scholarly journals An Interpolation-based Compiler and Optimizer for Relational Queries (System design Report)

10.29007/53fk ◽  
2018 ◽  
Author(s):  
David Toman ◽  
Grant Weddell
Keyword(s):  
System Design ◽  
Cost Model ◽  
Interpolation Theorem ◽  
Database Schema ◽  
External Cost ◽  
First Order ◽  
User Query ◽  
Order Constraints ◽  
Proof Procedure ◽  
Distinguished Subset

We outline the implementation of a query compiler for relational queries that generates query plans with respect to a database schema, that is, a set of arbitrary first-order constraints, and a distinguished subset of predicate symbols from the underlying signature that correspond to access paths. The compiler is based on a variant of the Craig interpolation theorem, with reasoning realized via a modified analytic tableau proof procedure. This procedure decouples the generation of candidate plans that are interpolants from the tableau proof procedure, and applies A*-based search with respect to an external cost model to arbitrate among the alternative candidate plans. The tableau procedure itself is implemented as a virtual machine that operates on a compiled and optimized byte-code that faithfully implements reasoning with respect to the database schema constraints and a user query.

scholarly journals On an interpretation of second order quantification in first order intuitionistic propositional logic

1992 ◽  
Vol 57 (1) ◽  
pp. 33-52 ◽  
Author(s):  
Andrew M. Pitts
Keyword(s):  
Propositional Logic ◽  
Heyting Algebra ◽  
Propositional Calculus ◽  
Interpolation Theorem ◽  
Second Order ◽  
Heyting Algebras ◽  
Intuitionistic Propositional Logic ◽  
First Order ◽  
Intuitionistic Propositional Calculus ◽  
Algebraic Counterpart

AbstractWe prove the following surprising property of Heyting's intuitionistic propositional calculus, IpC. Consider the collection of formulas, ϕ, built up from propositional variables (p, q, r, …) and falsity (⊥) using conjunction (∧), disjunction (∨) and implication (→). Write ⊢ϕ to indicate that such a formula is intuitionistically valid. We show that for each variable p and formula ϕ there exists a formula Apϕ (effectively computable from ϕ), containing only variables not equal to p which occur in ϕ, and such that for all formulas ψ not involving p, ⊢ψ → Apϕ if and only if ⊢ψ → ϕ. Consequently quantification over propositional variables can be modelled in IpC, and there is an interpretation of the second order propositional calculus, IpC2, in IpC which restricts to the identity on first order propositions.An immediate corollary is the strengthening of the usual interpolation theorem for IpC to the statement that there are least and greatest interpolant formulas for any given pair of formulas. The result also has a number of interesting consequences for the algebraic counterpart of IpC, the theory of Heyting algebras. In particular we show that a model of IpC2 can be constructed whose algebra of truth-values is equal to any given Heyting algebra.

Uniqueness, definability and interpolation

1988 ◽  
Vol 53 (2) ◽  
pp. 554-570 ◽  
Author(s):  
Kosta Došen ◽  
Peter Schroeder-Heister
Keyword(s):  
Proof Theory ◽  
Interpolation Theorem ◽  
Order Logic ◽  
First Order Logic ◽  
General Notion ◽  
Structural Part ◽  
First Order ◽  
First Case ◽  
Additional Constant ◽  
Special Case

This paper is meant to be a comment on Beth's definability theorem. In it we shall make the following points.Implicit definability as mentioned in Beth's theorem for first-order logic is a special case of a more general notion of uniqueness. If α is a nonlogical constant, Tα a set of sentences, α* an additional constant of the same syntactical category as α and Tα, a copy of Tα with α* instead of α, then for implicit definability of α in Tα one has, in the case of predicate constants, to derive α(x1,…,xn) ↔ α*(x1,…,xn) from Tα ∪ Tα*, and similarly for constants of other syntactical categories. For uniqueness one considers sets of schemata Sα and derivability from instances of Sα ∪ Sα* in the language with both α and α*, thus allowing mixing of α and α* not only in logical axioms and rules, but also in nonlogical assumptions. In the first case, but not necessarily in the second one, explicit definability follows. It is crucial for Beth's theorem that mixing of α and α* is allowed only inside logic, not outside. This topic will be treated in §1.Let the structural part of logic be understood roughly in the sense of Gentzen-style proof theory, i.e. as comprising only those rules which do not specifically involve logical constants. If we restrict mixing of α and α* to the structural part of logic which we shall specify precisely, we obtain a different notion of implicit definability for which we can demonstrate a general definability theorem, where a is not confined to the syntactical categories of nonlogical expressions of first-order logic. This definability theorem is a consequence of an equally general interpolation theorem. This topic will be treated in §§2, 3, and 4.

