Tuple Interpretations for Higher-Order Complexity

Kop, Cynthia; Vale, Deivid

doi:10.4230/lipics.fscd.2021.31

Cited by 5 publications

(10 citation statements)

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“…We denote this one by nat_CompatRel. This type class essentially models the notion of extended well-founded set introduced in [19]. An extended well-founded set is a set together with compatible orders >, ≥ such that > is well-founded and ≥ is a quasi-ordering.…”

Section: Interpreting Types and Termsmentioning

confidence: 99%

“…The approach we use here is slightly different from what is usually done in higher-order rewriting. In [8,19,26], for instance, context information is lifted to the meta-level and variables are interpreted using the notion of valuations. In contrast, in our setting, the typing context lives at the syntactic level and variables are interpreted as weakly monotonic functions.…”

Section: ▶ Definition 32 (Monotonicmapsv) Weakly Monotone Maps Are De...mentioning

confidence: 99%

“…There are two ways to deal with this. One way is by interpreting function types as strictly monotonic maps [19]. In this approach, this interpretation of application is valid.…”

Section: ▶ Definition 37 (Sem_var) We Interpret Variables With the Fo...mentioning

confidence: 99%

“…Not only do many benchmarks contain hundreds of rules, modern termination tools make use of various proof methods that have been developed for decades. Indeed, a single termination proof might for instance make use of a combination of dependency pairs [3,9,20], recursive path orders [14,21], rule removal, and multiple kinds of interpretations [8,19,25,33]. This makes bugs very difficulty to find.…”

Section: Introductionmentioning

confidence: 99%

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Certifying Higher-Order Polynomial Interpretations

Weide¹,

Vale²,

Kop³

2023

Preprint

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Higher-order rewriting is a framework in which one can write higher-order programs and study their properties. One such property is termination: the situation that for all inputs, the program eventually halts its execution and produces an output. Several tools have been developed to check whether higher-order rewriting systems are terminating. However, developing such tools is difficult and can be error-prone. In this paper, we present a way of certifying termination proofs of higher-order term rewriting systems. We formalize a specific method, namely the polynomial interpretation method, that is used to prove termination. In addition, we give a program that turns the output of Wanda, a termination analysis tool for higher-order rewriting systems, into a Coq script, so that we can check whether the output is a valid proof of termination.

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Section: Interpreting Types and Termsmentioning

confidence: 99%

Section: ▶ Definition 32 (Monotonicmapsv) Weakly Monotone Maps Are De...mentioning

confidence: 99%

“…There are two ways to deal with this. One way is by interpreting function types as strictly monotonic maps [19]. In this approach, this interpretation of application is valid.…”

Section: ▶ Definition 37 (Sem_var) We Interpret Variables With the Fo...mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Certifying Higher-Order Polynomial Interpretations

Weide¹,

Vale²,

Kop³

2023

Preprint

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“…Corollary 2 is useful when models over non-totally ordered carriers are considered. There are important methods (for termination) that crucially rely on such carriers: the matrix interpretations [5], or more generally the tuple interpretations [15,34].…”

Section: Corollary 2 S T Is R-unsatisfiable If and Only If There Exis...mentioning

confidence: 99%

Term Orderings for Non-reachability of (Conditional) Rewriting

Yamada

2022

Automated Reasoning

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We propose generalizations of reduction pairs, well-established techniques for proving termination of term rewriting, in order to prove unsatisfiability of reachability (infeasibility) in plain and conditional term rewriting. We adapt the weighted path order, a merger of the Knuth–Bendix order and the lexicographic path order, into the proposed framework. The proposed approach is implemented in the termination prover , and the strength of our approach is demonstrated through examples and experiments.

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On Basic Feasible Functionals and the Interpretation Method

Baillot,

Dal Lago,

Kop

et al. 2024

Lecture Notes in Computer Science

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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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Tuple Interpretations for Higher-Order Complexity

Cited by 5 publications

References 0 publications

Certifying Higher-Order Polynomial Interpretations

Certifying Higher-Order Polynomial Interpretations

Term Orderings for Non-reachability of (Conditional) Rewriting

On Basic Feasible Functionals and the Interpretation Method

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