A Sequent Calculus for First-Order Logic.
01 Jan 2019-Vol. 2019
TL;DR: The completeness of a one-sided sequent calculus for first-order logic is shown via a translation from a complete semantic tableau calculus, the proof of which is based on the First-Order Logic According to Fitting theory.
Abstract: This work formalizes soundness and completeness of a one-sided sequent calculus for first-order logic. The completeness is shown via a translation from a complete semantic tableau calculus, the proof of which is based on the First-Order Logic According to Fitting theory. The calculi and proof techniques are taken from Ben-Ari’s Mathematical Logic for Computer Science [1].
Citations
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TL;DR: This work certifies in the proof assistant Isabelle/HOL the soundness of a declarative first-order prover with equality and shows examples of proofs and how they are made in the prover.
Abstract: We certify in the proof assistant Isabelle/HOL the soundness of a declarative first-order prover with equality. The LCF-style prover is a translation we have made, to Standard ML, of a prover in John Harrison’s Handbook of Practical Logic and Automated Reasoning. We certify it by replacing its kernel with a certified version that we program, certify and generate code from; all in Isabelle/HOL. In a declarative proof each step of the proof is declared, similar to the sentences in a thorough paper proof. The prover allows proofs to mix the declarative style with automatic theorem proving by using a tableau prover. Our motivation is teaching how automated and declarative provers work and how they are used. The prover allows studying concrete code and a formal verification of correctness. We show examples of proofs and how they are made in the prover. The entire development runs in Isabelle’s ML environment as an interactive application or can be used standalone in OCaml or Standard ML (or in other functional programming languages like Haskell and Scala with some additional work).
19 citations
Cites background from "A Sequent Calculus for First-Order ..."
...Persson’s constructive completeness of intuitionistic predicate logic [33], Braselmann, Koepke and Schlöder’s sequent calculus for uncountable languages [4, 5, 38], Berghofer’s natural deduction [1], Ilik’s constructive completeness results for classical and intuitionistic logic [15], Blanchette, Popescu and Traytel’s abstract completeness library [3], Schlichtkrull’s resolution calculus [36], Peltier’s superposition calculus [32], and Paulson’s proof of Gödel’s incompleteness theorems [30]....
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TL;DR: A formalization in Isabelle/HOL of the resolution calculus for first-order logic with formal soundness and completeness proofs and a thorough overview of formalizations of first- order logic found in the literature is given.
Abstract: A formalization in Isabelle/HOL of the resolution calculus for first-order logic is presented. Its soundness and completeness are formally proven using the substitution lemma, semantic trees, Herbrand’s theorem, and the lifting lemma. In contrast to previous formalizations of resolution, it considers first-order logic with full first-order terms, instead of the propositional case.
17 citations
Cites background from "A Sequent Calculus for First-Order ..."
...Braselmann and Koepke [14,15] proved, inMizar, a sequent calculus for first-order logic sound and complete....
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01 Jan 2018
TL;DR: This thesis describes formalizations in Isabelle of several logics as well as tools built upon these, including the Natural Deduction Assistant (NaDeA), which is a tool for teaching first-order logic that allows users to build proofs in natural deduction.
Abstract: Isabelle is a proof assistant, i.e. a computer program that helps its user to define concepts in mathematics and computer science as well as to prove properties about them. This process is called formalization. Proof assistants aid their users by ensuring that proofs are constructed correctly and by conducting parts of the proofs automatically. A logical calculus is a set of rules and axioms that can be applied to construct theorems of the calculus. Logical calculi are employed in e.g. tools for formal verification of computer programs. Two important properties of logical calculi are soundness and completeness, since they state, respectively, that all theorems of a given calculus are valid, and that all valid statements are theorems of the calculus. Validity is defined by a semantics, which gives meaning to formulas. This thesis describes formalizations in Isabelle of several logics as well as tools built upon these. Specifically this thesis explains and discusses the following contributions of my PhD project: • A formalization of the resolution calculus for first-order logic, Herbrand’s theorem and the soundness and completeness of the calculus. • A formalization of the ordered resolution calculus for first-order logic, an abstract prover based on it and the prover’s soundness and completeness. • A verified automatic theorem prover for first-order logic. The prover is a refinement of the above formalization of an abstract prover. This explicitly shows that the abstract notion of a prover can describe concrete computer programs. • The Natural Deduction Assistant (NaDeA), which is a tool for teaching first-order logic that allows users to build proofs in natural deduction. The tool is based on a formalization of natural deduction and its soundness and completeness. • A verified proof assistant for first-order logic with equality. It is based on an axiomatic system and constitutes a tool for teaching logic and proof assistants. • A formalization of the propositional fragment of a paraconsistent infinite-valued higherorder logic. Theorems about the necessity of having infinitely many truth values are proved and formalized. Proof assistants are built to reject proofs that contain gaps or mistakes. Therefore, the formalized results are highly trustworthy. The tools based on formalized calculi consequently have an increased trustworthiness. The above formalizations revealed flaws and mistakes in the literature. In addition to the formalizations and tools themselves, my PhD project contributes solutions that repair these flaws and mistakes.
13 citations
28 Feb 2020
TL;DR: In this article, the authors present soundness and completeness proofs of a sequent calculus for first-order logic, formalized in the interactive proof assistant Isabelle/HOL, based on work by Stefan Berghofer, which has since updated to use Isabelle's declarative proof style Isar.
Abstract: 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.
