scholarly journals Experiments with ZF Set Theory in HOL and Isabelle

1995 ◽  
Vol 2 (37) ◽  
Author(s):  
Sten Agerholm ◽  
Mike Gordon
Keyword(s):  
Set Theory ◽  
Lambda Calculus ◽  
General Purpose ◽  
Higher Order ◽  
Proof Assistants ◽  
Higher Order Logic ◽  
First Order ◽  
Advantages And Disadvantages ◽  
Theory Versus

Most general purpose proof assistants support versions of<br />typed higher order logic. Experience has shown that these logics are capable<br />of representing most of the mathematical models needed in Computer<br />Science. However, perhaps there exist applications where ZF-style<br />set theory is more natural, or even necessary. Examples may include<br />Scott's classical inverse-limit construction of a model of the untyped lambda-calculus<br /> (D_inf) and the semantics of parts of the Z specification notation.<br /><br />This paper compares the representation and use of ZF set theory within<br />both HOL and Isabelle. The main case study is the construction of D_inf.<br />The advantages and disadvantages of higher-order set theory versus first-order<br />set theory are explored experimentally. This study also provides a<br />comparison of the proof infrastructure of HOL and Isabelle.

Resolution in type theory

1971 ◽  
Vol 36 (3) ◽  
pp. 414-432 ◽  
Author(s):  
Peter B. Andrews
Keyword(s):  
Set Theory ◽  
Type Theory ◽  
Higher Order ◽  
Order Logic ◽  
First Order Logic ◽  
Higher Order Logic ◽  
First Order ◽  
Different Types ◽  
Axiomatic Set Theory ◽  
Order Predicate Calculus

In [8] J. A. Robinson introduced a complete refutation procedure called resolution for first order predicate calculus. Resolution is based on ideas in Herbrand's Theorem, and provides a very convenient framework in which to search for a proof of a wff believed to be a theorem. Moreover, it has proved possible to formulate many refinements of resolution which are still complete but are more efficient, at least in many contexts. However, when efficiency is a prime consideration, the restriction to first order logic is unfortunate, since many statements of mathematics (and other disciplines) can be expressed more simply and naturally in higher order logic than in first order logic. Also, the fact that in higher order logic (as in many-sorted first order logic) there is an explicit syntactic distinction between expressions which denote different types of intuitive objects is of great value where matching is involved, since one is automatically prevented from trying to make certain inappropriate matches. (One may contrast this with the situation in which mathematical statements are expressed in the symbolism of axiomatic set theory.).

scholarly journals POPLMark reloaded: Mechanizing proofs by logical relations

2019 ◽  
Vol 29 ◽  
Author(s):  
ANDREAS ABEL ◽  
GUILLAUME ALLAIS ◽  
ALIYA HAMEER ◽  
BRIGITTE PIENTKA ◽  
ALBERTO MOMIGLIANO ◽  
...  
Keyword(s):  
Programming Languages ◽  
State Of The Art ◽  
General Purpose ◽  
Higher Order ◽  
Lessons Learned ◽  
Benchmark Problems ◽  
Proof Assistants ◽  
Abstract Syntax ◽  
The Masses

Abstract We propose a new collection of benchmark problems in mechanizing the metatheory of programming languages, in order to compare and push the state of the art of proof assistants. In particular, we focus on proofs using logical relations (LRs) and propose establishing strong normalization of a simply typed calculus with a proof by Kripke-style LRs as a benchmark. We give a modern view of this well-understood problem by formulating our LR on well-typed terms. Using this case study, we share some of the lessons learned tackling this problem in different dependently typed proof environments. In particular, we consider the mechanization in Beluga, a proof environment that supports higher-order abstract syntax encodings and contrast it to the development and strategies used in general-purpose proof assistants such as Coq and Agda. The goal of this paper is to engage the community in discussions on what support in proof environments is needed to truly bring mechanized metatheory to the masses and engage said community in the crafting of future benchmarks.

