The class of type-two basic feasible functionals ($\mathtt{BFF}_2$) is the analogue of $\mathtt{FP}$ (polynomial time functions) for type-2 functionals, that is, functionals that can take (first-order) functions as arguments. $\mathtt{BFF}_2$ can be defined through Oracle Turing machines with running time bounded by second-order polynomials. On the other hand, higher-order term rewriting provides an elegant formalism for expressing higher-order computation. We address the problem of characterizing $\mathtt{BFF}_2$ by higher-order term rewriting. Various kinds of interpretations for first-order term rewriting have been introduced in the literature for proving termination and characterizing first-order complexity classes. In this paper, we consider a recently introduced notion of cost-size interpretations for higher-order term rewriting and see second order rewriting as ways of computing type-2 functionals. We then prove that the class of functionals represented by higher-order terms admitting polynomially bounded cost-size interpretations exactly corresponds to $\mathtt{BFF}_2$.

Abstract Logically constrained term rewriting systems (LCTRSs) are a formalism for program analysis with support for data types that are not (co)inductively defined. Only imperative programs have been considered through the lens of LCTRSs so far since LCTRSs were introduced as a first-order formalism. In this paper, we propose logically constrained simply-typed term rewriting systems (LCSTRSs), a higher-order generalization of LCTRSs, which suits the needs of representing and analyzing functional programs. We also study the termination problem of LCSTRSs and define a variant of the higher-order recursive path ordering (HORPO) for the newly proposed formalism.

Abstract The class of basic feasible functionals ( $$\texttt{BFF}$$ BFF ) is the analog of $$\texttt{FP}$$ FP (polynomial time functions) for type-2 functionals, that is, functionals that can take (first-order) functions as arguments. $$\texttt{BFF}$$ BFF can be defined through Oracle Turing machines with running time bounded by second-order polynomials. On the other hand, higher-order term rewriting provides an elegant formalism for expressing higher-order computation. We address the problem of characterizing $$\texttt{BFF}$$ BFF by higher-order term rewriting. Various kinds of interpretations for first-order term rewriting have been introduced in the literature for proving termination and characterizing (first-order) complexity classes. In this paper, we consider a recently introduced notion of cost–size interpretations for higher-order term rewriting and see definitions as ways of computing functionals. We then prove that the class of functionals represented by higher-order terms admitting a certain kind of cost–size interpretation is exactly $$\texttt{BFF}$$ BFF .

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We develop a class of algebraic interpretations for many-sorted and higher-order term rewriting systems that takes type information into account. Specifically, base-type terms are mapped to \emph{tuples} of natural numbers and higher-order terms to functions between those tuples. Tuples may carry information relevant to the type; for instance, a term of type $\mathsf{nat}$ may be associated to a pair $(\mathsf{cost}, \mathsf{size})$ representing its evaluation cost and size. This class of interpretations results in a more fine-grained notion of complexity than runtime or derivational complexity, which makes it particularly useful to obtain complexity bounds for higher-order rewriting systems. We show that rewriting systems compatible with tuple interpretations admit finite bounds on derivation height. Furthermore, we demonstrate how to mechanically construct tuple interpretations and how to orient $β$ and $η$ reductions within our technique. Finally, we relate our method to runtime complexity and prove that specific interpretation shapes imply certain runtime complexity bounds.

Wanda is a fully automatic termination analysis tool for higher-order term rewriting. In this paper, we will discuss the methodology used in Wanda. Most pertinently, this includes a higher-order dependency pair framework and a variation of the higher-order recursive path ordering, as well as some non-termination analysis techniques and delegation to a first-order tool. Additionally, we will discuss Wanda’s internal rewriting formalism, and how to use Wanda in practice for systems in two different formalisms. We also present experimental results that consider both formalisms.

We revisit the static dependency pair method for proving termination of higher-order term rewriting and extend it in a number of ways: (1) We introduce a new rewrite formalism designed for general applicability in termination proving of higher-order rewriting, Algebraic Functional Systems with Meta-variables. (2) We provide a syntactically checkable soundness criterion to make the method applicable to a large class of rewrite systems. (3) We propose a modular dependency pair framework for this higher-order setting. (4) We introduce a fine-grained notion of formative and computable chains to render the framework more powerful. (5) We formulate several existing and new termination proving techniques in the form of processors within our framework. The framework has been implemented in the (fully automatic) higher-order termination tool WANDA .

This article aims to develop a verification method for procedural programs via a transformation into logically constrained term rewriting systems (LCTRSs). To this end, we extend transformation methods based on integer term rewriting systems to handle arbitrary data types, global variables, function calls, and arrays, and to encode safety checks. Then we adapt existing rewriting induction methods to LCTRSs and propose a simple yet effective method to generalize equations. We show that we can automatically verify memory safety and prove correctness of realistic functions. Our approach proves equivalence between two implementations; thus, in contrast to other works, we do not require an explicit specification in a separate specification language.

The termination method of weakly monotonic algebras, which has been defined for higher-order rewriting in the HRS formalism, offers a lot of power, but has seen little use in recent years. We adapt and extend this method to the alternative formalism of algebraic functional systems, where the simply-typed lambda-calculus is combined with algebraic reduction. Using this theory, we define higher-order polynomial interpretations, and show how the implementation challenges of this technique can be tackled. A full implementation is provided in the termination tool Wanda.

We extend the termination method using dynamic dependency pairs to higher order rewriting systems with beta as a rewrite step, also called Algebraic Functional Systems (AFSs). We introduce a variation of usable rules, and use monotone algebras to solve the constraints generated by dependency pairs. This approach differs in several respects from those dealing with higher order rewriting modulo beta (e.g. HRSs).