scholarly journals Natural Deduction and the Isabelle Proof Assistant

2018 ◽  
Vol 267 ◽  
pp. 140-155 ◽  
Author(s):  
Jørgen Villadsen ◽  
Andreas Halkjær From ◽  
Anders Schlichtkrull
Keyword(s):  
Natural Deduction ◽  
Proof Assistant ◽  
Isabelle Proof Assistant

Natural deduction and semantic models of justification logic in the proof assistant Coq

2020 ◽  
Author(s):  
Jesús Mauricio Andrade Guzmán ◽  
Francisco Hernández Quiroz
Keyword(s):  
Natural Deduction ◽  
Axiomatic Approach ◽  
Proof Assistant ◽  
Justification Logic ◽  
Possible World ◽  
Possible World Semantics ◽  
Language Semantics ◽  
Semantic Models

Abstract The purpose of this paper is to present a formalization of the language, semantics and axiomatization of justification logic in Coq. We present proofs in a natural deduction style derived from the axiomatic approach of justification logic. Additionally, we present possible world semantics in Coq based on Fitting models to formalize the semantic satisfaction of formulas. As an important result, with this implementation, it is possible to give a proof of soundness for $\mathsf{L}\mathsf{P}$ with respect to Fitting models.

scholarly journals A MACHINE-ASSISTED PROOF OF GÖDEL’S INCOMPLETENESS THEOREMS FOR THE THEORY OF HEREDITARILY FINITE SETS

2014 ◽  
Vol 7 (3) ◽  
pp. 484-498 ◽  
Author(s):  
LAWRENCE C. PAULSON
Keyword(s):  
Set Theory ◽  
Incompleteness Theorem ◽  
Proof Assistant ◽  
Mechanical Verification ◽  
Finite Set ◽  
Technical Issues ◽  
Incompleteness Theorems ◽  
Isabelle Proof Assistant ◽  
Finite Set Theory ◽  
Gödel’S Incompleteness Theorems

AbstractA formalization of Gödel’s incompleteness theorems using the Isabelle proof assistant is described. This is apparently the first mechanical verification of the second incompleteness theorem. The work closely follows Świerczkowski (2003), who gave a detailed proof using hereditarily finite set theory. The adoption of this theory is generally beneficial, but it poses certain technical issues that do not arise for Peano arithmetic. The formalization itself should be useful to logicians, particularly concerning the second incompleteness theorem, where existing proofs are lacking in detail.

scholarly journals Reasoning about Partial Correctness Assertions in Isabelle/HOL

2020 ◽  
Vol 27 (3) ◽  
pp. 84-101
Author(s):  
Alfio Ricardo Martini
Keyword(s):  
Formal Verification ◽  
Broad Class ◽  
Object Oriented ◽  
Concurrent Programs ◽  
Proof Assistant ◽  
Hoare Logic ◽  
Partial Correctness ◽  
Human Understanding ◽  
Isabelle Proof Assistant

Hoare Logic has a long tradition in formal verification and has been continuously developed and used to verify a broad class of programs, including sequential, object-oriented and concurrent programs. The purpose of this work is to provide a detailed and accessible exposition of the several ways the user can conduct, explore and write proofs of correctness of sequential imperative programs with Hoare logic and the ISABELLE proof assistant. With the proof language Isar, it is possible to write structured, readable proofs that are suitable for human understanding and communication.

scholarly journals Certifying Proofs in the First-Order Theory of Rewriting

2021 ◽  
pp. 127-144
Author(s):  
Fabian Mitterwallner ◽  
Alexander Lochmann ◽  
Aart Middeldorp ◽  
Bertram Felgenhauer
Keyword(s):  
Decision Procedure ◽  
Order Theory ◽  
Proof Assistant ◽  
Tree Automata ◽  
Decision Tool ◽  
Rewrite Systems ◽  
First Order ◽  
Decidable Theory ◽  
First Order Theory ◽  
Isabelle Proof Assistant

AbstractThe first-order theory of rewriting is a decidable theory for linear variable-separated rewrite systems. The decision procedure is based on tree automata techniques and recently we completed a formalization in the Isabelle proof assistant. In this paper we present a certificate language that enables the output of software tools implementing the decision procedure to be formally verified. To show the feasibility of this approach, we present , a reincarnation of the decision tool with certifiable output, and the formally verified certifier .

scholarly journals Enhanced Realizability Interpretation for Program Extraction

2021 ◽  
Author(s):  
◽  
Olga Petrovska
Keyword(s):  
Fixed Point ◽  
Natural Deduction ◽  
Theoretical Research ◽  
Inference Rules ◽  
Proof Assistant ◽  
Basic Form ◽  
Program Extraction ◽  
Formal Systems ◽  
Proof Construction

This thesis presents Intuitionistic Fixed Point Logic (IFP), a schema for formal systems aimed to work with program extraction from proofs. IFP in its basic form allows proof construction based on natural deduction inference rules, extended by induction and coinduction. The corresponding system RIFP (IFP with realiz-ers) enables transforming logical proofs into programs utilizing the enhanced re-alizability interpretation. The theoretical research is put into practice in PRAWF1, a Haskell-based proof assistant for program extraction.

Verification of Content-Centric Networking Using Proof Assistant

2016 ◽  
Vol E99.B (11) ◽  
pp. 2297-2304
Author(s):  
Sosuke MORIGUCHI ◽  
Takashi MORISHIMA ◽  
Mizuki GOTO ◽  
Kazuko TAKAHASHI
Keyword(s):  
Proof Assistant ◽  
Content Centric Networking

Suppose and Tell

2020 ◽  
Author(s):  
Timothy Williamson
Keyword(s):  
Conditional Probability ◽  
Natural Deduction ◽  
Special Kind ◽  
Context Sensitivity ◽  
Modal Operator ◽  
Counterfactual Conditionals ◽  
Deduction Rules ◽  
Psychological Sense

The book argues that our use of conditionals is governed by imperfectly reliable heuristics, in the psychological sense of fast and frugal (or quick and dirty) ways of assessing them. The primary heuristic is this: to assess ‘If A, C’, suppose A and on that basis assess C; whatever attitude you take to C conditionally on A (such as acceptance, rejection, or something in between) take unconditionally to ‘If A, C’. This heuristic yields both the equation of the probability of ‘If A, C’ with the conditional probability of C on A and standard natural deduction rules for the conditional. However, these results can be shown to make the heuristic implicitly inconsistent, and so less than fully reliable. There is also a secondary heuristic: pass conditionals freely from one context to another under normal conditions for acceptance of sentences on the basis of memory and testimony. The effect of the secondary heuristic is to undermine interpretations on which ‘if’ introduces a special kind of context-sensitivity. On the interpretation which makes best sense of the two heuristics, ‘if’ is simply the truth-functional conditional. Apparent counterexamples to truth-functionality are artefacts of reliance on the primary heuristic in cases where it is unreliable. The second half of the book concerns counterfactual conditionals, as expressed with ‘if’ and ‘would’. It argues that ‘would’ is an independently meaningful modal operator for contextually restricted necessity: the meaning of counterfactuals is simply that derived compositionally from the meanings of their constituents, including ‘if’ and ‘would’, making them contextually restricted strict conditionals.

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