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Type classes and overloading in higher-order logic
[chapter]
1997
Lecture Notes in Computer Science
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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]
2010
Lecture Notes in Computer Science
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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
2007
Journal of automated reasoning
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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]
1992
Lecture Notes in Computer Science
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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]
2016
arXiv
pre-print
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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
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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]
2018
arXiv
pre-print
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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]
2009
Lecture Notes in Computer Science
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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
2004
Journal of automated reasoning
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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
2013
ACM/IEEE International Conference on Model Driven Engineering Languages and Systems
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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]
2016
Lecture Notes in Computer Science
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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
2002
Electronical Notes in Theoretical Computer Science
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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
2015
European Conference on Object-Oriented Programming
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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
2002
Proceedings of the 29th ACM SIGPLAN-SIGACT symposium on Principles of programming languages - POPL '02
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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
2002
SIGPLAN notices
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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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