scholarly journals Verified Interactive Computation of Definite Integrals

2021 ◽  
pp. 485-503
Author(s):  
Runqing Xu ◽  
Liming Li ◽  
Bohua Zhan
Keyword(s):  
General Framework ◽  
Interactive Theorem Proving ◽  
Order Logic ◽  
Computer Algebra Systems ◽  
Science And Engineering ◽  
Higher Order Logic ◽  
Undergraduate Level ◽  
Definite Integrals ◽  
Proof Automation ◽  
Interactive Computation

AbstractSymbolic computation is involved in many areas of mathematics, as well as in analysis of physical systems in science and engineering. Computer algebra systems present an easy-to-use interface for performing these calculations, but do not provide strong guarantees of correctness. In contrast, interactive theorem proving provides much stronger guarantees of correctness, but requires more time and expertise. In this paper, we propose a general framework for combining these two methods, and demonstrate it using computation of definite integrals. It allows the user to carry out step-by-step computations in a familiar user interface, while also verifying the computation by translating it to proofs in higher-order logic. The system consists of an intermediate language for recording computations, proof automation for simplification and inequality checking, and heuristic integration methods. A prototype is implemented in Python based on HolPy, and tested on a large collection of examples at the undergraduate level.

scholarly journals Formal verification of Matrix based MATLAB models using interactive theorem proving

2021 ◽  
Vol 7 ◽  
pp. e440
Author(s):  
Ayesha Gauhar ◽  
Adnan Rashid ◽  
Osman Hasan ◽  
João Bispo ◽  
João M.P. Cardoso
Keyword(s):  
Formal Verification ◽  
Finite Impulse Response ◽  
Formal Analysis ◽  
Digital Signal ◽  
Interactive Theorem Proving ◽  
Higher Order ◽  
Order Logic ◽  
Theorem Prover ◽  
Formal Models ◽  
Higher Order Logic

MATLAB is a software based analysis environment that supports a high-level programing language and is widely used to model and analyze systems in various domains of engineering and sciences. Traditionally, the analysis of MATLAB models is done using simulation and debugging/testing frameworks. These methods provide limited coverage due to their inherent incompleteness. Formal verification can overcome these limitations, but developing the formal models of the underlying MATLAB models is a very challenging and time-consuming task, especially in the case of higher-order-logic models. To facilitate this process, we present a library of higher-order-logic functions corresponding to the commonly used matrix functions of MATLAB as well as a translator that allows automatic conversion of MATLAB models to higher-order logic. The formal models can then be formally verified in an interactive theorem prover. For illustrating the usefulness of the proposed library and approach, we present the formal analysis of a Finite Impulse Response (FIR) filter, which is quite commonly used in digital signal processing applications, within the sound core of the HOL Light theorem prover.

scholarly journals Composable Packages for Higher Order Logic Theories

10.29007/7gg3 ◽  
2018 ◽  
Author(s):  
Joe Hurd
Keyword(s):  
Theory Development ◽  
Interactive Theorem Proving ◽  
Higher Order ◽  
Order Logic ◽  
Theorem Prover ◽  
Higher Order Logic ◽  
Domain Specific ◽  
Language Functions ◽  
Hol Light ◽  
Dependency Structures

Interactive theorem proving is tackling ever larger formalization and verification projects, and there is a critical need for theory engineering techniques to support these efforts. One such technique is effective package management, which has the potential to simplify the development of logical theories by precisely checking dependencies and promoting re-use. This paper introduces a domain-specific language for defining composable packages of higher order logic theories, which is designed to naturally handle the complex dependency structures that often arise in theory development. The package composition language functions as a module system for theories, and the paper presents a well-defined semantics for the supported operations. Preliminary tests of the package language and its toolset have been made by packaging the theories distributed with the HOL Light theorem prover. This experience is described, leading to some initial theory engineering discussion on the ideal properties of a reusable theory.

scholarly journals Closed Structure

2021 ◽  
Author(s):  
Peter Fritz ◽  
Harvey Lederman ◽  
Gabriel Uzquiano
Keyword(s):  
Formal Language ◽  
Syntactic Structure ◽  
Natural Extension ◽  
Propositional Attitudes ◽  
Higher Order ◽  
Order Logic ◽  
Bertrand Russell ◽  
Structured Propositions ◽  
Higher Order Logic ◽  
Closed Structure

AbstractAccording to the structured theory of propositions, if two sentences express the same proposition, then they have the same syntactic structure, with corresponding syntactic constituents expressing the same entities. A number of philosophers have recently focused attention on a powerful argument against this theory, based on a result by Bertrand Russell, which shows that the theory of structured propositions is inconsistent in higher order-logic. This paper explores a response to this argument, which involves restricting the scope of the claim that propositions are structured, so that it does not hold for all propositions whatsoever, but only for those which are expressible using closed sentences of a given formal language. We call this restricted principle Closed Structure, and show that it is consistent in classical higher-order logic. As a schematic principle, the strength of Closed Structure is dependent on the chosen language. For its consistency to be philosophically significant, it also needs to be consistent in every extension of the language which the theorist of structured propositions is apt to accept. But, we go on to show, Closed Structure is in fact inconsistent in a very natural extension of the standard language of higher-order logic, which adds resources for plural talk of propositions. We conclude that this particular strategy of restricting the scope of the claim that propositions are structured is not a compelling response to the argument based on Russell’s result, though we note that for some applications, for instance to propositional attitudes, a restricted thesis in the vicinity may hold some promise.

Adapting functional programs to higher order logic

2008 ◽  
Vol 21 (4) ◽  
pp. 377-409 ◽  
Author(s):  
Scott Owens ◽  
Konrad Slind
Keyword(s):  
Higher Order ◽  
Order Logic ◽  
Higher Order Logic

scholarly journals “How did the serpent of inconsistency enter Frege’s paradise?”

2019 ◽  
pp. 411-436
Author(s):  
Crispin Wright
Keyword(s):  
Cardinal Number ◽  
Ordinal Number ◽  
Higher Order ◽  
Order Logic ◽  
Higher Order Logic ◽  
Indefinite Extensibility ◽  
Basic Law ◽  
Per Se

The paper explores the alleged connection between indefinite extensibility and the classic paradoxes of Russell, Burali-Forti, and Cantor. It is argued that while indefinite extensibility is not per se a source of paradox, there is a degenerate subspecies—reflexive indefinite extensibility—which is. The result is a threefold distinction in the roles played by indefinite extensibility in generating paradoxes for the notions of ordinal number, cardinal number, and set respectively. Ordinal number, intuitively understood, is a reflexively indefinitely extensible concept. Cardinal number is not. And Set becomes so only in the setting of impredicative higher-order logic—so that Frege’s Basic Law V is guilty at worst of partnership in crime, rather than the sole offender.

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