, achieved through an incorporation of a typed λ-calculus into the modal µ-calculus. Its model checking problem on finite transition systems is decidable, albeit of high complexity, namely k-EXPTIME-complete for formulas that use functions of type order at most k > 0. In this paper we present a fragment with a presumably easier model checking problem. We show that so-called tail-recursive formulas of type order k can be model checked in (k − 1)-EXPSPACE, and also give matching lower bounds. This yields generic results for the complexity of bisimulation-invariant non-regular properties, as these can typically be defined in HFL.

On the relationship between higher-order recursion schemes and higher-order fixpoint logic

We study the relationship between two kinds of higher-order extensions of model checking: HORS model checking, where models are extended to higher-order recursion schemes, and HFL model checking, where the logic is extedned to higher-order modal fixpoint logic. Those extensions have been independently studied until recently, and the former has been applied to higher-order program verification. We show that there exist (arguably) natural reductions between the two problems. To prove the correctness of the translation from HORS to HFL model checking, we establish a type-based characterization of HFL model checking, which should be of independent interest. The results reveal a close relationship between the two problems, enabling cross-fertilization of the two research threads.

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.

Temporal Logics Beyond Regularity

Non-regular program correctness properties play an important role in the specification of unbounded buffers, recursive procedures, etc. This thesis surveys results about the relative expressive power and complexity of temporal logics which are capable of defining non-regular properties. In particular, it features Propositional Dynamic Logic of Context-Free Programs, Fixpoint Logic with Chop, the Modal Iteration Calculus, and Higher-Order Fixpoint Logic.<br /> <br />Regarding expressive power we consider two classes of structures: arbitrary transition systems as well as finite words as a subclass of the former. The latter is meant to give an intuitive account of the logics' expressive powers by relating them to known language classes defined in terms of grammars or Turing Machines. <br /> <br /> Regarding the computational complexity of temporal logics beyond regularity we focus on their model checking problems since their satisfiability problems are all highly undecidable. Their model checking complexities range between polynomial time and non-elementary.