On the Unification Problem in Physics

In this paper we investigate the type inference problem for a large class of type assignment systems for the λ-calculus. This is the problem of determining if a term has a type in a given system. We discuss, in particular, a collection of type assignment systems which correspond to the typed systems of Barendregt’s “cube”. Type dependencies being shown redundant, we focus on the strongest of all, Fω, the type assignment version of the system Fω of Girard. In order to manipulate uniformly type inferences we give a syntax directed presentation of Fω and introduce the notions of scheme and of principal type scheme. Making essential use of them, we succeed in reducing the type inference problem for Fω to a restriction of the higher order semi-unification problem and in showing that the conditional type inference problem for Fω is undecidable. Throughout the paper we call attention to open problems and formulate some conjectures.

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The unification problem

Biolinguistics ◽

10.1017/cbo9780511605765.003 ◽

2000 ◽

pp. 15-56

Keyword(s):

Unification Problem

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Analysis of Service Compatibility

Services and Business Computing Solutions with XML ◽

10.4018/978-1-60566-330-2.ch009 ◽

2010 ◽

pp. 136-155

Author(s):

Ken Q. Pu

Keyword(s):

Relational Database ◽

Type System ◽

Large Family ◽

Service Description ◽

Worst Case ◽

Database Queries ◽

Unification Algorithm ◽

Data Services ◽

Wide Range ◽

Unification Problem

In this chapter, the authors apply type-theoretic techniques to the service description and composition verification. A flexible type system is introduced for modeling instances and mappings of semi-structured data, and is demonstrated to be effective in modeling a wide range of data services, ranging from relational database queries to web services for XML. Type-theoretic analysis and verification are then reduced to the problem of type unification. Some (in)tractability results of the unification problem and the expressiveness of their proposed type system are presented in this chapter. Finally, the auhtors construct a complete unification algorithm which runs in EXP-TIME in the worst case, but runs in polynomial time for a large family of unification problems rising from practical type analysis of service compositions.

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A decidable case of the semi-unification problem

Mathematical Foundations of Computer Science 1991 - Lecture Notes in Computer Science ◽

10.1007/3-540-54345-7_75 ◽

1991 ◽

pp. 318-327 ◽

Cited By ~ 2

Author(s):

Hans Leiß ◽

Fritz Henglein

Keyword(s):

Unification Problem

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A parallel algorithm for the monadic unification problem

BIT Numerical Mathematics ◽

10.1007/bf01934375 ◽

1985 ◽

Vol 25 (2) ◽

pp. 302-306 ◽

Cited By ~ 2

Author(s):

Ivan E. Auger ◽

M. S. Krishnamoorthy

Keyword(s):

Parallel Algorithm ◽

Unification Problem

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Unification problem in equational theories

Cybernetics and Systems Analysis ◽

10.1007/bf02733227 ◽

1997 ◽

Vol 33 (6) ◽

pp. 874-899

Author(s):

S. L. Kryvyi

Keyword(s):

Equational Theories ◽

Unification Problem

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A framework for improving error messages in dependently-typed languages

Open Computer Science ◽

10.1515/comp-2019-0001 ◽

2019 ◽

Vol 9 (1) ◽

pp. 1-32 ◽

Cited By ~ 1

Author(s):

Joseph Eremondi ◽

Wouter Swierstra ◽

Jurriaan Hage

Keyword(s):

Programming Languages ◽

Type System ◽

Higher Order ◽

Type Checking ◽

Unification Algorithm ◽

Implicit Arguments ◽

Unification Problem ◽

Heuristic Analysis ◽

Type Check ◽

Typed Languages

AbstractDependently-typed programming languages provide a powerful tool for establishing code correctness. However, it can be hard for newcomers to learn how to employ the advanced type system of such languages effectively. For simply-typed languages, several techniques have been devised to generate helpful error messages and suggestions for the programmer. We adapt these techniques to dependently-typed languages, to facilitate their more widespread adoption. In particular, we modify a higher-order unification algorithm that is used to resolve and type-check implicit arguments. We augment this algorithm with replay graphs, allowing for a global heuristic analysis of a unification problem-set, error-tolerant typing, which allows type-checking to continue after errors are found, and counter-factual unification, which makes error messages less affected by the order in which types are checked. A formalization of our algorithm is presented with an outline of its correctness. We implement replay graphs, and compare the generated error messages to those from existing languages, highlighting the improvements we achieved.

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