16 research outputs found

    Superposition for Higher-Order Logic

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    Efficient Full Higher-Order Unification

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    We developed a procedure to enumerate complete sets of higher-order unifiers based on work by Jensen and Pietrzykowski. Our procedure removes many redundant unifiers by carefully restricting the search space and tightly integrating decision procedures for fragments that admit a finite complete set of unifiers. We identify a new such fragment and describe a procedure for computing its unifiers. Our unification procedure, together with new higher-order term indexing data structures, is implemented in the Zipperposition theorem prover. Experimental evaluation shows a clear advantage over Jensen and Pietrzykowski's procedure

    Superposition for Lambda-Free Higher-Order Logic

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    We introduce refutationally complete superposition calculi for intentional and extensional clausal \lambda-free higher-order logic, two formalisms that allow partial application and applied variables. The calculi are parameterized by a term order that need not be fully monotonic, making it possible to employ the \lambda-free higher-order lexicographic path and Knuth-Bendix orders. We implemented the calculi in the Zipperposition prover and evaluated them on Isabelle/HOL and TPTP benchmarks. They appear promising as a stepping stone towards complete, highly efficient automatic theorem provers for full higher-order logic

    Superposition for Lambda-Free Higher-Order Logic

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    International audienceWe introduce refutationally complete superposition calculi for intentional and extensional 位-free higher-order logic, two formalisms that allow partial application and applied variables. The calculi are parameterized by a term order that need not be fully monotonic, making it possible to employ the 位-free higher-order lexicographic path and Knuth-Bendix orders. We implemented the calculi in the Zipperposition prover and evaluated them on TPTP benchmarks. They appear promising as a stepping stone towards complete, efficient automatic theorem provers for full higher-order logic

    A Formal Proof of PAC Learnability for Decision Stumps

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    We present a formal proof in Lean of probably approximately correct (PAC) learnability of the concept class of decision stumps. This classic result in machine learning theory derives a bound on error probabilities for a simple type of classifier. Though such a proof appears simple on paper, analytic and measure-theoretic subtleties arise when carrying it out fully formally. Our proof is structured so as to separate reasoning about deterministic properties of a learning function from proofs of measurability and analysis of probabilities.Comment: 13 pages, appeared in Certified Programs and Proofs (CPP) 202

    The Embedding Path Order for Lambda-Free Higher-Order Terms

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    The embedding path order, introduced in this article, is a variant of the recursive path order (RPO) for untyped 位-free higher-order terms (also called applicative first-order terms). Unlike other higher-order variants of RPO, it is a ground-total and well-founded simplification order, making it more suitable for the superposition calculus. I formally proved the order鈥檚 theoretical properties in Isabelle/HOL and evaluated the order in a prototype based on the superposition prover Zipperposition

    Privacy accounting \varepsilonconomics: Improving differential privacy composition via a posteriori bounds

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    Differential privacy (DP) is a widely used notion for reasoning about privacy when publishing aggregate data. In this paper, we observe that certain DP mechanisms are amenable to a posteriori privacy analysis that exploits the fact that some outputs leak less information about the input database than others. To exploit this phenomenon, we introduce output differential privacy (ODP) and a new composition experiment, and leverage these new constructs to obtain significant privacy budget savings and improved privacy-utility tradeoffs under composition. All of this comes at no cost in terms of privacy; we do not weaken the privacy guarantee. To demonstrate the applicability of our a posteriori privacy analysis techniques, we analyze two well-known mechanisms: the Sparse Vector Technique and the Propose-Test-Release framework. We then show how our techniques can be used to save privacy budget in more general contexts: when a differentially private iterative mechanism terminates before its maximal number of iterations is reached, and when the output of a DP mechanism provides unsatisfactory utility. Examples of the former include iterative optimization algorithms, whereas examples of the latter include training a machine learning model with a large generalization error. Our techniques can be applied beyond the current paper to refine the analysis of existing DP mechanisms or guide the design of future mechanisms.Comment: 25 pages, 2 figures. To be published in PoPETs 2022. The formal proof and the code for generating the plots can be found at https://doi.org/10.6084/m9.figshare.1933064

    Superposition for Higher-Order Logic

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    We recently designed two calculi as stepping stones towards superposition for full higher-order logic: Boolean-free 位-superposition and superposition for first-order logic with interpreted Booleans. Stepping on these stones, we finally reach a sound and refutationally complete calculus for higher-order logic with polymorphism, extensionality, Hilbert choice, and Henkin semantics. In addition to the complexity of combining the calculus鈥檚 two predecessors, new challenges arise from the interplay between 位-terms and Booleans. Our implementation in Zipperposition outperforms all other higher-order theorem provers and is on a par with an earlier, pragmatic prototype of Booleans in Zipperposition
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