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Rewriting for Fitch Style Natural Deductions [chapter]

Herman Geuvers, Rob Nederpelt
2004 Lecture Notes in Computer Science  
Logical systems in natural deduction style are usually presented in the Gentzen style.  ...  We then describe three reduction relations on Fitch style natural deductions: one that removes garbage (subproofs that are not needed for the conclusion), one that removes repeats and one that unshares  ...  Introduction For Gentzen style natural deduction, ([Gentzen 1969] ) there is a well-known notion of rewriting: cut-elimination.  ... 
doi:10.1007/978-3-540-25979-4_10 fatcat:dspgpihdbzf2vef3hfd2y4agpq

Merging Procedural and Declarative Proof [chapter]

Cezary Kaliszyk, Freek Wiedijk
2009 Lecture Notes in Computer Science  
There are two different styles for writing natural deduction proofs: the 'Gentzen' style in which a proof is a tree with the conclusion at the root and the assumptions at the leaves, and the 'Fitch' style  ...  In this paper we give an algorithm for converting tree style proofs to flag style proofs. We then present a rewrite system that simplifies the results.  ...  Theorem 3 . 3 The rewrite system including the above rewrite rule is confluent. Fig. 5 . 5 Rewrite rule for eliminating explicit cuts from a Fitch deduction.  ... 
doi:10.1007/978-3-642-02444-3_13 fatcat:hhsb5djxi5cbjmzwrmyxbvkcuy

Dual-context Modal Logic as Left Adjoint of Fitch-style Modal Logic

Yoshihiko Kakutani, Yuito Murase, Yuichi Nishiwaki
2019 Journal of Information Processing  
In the stream of studies on intuitionistic modal logic, we can find mainly three kinds of natural deduction systems.  ...  In our previous studies, categorical semantics for Fitch-style modal logic is proposed and usefulness of levels is noticed.  ...  The authors thank anonymous reviewers for helpful comments.  ... 
doi:10.2197/ipsjjip.27.77 fatcat:k5zts37c35deffthvdtvrs3jhe

Natural deduction via graphs: formal definition and computation rules

HERMAN GEUVERS, IRIS LOEB
2007 Mathematical Structures in Computer Science  
We introduce the formalism of deduction graphs as a generalization of both Gentzen-Prawitz style natural deduction and Fitch style flag deduction.  ...  The advantage of this formalism is that subproofs can be shared, like in flag deductions (and unlike natural deduction), but also that the linearisation used in flag deductions is avoided.  ...  I n tr o d u c tio n Gentzen-Prawitz (Gentzen 1969; Prawitz 1965) style deduction and Fitch (Fitch 1952) style deduction are two popular ways of doing "natural deduction" .  ... 
doi:10.1017/s0960129507006123 fatcat:as4552prq5hdldraymxzi2mzjm

Simulation of Natural Deduction and Gentzen Sequent Calculus

Daniil Kozhemiachenko
2017 Logic and Logical Philosophy  
We show that general deduction Frege systems and Gentzen-style natural calculi provide at most quadratic speedup over nested deduction Frege systems and Fitch-style natural calculi and at most cubic speedup  ...  We consider four natural deduction systems: Fitch-style systems, Gentzen-style systems (in the form of dags), general deduction Frege systems and nested deduction Frege systems, as well as dag-like Gentzenstyle  ...  Zaitsev for their much-appreciated help and advice during the work on this paper.  ... 
doi:10.12775/llp.2017.009 fatcat:iuugcb6amrdkbegcuhyutvz2he

Modality via Iterated Enrichment [article]

Yuichi Nishiwaki, Yoshihiko Kakutani, Yuito Murase
2018 arXiv   pre-print
We emphasize that this paper answers the question raised in the survey paper by de Paiva and Ritter in 2011, what a categorical model for Fitch-style type theory is like.  ...  Interestingly, this interpretation suggests the fact that Fitch-style modal type theory is the right adjoint of dual-context calculus.  ...  Acknowledgement The authors thank their colleague Hiroki Kobayashi for helpful discussions and comments to an early draft of this paper.  ... 
arXiv:1804.02809v2 fatcat:kw5onsgl7vd5lezi7i6irb2b5u

A Mobile Application for Self-Guided Study of Formal Reasoning

David M. Cerna, Rafael P.D. Kiesel, Alexandra Dzhiganskaya
2020 Electronic Proceedings in Theoretical Computer Science  
Future developments will include rewrite rules which take a set of expressions and return a set of expressions, as well as a quantified first-order extension.  ...  Unlike most of the existing study aids for formal reasoning, AXolotl is an Android-based application with a simple touch-based interface.  ...  Unlike most other mobile apps, Natural Deduction provides a natural deduction fitch style proof system as well as a theorem prover.  ... 
doi:10.4204/eptcs.313.3 fatcat:ssvlfdr76ra6xjso7nhkqwgfmq

A Teaching Tool for Proving Equivalences between Logical Formulae [chapter]

