Streett Automata Model Checking of Higher-Order Recursion Schemes

We propose a practical algorithm for Streett automata model checking of higher-order recursion schemes (HORS), which checks whether the tree generated by a given HORS is accepted by a given Streett automaton. The Streett automata model checking of HORS is useful in the context of liveness verification of higher-order functional programs. The previous approach to Streett automata model checking converted Streett automata to parity automata and then invoked a parity tree automata model checker. We show through experiments that our direct approach outperforms the previous approach. Besides being able to directly deal with Streett automata, our algorithm is the first practical Streett or parity automata model checking algorithm that runs in time polynomial in the size of HORS, assuming that the other parameters are fixed. Previous practical fixed-parameter polynomial time algorithms for HORS could only deal with the class of trivial tree automata. We have confirmed through experiments that (a parity automata version of) our model checker outperforms previous parity automata model checkers for HORS

Higher-Order Model Checking Step by Step

We show a new simple algorithm that solves the model-checking problem for recursion schemes: check whether the tree generated by a given higher-order recursion scheme is accepted by a given alternating parity automaton. The algorithm amounts to a procedure that transforms a recursion scheme of order n to a recursion scheme of order n-1, preserving acceptance, and increasing the size only exponentially. After repeating the procedure n times, we obtain a recursion scheme of order 0, for which the problem boils down to solving a finite parity game. Since the size grows exponentially at each step, the overall complexity is n-EXPTIME, which is known to be optimal. More precisely, the transformation is linear in the size of the recursion scheme, assuming that the arity of employed nonterminals and the size of the automaton are bounded by a constant; this results in an FPT algorithm for the model-checking problem. Our transformation is a generalization of a previous transformation of the author (2020), working for reachability automata in place of parity automata. The step-by-step approach can be opposed to previous algorithms solving the considered problem "in one step", being compulsorily more complicated

On Average-Case Hardness of Higher-Order Model Checking

We study a mixture between the average case and worst case complexities of higher-order model checking, the problem of deciding whether the tree generated by a given ? Y-term (or equivalently, a higher-order recursion scheme) satisfies the property expressed by a given tree automaton. Higher-order model checking has recently been studied extensively in the context of higher-order program verification. Although the worst-case complexity of the problem is k-EXPTIME complete for order-k terms, various higher-order model checkers have been developed that run efficiently for typical inputs, and program verification tools have been constructed on top of them. One may, therefore, hope that higher-order model checking can be solved efficiently in the average case, despite the worst-case complexity. We provide a negative result, by showing that, under certain assumptions, for almost every term, the higher-order model checking problem specialized for the term is k-EXPTIME hard with respect to the size of automata. The proof is based on a novel intersection type system that characterizes terms that do not contain any useless subterms

LambdaY-Calculus With Priorities

International audienceThe lambdaY-calculus with priorities is a variant of the simply-typed lambda calculus designed for higher-order model-checking. The higher-order model-checking problem asks if a given parity tree automaton accepts the Böhm tree of a given term of the simply-typed lambda calculus with recursion. We show that this problem can be reduced to the same question but for terms of lambdaY-calculus with priorities and visibly parity automata; a subclass of parity automata. The latter question can be answered by evaluating terms in a simple powerset model with least and greatest fixpoints. We prove that the recognizing power of powerset models and visibly parity automata are the same. So, up to conversion to the lambdaY-calculus with priorities, powerset models with least and greatest fixpoints are indeed the right semantic framework for the model-checking problem. The reduction to lambdaY-calculus with priorities is also efficient algorithmically: it gives an algorithm of the same complexity as direct approaches to the higher-order model-checking problem. This indicates that the task of calculating the value of a term in a powerset model is a central algo-rithmic problem for higher-order model-checking

Symbolic Backwards-Reachability Analysis for Higher-Order Pushdown Systems

Higher-order pushdown systems (PDSs) generalise pushdown systems through the use of higher-order stacks, that is, a nested "stack of stacks" structure. These systems may be used to model higher-order programs and are closely related to the Caucal hierarchy of infinite graphs and safe higher-order recursion schemes. We consider the backwards-reachability problem over higher-order Alternating PDSs (APDSs), a generalisation of higher-order PDSs. This builds on and extends previous work on pushdown systems and context-free higher-order processes in a non-trivial manner. In particular, we show that the set of configurations from which a regular set of higher-order APDS configurations is reachable is regular and computable in n-EXPTIME. In fact, the problem is n-EXPTIME-complete. We show that this work has several applications in the verification of higher-order PDSs, such as linear-time model-checking, alternation-free mu-calculus model-checking and the computation of winning regions of reachability games

Computer Aided Verification

This open access two-volume set LNCS 10980 and 10981 constitutes the refereed proceedings of the 30th International Conference on Computer Aided Verification, CAV 2018, held in Oxford, UK, in July 2018. The 52 full and 13 tool papers presented together with 3 invited papers and 2 tutorials were carefully reviewed and selected from 215 submissions. The papers cover a wide range of topics and techniques, from algorithmic and logical foundations of verification to practical applications in distributed, networked, cyber-physical, and autonomous systems. They are organized in topical sections on model checking, program analysis using polyhedra, synthesis, learning, runtime verification, hybrid and timed systems, tools, probabilistic systems, static analysis, theory and security, SAT, SMT and decisions procedures, concurrency, and CPS, hardware, industrial applications