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COMPLEXITY HIERARCHIES BEYOND ELEMENTARY
"... Abstract. We introduce a hierarchy of fastgrowing complexity classes and show its suitability for completeness statements of many non elementary problems. This hierarchy allows the classification of many decision problems with a nonelementary complexity, which occur naturally in logic, combinato ..."
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Abstract. We introduce a hierarchy of fastgrowing complexity classes and show its suitability for completeness statements of many non elementary problems. This hierarchy allows the classification of many decision problems with a nonelementary complexity, which occur naturally in logic, combinatorics, formal languages, verification, etc., with complexities ranging from simple towers of exponentials to Ackermannian and beyond. 1.
Positive HigherOrder Queries
"... We investigate a higherorder query language that embeds operators of the positive relational algebra within the simplytyped λcalculus. Our language allows one to succinctly define ordinary positive relational algebra queries (conjunctive queries and unions of conjunctive queries) and, in addition ..."
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We investigate a higherorder query language that embeds operators of the positive relational algebra within the simplytyped λcalculus. Our language allows one to succinctly define ordinary positive relational algebra queries (conjunctive queries and unions of conjunctive queries) and, in addition, secondorder query functionals, which allow the transformation of CQs and UCQs in a generic (i.e., syntaxindependent) way. We investigate the equivalence and containment problems for this calculus, which subsumes traditional CQ/UCQ containment. Query functionals are said to be equivalent if the output queries are equivalent, for each possible input query, and similarly for containment. These notions of containment and equivalence depend on the class of (ordinary relational algebra) queries considered. We show that containment and equivalence are decidable when query variables are restricted to positive relational algebra and we identify the precise complexity of the problem. We also identify classes of functionals where containment is tractable. Finally, we provide upper bounds to the complexity of the containment problem when functionals act over other classes.
Extracting Herbrand Disjunctions by Functional Interpretation
"... Abstract. Carrying out a suggestion by Kreisel, we adapt Gödel’s functional interpretation to ordinary firstorder predicate logic(PL) and thus devise an algorithm to extract Herbrand terms from PLproofs. The extraction is carried out in an extension of PL to higher types. The algorithm consists of ..."
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Abstract. Carrying out a suggestion by Kreisel, we adapt Gödel’s functional interpretation to ordinary firstorder predicate logic(PL) and thus devise an algorithm to extract Herbrand terms from PLproofs. The extraction is carried out in an extension of PL to higher types. The algorithm consists of two main steps: first we extract a functional realizer, next we compute the βnormalform of the realizer from which the Herbrand terms can be read off. Even though the extraction is carried out in the extended language, the terms are ordinary PLterms. In contrast to approaches to Herbrand’s theorem based on cut elimination or εelimination this extraction technique is, except for the normalization step, of low polynomial complexity, fully modular and furthermore allows an analysis of the structure of the Herbrand terms, in the spirit of Kreisel ([13]), already prior to the normalization step. It is expected that the implementation of functional interpretation in Schwichtenberg’s MINLOG system can be adapted to yield an efficient Herbrandterm extraction tool. 1.
On the computational complexity of cutreduction
, 2009
"... Using appropriate notation systems for proofs, cutreduction can often be rendered feasible on these notations. Explicit bounds can be given. Developing a suitable notation system for Bounded Arithmetic, and applying these bounds, all the known results on definable functions of certain such theorie ..."
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Using appropriate notation systems for proofs, cutreduction can often be rendered feasible on these notations. Explicit bounds can be given. Developing a suitable notation system for Bounded Arithmetic, and applying these bounds, all the known results on definable functions of certain such theories can be reobtained in a uniform way. 1 Introduction and Related Work Since Gentzen’s invention of the “Logik Kalkül ” LK and the proof of his “Hauptsatz ” [Gen35a, Gen35b], cutelimination has been studied in many papers on proof theory. Mints ’ invention of continuous normalisation [Min78, KMS75] isolates operational aspects of normalisation, that is, the manipulations on (in
Higherorder functions and structured datatypes
 In WebDB
, 2012
"... Recent proposals from the World Wide Web consortium propose adding support for higherorder functions within the XQuery standard. In this work we explore languages adding higherorder features on top of XML and other structured datatypes. We define a higherorder extension for Core XQuery, along wit ..."
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Recent proposals from the World Wide Web consortium propose adding support for higherorder functions within the XQuery standard. In this work we explore languages adding higherorder features on top of XML and other structured datatypes. We define a higherorder extension for Core XQuery, along with a higherorder algebra over complex values which has the same complexity as the XMLbased language. We discuss our language and its relation with proposed extensions to the XQuery standard, study the complexity of evaluation, and briefly discuss our approach to implementing the language. 1.
Shredding higherorder nested queries
"... We present a modular account of query shredding, the simulation of a single nested relational query by a number of flat relational queries, applicable to both set and multiset semantics. Our key insight is that shredding can be greatly simplified by first rewriting the input query into a canonical n ..."
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We present a modular account of query shredding, the simulation of a single nested relational query by a number of flat relational queries, applicable to both set and multiset semantics. Our key insight is that shredding can be greatly simplified by first rewriting the input query into a canonical normal form. Normalisation allows us to define shredding translations on types and terms independently of one another, unlike previous work. An added benefit of normalisation is that we support higherorder terms for free, provided that the result type is a plain nested relation type (without higher order components). In order to generate SQL we consider several alternatives for generating indexes, focusing on a lightweight use of SQL OLAP features. We prove correctness of our translations, focusing on the central shredding step: shredding a nested query, running the shredded queries, and stitching the results back together yields the same results as running the nested query directly. 1.
Abstract
, 2006
"... We investigate the relation between intermediate predicate logics based on countable linear Kripke frames with constant domains and Gödel logics. We show that for any such Kripke frame there is a Gödel logic which coincides with the logic defined by this Kripke frame on constant domains and vice ver ..."
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We investigate the relation between intermediate predicate logics based on countable linear Kripke frames with constant domains and Gödel logics. We show that for any such Kripke frame there is a Gödel logic which coincides with the logic defined by this Kripke frame on constant domains and vice versa. This allows us to transfer several recent results on Gödel logics to logics based on countable linear Kripke frames with constant domains: We obtain a complete characterisation of axiomatisability of logics based on countable linear Kripke frames with constant domains. 1 Furthermore, we obtain that the total number of logics defined by countable linear Kripke frames on constant domains is countable. 1
SWANSEA UNIVERSITY REPORT SERIES On the computational complexity of cutreduction
, 2007
"... We investigate the complexity of cutreduction on proof notations, in particular identifying situations where cutreduction operates feasibly, i.e., subexponential, on proof notations. We then apply the machinery to characterise definable search problem in Bounded Arithmetic. To explain our results ..."
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We investigate the complexity of cutreduction on proof notations, in particular identifying situations where cutreduction operates feasibly, i.e., subexponential, on proof notations. We then apply the machinery to characterise definable search problem in Bounded Arithmetic. To explain our results with an example, let E(d) denote Mints ’ continuous cutreduction operator which reduces the complexity of all cuts of a propositional derivation d by one level. We will show that if all subproofs of d can be denoted with notations of size s, and the height of d is h, then subproofs of the derivation E(d) can be denoted by notations of size h · (s + O(1)). Together with the observation that determining the last inference of a denoted derivation as well as determining notations for immediate subderivations is easy (i.e., polynomial time computable), we can apply this result to reobtain that the Σ b idefinable functions of the Bounded Arithmetic theory S i 2 are in the ith level of the polynomial time