Josje Lodder, Bastiaan Heeren
2011 Lecture Notes in Computer Science  
This tool is based on an earlier tool for rewriting logical formulae into disjunctive normal form (DNF).  ...  In this paper we describe a teaching tool for proving equivalences between propositional logic formulae, using rewrite rules such as De Morgan's laws and double negation.  ...  We thank the anonymous reviewers for their constructive comments. Discussions with Daniel Herding about searching proofs for set algebra are gratefully acknowledged.  ... 
doi:10.1007/978-3-642-21350-2_18 fatcat:gd7tm6i7njatreut67244xk6lm

A new graphical calculus of proofs

Sandra Alves, Maribel Fernández, Ian Mackie
2011 Electronic Proceedings in Theoretical Computer Science  
We offer a simple graphical representation for proofs of intuitionistic logic, which is inspired by proof nets and interaction nets (two formalisms originating in linear logic).  ...  Deduction graphs are a generalisation of natural deduction and Fitch style flag deduction; they have both nodes and boxes.  ...  Figure 1 : 1 Intuitionistic Natural Deduction Simplified to the linear case.  ... 
doi:10.4204/eptcs.48.8 fatcat:hnxcy5nyhrccjadw2qxdzr56oe

An Approximation Based Approach to Infinitary Lambda Calculi [chapter]

Stefan Blom
2004 Lecture Notes in Computer Science  
for Fitch style natural deductions H.  ...  Blom 17:10 -17:45 Böhm-Like Trees for Term Rewriting Systems J.  ...  We recommend visits to the cathedral treasury, the town hall and the Ludwig Forum for International Art.  ... 
doi:10.1007/978-3-540-25979-4_16 fatcat:3cjahpq7ingsjezeojlru6jrtu

A Synthesis of the Procedural and Declarative Styles of Interactive Theorem Proving

Freek Wiedijk, Pawel Urzyczyn
2012 Logical Methods in Computer Science  
in a controlled natural language, like in Isabelle/Isar).  ...  We propose a synthesis of the two proof styles of interactive theorem proving: the procedural style (where proofs are scripts of commands, like in Coq) and the declarative style (where proofs are texts  ...  Acknowledgements Thanks to Henk Barendregt for his vision of a luxury mathmode for interactive theorem provers, which led directly to the approach from this paper.  ... 
doi:10.2168/lmcs-8(1:30)2012 fatcat:nlidbpgmgvhzfd6rsmei4iwv6u

Tackling Incomplete System Specifications Using Natural Deduction in the Paracomplete Setting

Alexander Bolotov, Vasilyi Shangin
2014 2014 IEEE 38th Annual Computer Software and Applications Conference  
Note that for many researchers, one of the core features of natural deduction, the opportunity to introduce arbitrary formulae as assumptions, has been a point of great scepticism regarding the very possibility  ...  As a consequence, for example, the law of excluded middle is not valid.  ...  Therefore, proofs by our search algorithms in the classical case can be linearly simulated by classical Frege style formulation which puts our Fitch-style natural deduction construction in line of propositional  ... 
doi:10.1109/compsac.2014.15 dblp:conf/compsac/BolotovS14 fatcat:f2kl6mwrynarbkzsqkknwcet4u

Introduction to Type Theory [chapter]

Herman Geuvers
2009 Lecture Notes in Computer Science  
The lecture was meant as an introduction to types λ-calculus for PhD. students that have some (but possibly not much) familiarity with logic or functional programming.  ...  In the lecture I have focussed on the use of type theory for compile-time checking of functional programs and on the use of types in proof assistants (theorem provers).  ...  We therefor introduce a Fitch style representation of typing derivations, named after the logician Fitch, who has developed a natural deduction system for logic in this format, also called flag deduction  ... 
doi:10.1007/978-3-642-03153-3_1 fatcat:qm3uz2nzjvh6tm5ca4nzvydjya

A Survey of Nonstandard Sequent Calculi

Andrzej Indrzejczak
2014 Studia Logica: An International Journal for Symbolic Logic  
modal and temporal logics. 5 General theory of labelled systems is presented in Gabbay [19]; for a survey of approaches developed for modal logics see Indrzejczak [30].  ...  Hence we will simply call a sequent/rule an infinite set of sequents/rules covered by a schema as well as a particular instance. 2 There are popular systems defined for multi-valued logics with sequents  ...  Moreover, in contrast to Natural Deduction systems in Jaśkowski style 7 , it does not have to use any bookkeeping techniques, like lines or boxes, for showing the dependancy of formulae in the proof from  ... 
doi:10.1007/s11225-014-9567-y fatcat:jp5yu3ueojaarjlclhjpfdympy

Variable declarations in natural deduction

Daniel J. Velleman
2006 Annals of Pure and Applied Logic  
(For more on the history of natural deduction, see [1] and [16] .) The formulation of the quantifier rules for natural deduction proved to be particularly difficult.  ...  As Irving Anellis has observed: It is clear that the difficulties involved in developing a set of quantification rules for systems of natural deduction which are at once both simple, i.e.  ...  Acknowledgements I would like to thank Charles Silver, Joseph Moore, Kit Fine, George Weaver, and two anonymous referees for helpful comments on earlier drafts of this paper.  ... 
doi:10.1016/j.apal.2006.05.009 fatcat:qvqkuf4hqjb2pfcbcckkjpaatq
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