A Note on the Interpolation Theorem in First Order Logic

1982 ◽  
Vol 28 (14-18) ◽  
pp. 215-218 ◽  
Author(s):  
George Weaver
Keyword(s):  
Interpolation Theorem ◽  
Order Logic ◽  
First Order Logic ◽  
First Order

scholarly journals Variational and quasivariational inequalities with first order constraints

2013 ◽  
Vol 397 (2) ◽  
pp. 738-756 ◽  
Author(s):  
Assis Azevedo ◽  
Fernando Miranda ◽  
Lisa Santos
Keyword(s):  
Quasivariational Inequalities ◽  
First Order ◽  
Order Constraints

An Interpolation Theorem

2000 ◽  
Vol 6 (4) ◽  
pp. 447-462 ◽  
Author(s):  
Martin Otto
Keyword(s):  
Interpolation Theorem ◽  
Order Logic ◽  
First Order Logic ◽  
Elementary Model ◽  
Relation Symbol ◽  
First Order ◽  
Theoretic Proof ◽  
Universal Quantification ◽  
The Given ◽  
Interpolation Result

AbstractLyndon's Interpolation Theorem asserts that for any valid implication between two purely relational sentences of first-order logic, there is an interpolant in which each relation symbol appears positively (negatively) only if it appears positively (negatively) in both the antecedent and the succedent of the given implication. We prove a similar, more general interpolation result with the additional requirement that, for some fixed tuple of unary predicates U, all formulae under consideration have all quantifiers explicitly relativised to one of the U. Under this stipulation, existential (universal) quantification over U contributes a positive (negative) occurrence of U.It is shown how this single new interpolation theorem, obtained by a canonical and rather elementary model theoretic proof, unifies a number of related results: the classical characterisation theorems concerning extensions (substructures) with those concerning monotonicity, as well as a many-sorted interpolation theorem focusing on positive vs. negative occurrences of predicates and on existentially vs. universally quantified sorts.

Automated complexity analysis of Nuprl extracted programs

2001 ◽  
Vol 11 (1) ◽  
pp. 3-31 ◽  
Author(s):  
RALPH BENZINGER
Keyword(s):  
Abstract Interpretation ◽  
Complexity Analysis ◽  
Cost Model ◽  
Functional Program ◽  
First Order ◽  
Value Evaluation ◽  
Recursive Programs ◽  
Symbolic Evaluation ◽  
Primitive Recursive ◽  
Higher Order Functions

This paper describes the Automated Complexity Analysis Prototype (ACAp) system for automated complexity analysis of functional programs synthesized with the Nuprl proof development system. We define a simple abstract cost model for NUPRL's term language based on the current call-by-name evaluator. The framework uses abstract functions and abstract lists to facilitate reasoning about primitive recursive programs with first-order functions, lazy lists and a subclass of higher-order functions. The ACAp system automatically derives upper bounds on the time complexity of NUPRL extracts relative to a given profiling semantics. Analysis proceeds by abstract interpretation of the extract, where symbolic evaluation rules extend standard evaluation to terms with free variables. Symbolic evaluation of recursive programs generates systems of multi-variable difference equations, which are solved using the MATHEMATICA computer algebra system. The use of the system is exemplified by analyzing a proof extract that computes the maximum segment sum of a list and a functional program that determines the minimum of a list via sorting. For both results, we compare call-by-name to call-by-value evaluation.

scholarly journals Finite Controllability for Ontology-Mediated Query Answering of CRPQ

2020 ◽  
Author(s):  
Diego Figueira ◽  
Santiago Figueira ◽  
Edwin Pin Baque
Keyword(s):  
Query Answering ◽  
Graph Structure ◽  
First Order ◽  
Underlying Graph ◽  
Graph Classes ◽  
Regular Path ◽  
Order Constraints ◽  
Simple Recursion ◽  
Regular Path Queries ◽  
Path Queries

Finite ontology mediated query answering (FOMQA) is the variant of ontology mediated query answering (OMQA) where the represented world is assumed to be finite, and thus only finite models of the ontology are considered. We study the property of finite-controllability, that is, whether FOMQA and OMQA are equivalent, for fragments of C2RPQ. C2RPQ is the language of conjunctive two-way regular path queries, which can be regarded as the result of adding simple recursion to Conjunctive Queries. For graph classes S, we consider fragments C2RPQ(S) of C2RPQ as the queries whose underlying graph structure is in S. We completely classify the finitely controllable and non-finitely controllable fragments under: inclusion dependencies, (frontier-)guarded rules, frontier-one rules (either with or without constants), and more generally under guarded-negation first-order constraints. For the finitely controllable fragments, we show a reduction to the satisfiability problem for guarded-negation first-order logic, yielding a 2EXPTIME algorithm (in combined complexity) for the corresponding (F)OMQA problem.

External Cost Model and Application of International Container Intermodal Transport

2013 ◽  
Vol 5 (5) ◽  
pp. 510-517
Author(s):  
Maozeng Xu ◽  
Lixia Yang ◽  
Bo Zheng ◽  
Mengying Feng
Keyword(s):  
Cost Model ◽  
External Cost ◽  
Intermodal Transport

Streaming of Continuous Media for Distance Education Systems

2011 ◽  
pp. 190-216
Author(s):  
Ali Dashti ◽  
Maytham Safar
Keyword(s):  
Distance Education ◽  
Streaming Media ◽  
Cost Model ◽  
User Profile ◽  
Continuous Media ◽  
Large Size ◽  
User Query ◽  
Proposed Model ◽  
New Challenges

Distance education created new challenges regarding the delivery of large size isochronous continuous streaming media (SM) objects. In this paper, we consider the design of a framework for customized SM presentations, where each presentation consists of a number of SM objects that should be retrieved and displayed to the user in a coherent fashion. We describe a retrieval optimizer (Prime) that captures the flexibilities and requirements imposed by the user query, user profile, and session profile. Then, it determines how this query script should be imposed against the continuous media (CM) server to reduce contention. We also provide a cost model to evaluate each proposed plan. Finally, we explain the role of memory buffering in alleviating the server bandwidth fragmentation problem. Our preliminary experimental results show the feasibility and effectiveness of our proposed model and techniques in generating near optimal retrieval.

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