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Type classes and overloading in higher-order logic [chapter]

Markus Wenzel
1997 Lecture Notes in Computer Science  
Type classes and overloading are shown to be independent concepts that can both be added to simple higher-order logics in the tradition of Church and Gordon, without demanding more logical expressiveness  ...  The latter can be used to describe simple abstract theories with a single carrier type and a fixed signature of operations.  ...  Olaf Müller, Wolfgang Naraschewski, David von Oheimb and Oscar Slotosch commented draft versions of the paper.  ... 
doi:10.1007/bfb0028402 fatcat:a7tffdkm2vgblfuk6e5grjqqea

A Mechanized Translation from Higher-Order Logic to Set Theory [chapter]

Alexander Krauss, Andreas Schropp
2010 Lecture Notes in Computer Science  
In order to make existing formalizations available for settheoretic developments, we present an automated translation of theories from Isabelle/HOL to Isabelle/ZF.  ...  This covers all fundamental primitives, particularly type classes. The translation produces LCF-style theorems that are checked by Isabelle's inference kernel.  ...  Sascha Böhme and Armin Heller read a draft of this paper on short notice and made helpful comments.  ... 
doi:10.1007/978-3-642-14052-5_23 fatcat:p2bsnjmo6nd3bp5vrokabcfwdi

Translating Higher-Order Clauses to First-Order Clauses

Jia Meng, Lawrence C. Paulson
2007 Journal of automated reasoning  
Interactive provers typically use higher-order logic, while automatic provers typically use first-order logic.  ...  In order to integrate interactive provers with automatic ones, it is necessary to translate higher-order formulae to first-order form. The translation should ideally be both sound and practical.  ...  Christoph Benzmüller and various referees made many useful suggestions for improving this paper.  ... 
doi:10.1007/s10817-007-9085-y fatcat:4uluysnq7rb7vjxzbcnvsgpcci

Isabelle-91 [chapter]

Tobias Nipkow, Lawrence C. Paulson
1992 Lecture Notes in Computer Science  
The latest incarnation of Isabelle, Isabelle-91, features a type system based on order-sorted unification; this supports polymorphism and overloading in logic definitions.  ...  Object-logics are formalized within higher-order logic, which is Isabelle's meta-logic. Proofs are performed by a generalization of resolution, using higher-order unification.  ...  This rules out quantification over predicates and functions, thus ensuring that FOL is indeed a first-order logic and not a higher-order one in disguise.  ... 
doi:10.1007/3-540-55602-8_201 fatcat:ontrfl4pmjcyxbrufswzsbwxqu

CoALP-Ty'16 [article]

Ekaterina Komendantskaya, František Farka
2016 arXiv   pre-print
This volume constitutes the pre-proceedings of the Workshop on Coalgebra, Horn Clause Logic Programming and Types (CoALP-Ty'16), held on 28--29 November 2016 in Edinburgh as a mark of the end of the EPSRC  ...  Clause Logic for Type Inference in Functional Languages and Beyond After discussion at the workshop authors of the extended abstracts will be invited to submit a full paper to go through a second round  ...  This in turn gives us a universe of discourse for exploring other semantics and proof systems for mixed inductive-coinductive logic programs.  ... 
arXiv:1612.03032v1 fatcat:d6gug5imufgwbcslntnyts4nim

Page 7800 of Mathematical Reviews Vol. , Issue 97M [page]

1997 Mathematical Reviews  
The authors introduce the class of negative logic pro- grams and a simple graph representation, the rule graph.  ...  Logic Programming 1997, Article 2, 19 pp. (electronic). Summary: “We present a minimal function graph semantics for a higher-order functional language with applicative evaluation or- der.  ... 

Formalising Mathematics In Simple Type Theory [article]

Lawrence C. Paulson
2018 arXiv   pre-print
Despite the considerable interest in new dependent type theories, simple type theory (which dates from 1940) is sufficient to formalise serious topics in mathematics.  ...  Harrison's technique for formalising Euclidean spaces is contrasted with an approach using Isabelle/HOL's axiomatic type classes.  ...  Many thanks to Jeremy Avigad, Johannes Hölzl, Andrew Pitts and the anonymous referee for their comments.  ... 
arXiv:1804.07860v1 fatcat:zkrhn2d5nzawrfi2rh62fxghqy

Local Theory Specifications in Isabelle/Isar [chapter]

Florian Haftmann, Makarius Wenzel
2009 Lecture Notes in Computer Science  
Here we introduce targets for locales, type-classes, and class instantiations.  ...  For example, a recursive function may be defined depending on locale parameters and assumptions, or an inductive predicate definition may provide the witness in a type-class instantiation.  ...  Amine Chaieb convinced the authors that serious integration of locales and type-classes is really needed for advanced algebraic proof tools.  ... 
doi:10.1007/978-3-642-02444-3_10 fatcat:ypqrtlwn35fpxp5o7wei37e5cy

