scholarly journals Efficient Full Higher-Order Unification

2021 ◽  
Vol Volume 17, Issue 4 ◽  
Author(s):  
Petar Vukmirović ◽  
Alexander Bentkamp ◽  
Visa Nummelin
Keyword(s):  
Data Structures ◽  
Experimental Evaluation ◽  
Order Term ◽  
Search Space ◽  
Decision Procedures ◽  
Higher Order ◽  
Theorem Prover ◽  
Clear Advantage ◽  
Complete Set ◽  
Higher Order Term

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.

Derivation of Skyrme Lagrangian and higher-order term

1990 ◽  
Vol 41 (9) ◽  
pp. 2930-2932
Author(s):  
Akihiro Ito
Keyword(s):  
Order Term ◽  
Higher Order ◽  
Higher Order Term

scholarly journals A rewriting calculus for cyclic higher-order term graphs

2007 ◽  
Vol 17 (3) ◽  
pp. 363-406 ◽  
Author(s):  
PAOLO BALDAN ◽  
CLARA BERTOLISSI ◽  
HORATIU CIRSTEA ◽  
CLAUDE KIRCHNER
Keyword(s):  
Order Term ◽  
Equational Theory ◽  
Term Rewriting ◽  
Higher Order ◽  
Graph Representation ◽  
First Order ◽  
Rewrite Rules ◽  
Evaluation Mechanism ◽  
Matching Constraints ◽  
Higher Order Term

The Rewriting Calculus (ρ-calculus, for short) was introduced at the end of the 1990s and fully integrates term-rewriting and λ-calculus. The rewrite rules, acting as elaborated abstractions, their application and the structured results obtained are first class objects of the calculus. The evaluation mechanism, which is a generalisation of beta-reduction, relies strongly on term matching in various theories.In this paper we propose an extension of the ρ-calculus, called ρg-calculus, that handles structures with cycles and sharing rather than simple terms. This is obtained by using recursion constraints in addition to the standard ρ-calculus matching constraints, which leads to a term-graph representation in an equational style. Like in the ρ-calculus, the transformations are performed by explicit application of rewrite rules as first-class entities. The possibility of expressing sharing and cycles allows one to represent and compute over regular infinite entities.We show that the ρg-calculus, under suitable linearity conditions, is confluent. The proof of this result is quite elaborate, due to the non-termination of the system and the fact that ρg-calculus-terms are considered modulo an equational theory. We also show that the ρg-calculus is expressive enough to simulate first-order (equational) left-linear term-graph rewriting and α-calculus with explicit recursion (modelled using a letrec-like construct).

Asymptotic expansions for laminar forced-convection heat and mass transfer Part 2. Boundary-layer flows

1966 ◽  
Vol 24 (2) ◽  
pp. 339-366 ◽  
Author(s):  
J. D. Goddard ◽  
Andreas Acrivos
Keyword(s):  
Heat Transfer ◽  
Boundary Layer ◽  
Mass Transfer ◽  
Prandtl Number ◽  
Order Term ◽  
Higher Order ◽  
Convection Heat ◽  
Boundary Layer Flows ◽  
Prandtl Numbers ◽  
Higher Order Term

This is the second of two articles by the authors dealing with asymptotic expansions for forced-convection heat or mass transfer to laminar flows. It is shown here how the method of the first paper (Acrivos & Goddard 1965), which was used to derive a higher-order term in the large Péclet number expansion for heat or mass transfer to small Reynolds number flows, can yield equally well higher-order terms in both the large and the small Prandtl number expansions for heat transfer to laminar boundary-layer flows. By means of this method an exact expression for the first-order correction to Lighthill's (1950) asymptotic formula for heat transfer at large Prandtl numbers, as well as an additional higher-order term for the small Prandtl number expansion of Morgan, Pipkin & Warner (1958), are derived. The results thus obtained are applicable to systems with non-isothermal surfaces and arbitrary planar or axisymmetric flow geometries. For the latter geometries a derivation is given of a higher-order term in the Péclet number expansion which arises from the curvature of the thermal layer for small Prandtl numbers. Finally, some applications of the results to ‘similarity’ flows are also presented.

Accurate Description of Near-Crack-Tip Fields for the Estimation of Inelastic Zone Extent in Quasi-Brittle Materials

2012 ◽  
Vol 525-526 ◽  
pp. 529-532 ◽  
Author(s):  
Václav Veselý ◽  
Jakub Sobek ◽  
Lucie Šestáková ◽  
Stanislav Seitl
Keyword(s):  
Finite Element ◽  
Order Term ◽  
Higher Order ◽  
Fe Analysis ◽  
Displacement Fields ◽  
Deterministic Method ◽  
Stress And Displacement Fields ◽  
Higher Order Terms ◽  
Higher Order Term ◽  
Selection Of

A description of stress and displacement fields by means of the Williams power series using also higher-order terms is the focus of this paper. Coefficients of this series are determined via the over-deterministic method from the results of conventional finite element (FE) analysis. A study is conducted into the selection of the FE node set whose results are processed in this regression technique. Coefficients up to the twelfth term were determined with high precision. The effect of the position of the FE node set on the accuracy of the values of the higher-order term coefficients is reported.

LANDAU-DE GENNES MODEL WITH THE INCLUSION OF DENSITY CHANGE FOR THE NEMATIC-ISOTROPIC PHASE TRANSITION

1996 ◽  
Vol 10 (16) ◽  
pp. 777-778 ◽  
Author(s):  
B. NANDI ◽  
P.K. MUKHERJEE ◽  
M. SAHA
Keyword(s):  
Phase Transition ◽  
Temperature Difference ◽  
Order Term ◽  
Higher Order ◽  
Density Change ◽  
Sixth Order ◽  
Isotropic Phase ◽  
Definite Improvement ◽  
Higher Order Term ◽  
Good Agreement

The Landau-de Gennes model of the nematic-isotropic phase transition with the inclusion of the density change and a higher order term (sixth-order term) in a simple way is examined. The temperature difference T NI −T* obtained is in good agreement with the experimental one. We also obtained the value of the ratio (Q*–Q NI )/Q NI . The value of (Q*–Q NI )/Q NI is not improved over the earlier attempt, but there is a definite improvement in the value of T NI –T* which is obtained as 1.045 K.

An intensional type theory: motivation and cut-elimination

2001 ◽  
Vol 66 (1) ◽  
pp. 383-400 ◽  
Author(s):  
Paul C Gilmore
Keyword(s):  
Type Theory ◽  
Proof Theory ◽  
Predicate Logic ◽  
Order Term ◽  
Higher Order ◽  
Cut Elimination ◽  
Consistency Proof ◽  
Completeness Proof ◽  
Higher Order Term

AbstractBy the theory TT is meant the higher order predicate logic with the following recursively defined types:(1) 1 is the type of individuals and [] is the type of the truth values:(2) [τ1…..τn] is the type of the predicates with arguments of the types τ1…..τn.The theory ITT described in this paper is an intensional version of TT. The types of ITT are the same as the types of TT, but the membership of the type 1 of individuals in ITT is an extension of the membership in TT. The extension consists of allowing any higher order term, in which only variables of type 1 have a free occurrence, to be a term of type 1. This feature of ITT is motivated by a nominalist interpretation of higher order predication.In ITT both well-founded and non-well-founded recursive predicates can be defined as abstraction terms from which all the properties of the predicates can be derived without the use of non-logical axioms.The elementary syntax, semantics, and proof theory for ITT are defined. A semantic consistency proof for ITT is provided and the completeness proof of Takahashi and Prawitz for a version of TT without cut is adapted for ITT: a consequence is the redundancy of cut.

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