Dual systems for all: Higher-order, role-based relational reasoning as a uniquely derived feature of human cognition

2019 ◽  
Vol 42 ◽  
Author(s):  
Daniel J. Povinelli ◽  
Gabrielle C. Glorioso ◽  
Shannon L. Kuznar ◽  
Mateja Pavlic
Keyword(s):  
Temporal Reasoning ◽  
Higher Order ◽  
Important Case ◽  
Human Cognition ◽  
Relational Reasoning ◽  
Dual Systems ◽  
First Order ◽  
Reasoning System ◽  
Role Based

Abstract Hoerl and McCormack demonstrate that although animals possess a sophisticated temporal updating system, there is no evidence that they also possess a temporal reasoning system. This important case study is directly related to the broader claim that although animals are manifestly capable of first-order (perceptually-based) relational reasoning, they lack the capacity for higher-order, role-based relational reasoning. We argue this distinction applies to all domains of cognition.

scholarly journals Making Automatic Theorem Provers more Versatile

10.29007/n6j7 ◽  
2018 ◽  
Author(s):  
Simon Cruanes
Keyword(s):  
Higher Order ◽  
Order Logic ◽  
Higher Order Logic ◽  
Research Directions ◽  
First Order ◽  
Theorem Provers ◽  
Smt Solvers ◽  
Automatic Theorem Provers ◽  
Automatic Theorem

We argue that automatic theorem provers should become more versatile and should be able to tackle problems expressed in richer input formats. Salient research directions include (i) developing tight combinations of SMT solvers and first-order provers; (ii) adding better handling of theories in first-order provers; (iii) adding support for inductive proving; (iv) adding support for user-defined theories and functions; and (v) bringing to the provers some basic abilities to deal with logics beyond first-order, such as higher-order logic.

An untyped higher order logic with Y combinator

2007 ◽  
Vol 72 (4) ◽  
pp. 1385-1404
Author(s):  
James H. Andrews
Keyword(s):  
Model Theory ◽  
Lambda Calculus ◽  
Higher Order ◽  
Order Logic ◽  
Proof System ◽  
Cut Elimination ◽  
Higher Order Logic ◽  
Consistent Model

AbstractWe define a higher order logic which has only a notion of sort rather than a notion of type, and which permits all terms of the untyped lambda calculus and allows the use of the Y combinator in writing recursive predicates. The consistency of the logic is maintained by a distinction between use and mention, as in Gilmore's logics. We give a consistent model theory, a proof system which is sound with respect to the model theory, and a cut-elimination proof for the proof system. We also give examples showing what formulas can and cannot be used in the logic.

scholarly journals Formalization of Linear Space Theory in the Higher-Order Logic Proving System

2013 ◽  
Vol 2013 ◽  
pp. 1-6 ◽  
Author(s):  
Jie Zhang ◽  
Danwen Mao ◽  
Yong Guan
Keyword(s):  
Linear Space ◽  
Theorem Proving ◽  
Predicate Logic ◽  
Higher Order ◽  
Order Logic ◽  
First Order Logic ◽  
Space Theory ◽  
Higher Order Logic ◽  
First Order ◽  
Important Approach

Theorem proving is an important approach in formal verification. Higher-order logic is a form of predicate logic that is distinguished from first-order logic by additional quantifiers and stronger semantics. Higher-order logic is more expressive. This paper presents the formalization of the linear space theory in HOL4. A set of properties is characterized in HOL4. This result is used to build the underpinnings for the application of higher-order logic in a wider spectrum of engineering applications.

scholarly journals Antirealism/realism of Hintikka’s game-theoretical semantics

2018 ◽  
Vol 26 (3) ◽  
Author(s):  
Heda Festini
Keyword(s):  
Higher Order ◽  
Order Logic ◽  
First Order Logic ◽  
Higher Order Logic ◽  
First Order ◽  
Theory Of Meaning ◽  
Truth Conditional ◽  
Context Dependent ◽  
Extended Notion

Hintikka’s game-theoretical semantics (GTS) is presented as an anti-Tarskian semantical approach to the context-dependent fragments of Englisch, which overcomes the usual notion of semantical realism. Analysing Hintikka’s critique of Tarski’s interpretation of the truth-conditional theory of meaning, its recursive fashion and the narrow notion of realism, Hintikka’s basic conception is presented in the following manner:1. the Context-Principle vs. the Frege Principle,2.First-order logic together with higher-order logic vs. the primacy of first-order logic,3.verificationist/falsificationist theory vs. Taraski’s narrow truth- conditional theory.Comparing some reviews of Hintikka’s GTS (M. Dummett, E. Itkonen, E. Saarinen, M. Hand) with a short examination of the antirealistic/realistic controversis by C. Wright and M. Dummett, the following was reached:Hintikka’s GTS introduces a new, more extended notion of realism, which embraces Taraski-type realistic semantics, Hintikka’s GTS and with this the question of the possibility to also include Dummett’s neoverificationism or other orientations, remains open.

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