Students' Proof Assistant (SPA)

Schlichtkrull, Anders; Villadsen, Jørgen; From, Asta Halkjær

doi:10.4204/eptcs.290.1

Cited by 9 publications

(8 citation statements)

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“…We briefly describe our route from axiomatic propositional logic [7] to firstorder logic with equality in our Students' Proof Assistant (SPA) [36] running inside Isabelle/HOL with a formally verified LCF-style prover kernel [31] and declarative proofs [41,42].…”

Section: Week Main Topicsmentioning

confidence: 99%

“…Here FV is the set of free variables in a formula and FVT is the set of free variables in a term. Note that the axiomatic system is substitutionless as it uses equality in a clever way to avoid the complications of substitution [20,36]. Amongst Pelletier's problems [32] for automated reasoning is problem 34, which is also known as Andrews's Challenge.…”

Section: Week Main Topicsmentioning

confidence: 99%

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Using Isabelle in Two Courses on Logic and Automated Reasoning

Villadsen¹,

Jacobsen²

2021

Formal Methods Teaching

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We present our experiences teaching two courses on formal methods and detail the contents of the courses and their positioning in the curriculum. The first course is a bachelor course on logical systems and logic programming, with a focus on classical first-order logic and automatic theorem proving. The second course is a master course on automated reasoning, with a focus on classical higher-order logic and interactive theorem proving. The proof assistant Isabelle is used in both courses, using Isabelle/Pure as well as Isabelle/HOL. We describe our online tools to be used with Isabelle/HOL, in particular the Sequent Calculus Verifier (SeCaV) and the Natural Deduction Assistant (NaDeA). We also describe our innovative Students' Proof Assistant which is formally verified in Isabelle/HOL and integrated in Isabelle/jEdit using Isabelle/ML.

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Section: Week Main Topicsmentioning

confidence: 99%

Section: Week Main Topicsmentioning

confidence: 99%

Using Isabelle in Two Courses on Logic and Automated Reasoning

Villadsen¹,

Jacobsen²

2021

Formal Methods Teaching

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“…• Jensen, Larsen, Schlichtkrull and Villadsen [31,64] formalized in Isabelle/HOL an axiomatic system for classical logic. • Raffali [52] formalized in Phox natural deduction for classical logic.…”

Section: Related Workmentioning

confidence: 99%

Interactive Theorem Proving for Logic and Information

Villadsen¹,

From²,

Jensen³

et al. 2021

Studies in Computational Intelligence

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Automated reasoning is the study of computer programs that can build proofs of theorems in a logic. Such programs can be either automatic theorem provers or interactive theorem provers. The latter are also called proof assistants because the user constructs the proofs with the help of the system. We focus on the Isabelle proof assistant. The system ensures that the proofs are correct, in contrast to pen-and-paper proofs which must be checked manually. We present applications to logical systems and models of information, in particular selected modal logics extending classical propositional logic. Epistemic logic allows intelligent systems to reason about the knowledge of agents. Public announcements can change the knowledge of the system and its agents. In order to account for this, epistemic logic can be extended to public announcement logic. An axiomatic system consists of axioms and rules of inference for deriving statements in a logic. Sound systems can only derive valid statements and complete systems can derive all valid statements. We describe formalizations of sound and complete axiomatic systems for epistemic logic and public announcement logic, thereby strengthening the foundations of automated reasoning for logic and information.

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“…Formalizations of other proof systems for first-order logic also appear, such as axiomatic systems for classical logic (in Isabelle by Jensen, Larsen, Schlichtkrull and Villadsen [10,20]), natural deduction for classical logic (in Isabelle/HOL by Berghofer [1,23,22] and in Phox by Raffali [15]), natural deduction for intuitionistic logic (in ALF by Persson [14]), resolution (by Schlichtkrull [18] in Isabelle/HOL and also in Isabelle/HOL by Schlichtkrull, Blanchette, Traytel and Waldmann [19]) and superposition (by Peltier [13] in Isabelle/HOL). Paulson's formalization in Isabelle/HOL of Gödel's Incompleteness Theorems [12] does not include a proof of completeness of a proof system.…”

Section: Related Workmentioning

confidence: 99%

Teaching a Formalized Logical Calculus

From

Jensen

Schlichtkrull

et al. 2020

Electron. Proc. Theor. Comput. Sci.

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Classical first-order logic is in many ways central to work in mathematics, linguistics, computer science and artificial intelligence, so it is worthwhile to define it in full detail. We present soundness and completeness proofs of a sequent calculus for first-order logic, formalized in the interactive proof assistant Isabelle/HOL. Our formalization is based on work by Stefan Berghofer, which we have since updated to use Isabelle's declarative proof style Isar (Archive of Formal Proofs, Entry FOL-Fitting, August 2007 / July 2018). We represent variables with de Bruijn indices; this makes substitution under quantifiers less intuitive for a human reader. However, the nature of natural numbers yields an elegant solution when compared to implementations of substitution using variables represented by strings. The sequent calculus considered has the special property of an always empty antecedent and a list of formulas in the succedent. We obtain the proofs of soundness and completeness for the sequent calculus as a derived result of the inverse duality of its tableau counterpart. We strive to not only present the results of the proofs of soundness and completeness, but also to provide a deep dive into a programming-like approach to the formalization of first-order logic syntax, semantics and the sequent calculus. We use the formalization in a bachelor course on logic for computer science and discuss our experiences. * Corresponding author: jovi@dtu.dk Teaching a Formalized Logical CalculusThe development described in this paper is available online: https://bitbucket.org/isafol/isafol/src/master/FOL_Berghofer/ 4253 lines FOL_Berghofer.thy 867 lines FOL_Tableau.thy A contribution of this paper 217 lines FOL_Sequent.thy A contribution of this paper 132 lines FOL_Appendix.thy A contribution of this paperThese numbers include blank lines and a few comments. All in all it takes around 5 seconds in real time to verify on a fairly standard computer. The entire formalization is based on the standard theory Main (the standard library of Isabelle/HOL which comes with many useful functions and facts about e.g. natural numbers and lists).We have recently named our system based on the formalization in Isabelle/HOL: SeCa  Sequent Calculus Verifier SeCaV formalizes first-order logic with constants and functionsSeCaV verifies one-sided sequent calculus proofs SeCaV uses the Isabelle proof assistantSeCaV is a tool for teaching logic Jørgen Villadsen Asta Halkjaer FromAlexander Birch Jensen Anders SchlichtkrullFor our bachelor course we focus on the fragment (499 lines) of the formalization available here:https://bitbucket.org/isafol/isafol/src/master/Sequent_Calculus/ Here only the soundness proof is included but in addition a small and a large proof in the sequent calculus is formalized. The structure of the paper is as follows. Section 2 explains the formalization of the syntax and the semantics of classical first-order logic. Section 3 describes the sequent calculus. Section 4 outlines the formalized soundness and completeness proof...

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Students' Proof Assistant (SPA)

Cited by 9 publications

References 8 publications

Using Isabelle in Two Courses on Logic and Automated Reasoning

Using Isabelle in Two Courses on Logic and Automated Reasoning

Interactive Theorem Proving for Logic and Information

Teaching a Formalized Logical Calculus

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