Organizing Numerical Theories Using Axiomatic Type Classes

Lawrence C. Paulson
2004 Journal of automated reasoning  
The approach recently adopted for Isabelle uses axiomatic type classes, an established approach to overloading.  ...  Abstractions such as semirings, rings, fields and their ordered counterparts are defined and theorems are proved algebraically. Types that meet the abstractions inherit the appropriate theorems.  ...  Tobias Nipkow and Markus Wenzel equipped Isabelle with axiomatic type classes. The type classes for ordered rings and fields are based on earlier work by Gertrud Bauer and Wenzel.  ... 
doi:10.1007/s10817-004-3997-6 fatcat:muybtjzpmbburkh3l5w5hfrm4y

On the Semantics of Object-Oriented Data Structures and Path Expressions

Achim D. Brucker, Delphine Longuet, Frédéric Tuong, Burkhart Wolff
2013 ACM/IEEE International Conference on Model Driven Engineering Languages and Systems  
In this paper, we present a formal semantics for object-oriented data models in which all data types and, thus, all class attributes and path expressions, support invalid and null.  ...  This has far-reaching consequences on both the logical and algebraic properties of OCL expressions as well as the path expressions over object-oriented data structures, i. e., class models.  ...  This research work has also been carried out under the leadership of the Technological Research Institute SystemX, and therefore granted with public funds within the scope of the French Program "Investissements  ... 
dblp:conf/models/BruckerLTW13 fatcat:tvts4fjd3nhm7ivdkx22sqdjwa

From Types to Sets by Local Type Definitions in Higher-Order Logic [chapter]

Ondřej Kunčar, Andrei Popescu
2016 Lecture Notes in Computer Science  
Types in Higher-Order Logic (HOL) are naturally interpreted as nonempty sets-this intuition is reflected in the type definition rule for the HOL-based systems (including Isabelle/HOL), where a new type  ...  We also address particularities of Isabelle/HOL and show how to perform the relativization in the presence of type classes.  ...  Acknowledgements We are indebted to the reviewers for useful comments and suggestions. We gratefully acknowledge support from DFG through grant Ni 491/13-3 and from EPSRC through grant EP/N019547/1.  ... 
doi:10.1007/978-3-319-43144-4_13 fatcat:woyf3gv5ffd7jlbb2ykyr6xvlq

Programmable Type Systems for Domain Specific Languages

Peter Thiemann
2002 Electronical Notes in Theoretical Computer Science  
In both cases we assess the guarantees obtained by the use of the typing and explore alternative solutions.  ...  We show that a component that ensures the generation of correct HTML documents can take good advantage of type-level functions, as implemented using functional logic overloading.  ...  Alternatively, a version limited to first-order kinds can be readily implemented using functional logic overloading [4] .  ... 
doi:10.1016/s1571-0661(04)80796-5 fatcat:5jg3coclvzb25aa222xucph33i

Trust, but Verify: Two-Phase Typing for Dynamic Languages

Panagiotis Vekris, Benjamin Cosman, Ranjit Jhala, Marc Herbstritt
2015 European Conference on Object-Oriented Programming  
Thus, to establish basic types, the analysis must reason precisely about values, but in the presence of higher-order functions and polymorphism, this reasoning itself can require basic types.  ...  Furthermore, the expressiveness of the basic type system and logics can be extended independently, e.g. to account for polymorphism, classes or new logical theories, directly yielding a more expressive  ...  This work was supported by NSF Grants CNS-1223850, CNS-0964702 and gifts from Microsoft Research.  ... 
doi:10.4230/lipics.ecoop.2015.52 dblp:conf/ecoop/VekrisCJ15 fatcat:ehfvzpml3zcrhofqddvitbfrfm

Functional logic overloading

Matthias Neubauer, Peter Thiemann, Martin Gasbichler, Michael Sperber
2002 Proceedings of the 29th ACM SIGPLAN-SIGACT symposium on Principles of programming languages - POPL '02  
Functional logic overloading is a novel approach to userdefined overloading that extends Haskell's concept of type classes in significant ways.  ...  Whereas type classes are conceptually predicates on types in standard Haskell, they are type functions in our approach.  ...  Embedding of Standard Type Classes It is easy to encode standard type classes in our system of type functions.  ... 
doi:10.1145/503272.503294 dblp:conf/popl/NeubauerTGS02 fatcat:hwv5233t3zgivhxkxyky3ysbjm

Functional logic overloading

Matthias Neubauer, Peter Thiemann, Martin Gasbichler, Michael Sperber
2002 SIGPLAN notices  
Functional logic overloading is a novel approach to userdefined overloading that extends Haskell's concept of type classes in significant ways.  ...  Whereas type classes are conceptually predicates on types in standard Haskell, they are type functions in our approach.  ...  Embedding of Standard Type Classes It is easy to encode standard type classes in our system of type functions.  ... 
doi:10.1145/565816.503294 fatcat:kiujqpbeeffr7a7yaczllzdxlq
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