11 citations
01 Jul 2020
TL;DR: In this article, soundness and completeness proofs for a Seligman-style tableau system for hybrid logic in the proof assistant Isabelle/HOL are formalized, showing how to lift certain rule restrictions, thereby simplifying the original unformalized proof.
Abstract: Hybrid logic is modal logic enriched with names for worlds. We formalize soundness and completeness proofs for a Seligman-style tableau system for hybrid logic in the proof assistant Isabelle/HOL. The formalization shows how to lift certain rule restrictions, thereby simplifying the original un-formalized proof. Moreover, the completeness proof we formalize is synthetic which suggests we can extend this work to prove a wider range of results about hybrid logic.
8 citations
References
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TL;DR: This work certifies in the proof assistant Isabelle/HOL the soundness of a declarative first-order prover with equality and shows examples of proofs and how they are made in the prover.
Abstract: We certify in the proof assistant Isabelle/HOL the soundness of a declarative first-order prover with equality. The LCF-style prover is a translation we have made, to Standard ML, of a prover in John Harrison’s Handbook of Practical Logic and Automated Reasoning. We certify it by replacing its kernel with a certified version that we program, certify and generate code from; all in Isabelle/HOL. In a declarative proof each step of the proof is declared, similar to the sentences in a thorough paper proof. The prover allows proofs to mix the declarative style with automatic theorem proving by using a tableau prover. Our motivation is teaching how automated and declarative provers work and how they are used. The prover allows studying concrete code and a formal verification of correctness. We show examples of proofs and how they are made in the prover. The entire development runs in Isabelle’s ML environment as an interactive application or can be used standalone in OCaml or Standard ML (or in other functional programming languages like Haskell and Scala with some additional work).
19 citations
TL;DR: A formalization in Isabelle/HOL of the resolution calculus for first-order logic with formal soundness and completeness proofs and a thorough overview of formalizations of first- order logic found in the literature is given.
Abstract: A formalization in Isabelle/HOL of the resolution calculus for first-order logic is presented. Its soundness and completeness are formally proven using the substitution lemma, semantic trees, Herbrand’s theorem, and the lifting lemma. In contrast to previous formalizations of resolution, it considers first-order logic with full first-order terms, instead of the propositional case.
17 citations
01 Jan 2018
TL;DR: This thesis describes formalizations in Isabelle of several logics as well as tools built upon these, including the Natural Deduction Assistant (NaDeA), which is a tool for teaching first-order logic that allows users to build proofs in natural deduction.
Abstract: Isabelle is a proof assistant, i.e. a computer program that helps its user to define concepts in mathematics and computer science as well as to prove properties about them. This process is called formalization. Proof assistants aid their users by ensuring that proofs are constructed correctly and by conducting parts of the proofs automatically. A logical calculus is a set of rules and axioms that can be applied to construct theorems of the calculus. Logical calculi are employed in e.g. tools for formal verification of computer programs. Two important properties of logical calculi are soundness and completeness, since they state, respectively, that all theorems of a given calculus are valid, and that all valid statements are theorems of the calculus. Validity is defined by a semantics, which gives meaning to formulas. This thesis describes formalizations in Isabelle of several logics as well as tools built upon these. Specifically this thesis explains and discusses the following contributions of my PhD project: • A formalization of the resolution calculus for first-order logic, Herbrand’s theorem and the soundness and completeness of the calculus. • A formalization of the ordered resolution calculus for first-order logic, an abstract prover based on it and the prover’s soundness and completeness. • A verified automatic theorem prover for first-order logic. The prover is a refinement of the above formalization of an abstract prover. This explicitly shows that the abstract notion of a prover can describe concrete computer programs. • The Natural Deduction Assistant (NaDeA), which is a tool for teaching first-order logic that allows users to build proofs in natural deduction. The tool is based on a formalization of natural deduction and its soundness and completeness. • A verified proof assistant for first-order logic with equality. It is based on an axiomatic system and constitutes a tool for teaching logic and proof assistants. • A formalization of the propositional fragment of a paraconsistent infinite-valued higherorder logic. Theorems about the necessity of having infinitely many truth values are proved and formalized. Proof assistants are built to reject proofs that contain gaps or mistakes. Therefore, the formalized results are highly trustworthy. The tools based on formalized calculi consequently have an increased trustworthiness. The above formalizations revealed flaws and mistakes in the literature. In addition to the formalizations and tools themselves, my PhD project contributes solutions that repair these flaws and mistakes.
13 citations
28 Feb 2020
TL;DR: In this article, the authors present soundness and completeness proofs of a sequent calculus for first-order logic, formalized in the interactive proof assistant Isabelle/HOL, based on work by Stefan Berghofer, which has since updated to use Isabelle's declarative proof style Isar.
Abstract: 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.
11 citations
01 Jul 2020
TL;DR: In this article, soundness and completeness proofs for a Seligman-style tableau system for hybrid logic in the proof assistant Isabelle/HOL are formalized, showing how to lift certain rule restrictions, thereby simplifying the original unformalized proof.
Abstract: Hybrid logic is modal logic enriched with names for worlds. We formalize soundness and completeness proofs for a Seligman-style tableau system for hybrid logic in the proof assistant Isabelle/HOL. The formalization shows how to lift certain rule restrictions, thereby simplifying the original un-formalized proof. Moreover, the completeness proof we formalize is synthetic which suggests we can extend this work to prove a wider range of results about hybrid logic.
8 citations