scholarly journals A Formalisation of Weak Normalisation (with Respect to Permutations) of Sequent Calculus Proofs

2000 ◽  
Vol 3 ◽  
pp. 1-26
Author(s):  
A.A. Adams
Keyword(s):  
Fixed Points ◽  
Intuitionistic Logic ◽  
Natural Deduction ◽  
Sequent Calculus ◽  
The State ◽  
Original Presentation

AbstractDyckhoff and Pinto present a weakly normalising system of reductions on derivations are characterised as the fixed points of the composition of the Prawitz translations into natural deduction and back. This paper presents a formalisation of the system, including a proof of the Weak normalisation property for the formalisation. More details can be found in earlier work by the author. The formalisation has been kept as closes as possible to the original presentation to allow an evaluation of the state of proof assistance for such methods, and to ensure similarity of methods, and not merely similarly of results. The formalisation is restricted to the implicational fragment of intuitionistic logic.

scholarly journals An Alternative Natural Deduction for the Intuitionistic Propositional Logic

2016 ◽  
Vol 45 (1) ◽  
Author(s):  
Mirjana Ilić
Keyword(s):  
Intuitionistic Logic ◽  
Propositional Logic ◽  
Natural Deduction ◽  
Sequent Calculus ◽  
Intuitionistic Propositional Logic ◽  
Deduction System ◽  
Natural Deduction System

A natural deduction system NI, for the full propositional intuitionistic logic, is proposed. The operational rules of NI are obtained by the translation from Gentzen’s calculus LJ and the normalization is proved, via translations from sequent calculus derivations to natural deduction derivations and back.

scholarly journals On Polymorphic Sessions and Functions

2021 ◽  
Vol 43 (2) ◽  
pp. 1-55
Author(s):  
Bernardo Toninho ◽  
Nobuko Yoshida
Keyword(s):  
Natural Deduction ◽  
Sequent Calculus ◽  
Linear Logic ◽  
Linear Formulation ◽  
Session Types ◽  
Logical Foundation ◽  
Π Calculus ◽  
Soundness And Completeness ◽  
System F ◽  
Algebraic Representations

This work exploits the logical foundation of session types to determine what kind of type discipline for the Λ-calculus can exactly capture, and is captured by, Λ-calculus behaviours. Leveraging the proof theoretic content of the soundness and completeness of sequent calculus and natural deduction presentations of linear logic, we develop the first mutually inverse and fully abstract processes-as-functions and functions-as-processes encodings between a polymorphic session π-calculus and a linear formulation of System F. We are then able to derive results of the session calculus from the theory of the Λ-calculus: (1) we obtain a characterisation of inductive and coinductive session types via their algebraic representations in System F; and (2) we extend our results to account for value and process passing, entailing strong normalisation.

A sequent calculus for type assignment

1977 ◽  
Vol 42 (1) ◽  
pp. 11-28 ◽  
Author(s):  
Jonathan P. Seldin
Keyword(s):  
Normal Form ◽  
Main Part ◽  
Natural Deduction ◽  
Sequent Calculus ◽  
Cut Elimination ◽  
Type Assignment ◽  
Image Position ◽  
Atomic Type

The sequent calculus formulation (L-formulation) of the theory of functionality without the rules allowing for conversion of subjects of [3, §14E6] fails because the (cut) elimination theorem (ET) fails. This can be most easily seen by the fact that it is easy to prove in the systemandbut not (as is obvious if α is an atomic type [an F-simple])The error in the “proof” of ET in [14, §3E6], [3, §14E6], and [7, §9C] occurs in Stage 3, where it is implicitly assumed that if [x]X ≡ [x] Y then X ≡ Y. In the above counterexample, we have [x]x ≡ ∣ ≡ [x](∣x) but x ≢ ∣x. Since the formulation of [2, §9F] is not really satisfactory (for reasons stated in [3, §14E]), a new seguent calculus formulation is needed for the case in which the rules for subject conversions are not present. The main part of this paper is devoted to presenting such a formulation and proving it equivalent to the natural deduction formulation (T-formulation). The paper will conclude in §6 with some remarks on the result that every ob (term) with a type (functional character) has a normal form.The conventions and definitions of [3], especially of §12D and Chapter 14, will be used throughout the paper.

Gentzen's Proof of Normalization for Natural Deduction

2008 ◽  
Vol 14 (2) ◽  
pp. 240-257 ◽  
Author(s):  
Jan von Plato
Keyword(s):  
Natural Deduction ◽  
Sequent Calculus ◽  
Direct Proof ◽  
Cut Elimination ◽  
Doctoral Thesis ◽  
Detailed Proof ◽  
Normalization Theorem ◽  
The One ◽  
The Right ◽  
Special Case

AbstractGentzen writes in the published version of his doctoral thesis Untersuchungen über das logische Schliessen (Investigations into logical reasoning) that he was able to prove the normalization theorem only for intuitionistic natural deduction, but not for classical. To cover the latter, he developed classical sequent calculus and proved a corresponding theorem, the famous cut elimination result. Its proof was organized so that a cut elimination result for an intuitionistic sequent calculus came out as a special case, namely the one in which the sequents have at most one formula in the right, succedent part. Thus, there was no need for a direct proof of normalization for intuitionistic natural deduction. The only traces of such a proof in the published thesis are some convertibilities, such as when an implication introduction is followed by an implication elimination [1934–35, II.5.13]. It remained to Dag Prawitz in 1965 to work out a proof of normalization. Another, less known proof was given also in 1965 by Andres Raggio.We found in February 2005 an early handwritten version of Gentzen's thesis, with exactly the above title, but with rather different contents: Most remarkably, it contains a detailed proof of normalization for what became the standard system of natural deduction. The manuscript is located in the Paul Bernays collection at the ETH-Zurichwith the signum Hs. 974: 271. Bernays must have gotten it well before the time of his being expelled from Göttingen on the basis of the racial laws in April 1933.

Gentzen-Style Sequent Calculus for Semi-intuitionistic Logic

Studia Logica ◽  
2016 ◽  
Vol 104 (6) ◽  
pp. 1245-1265 ◽  
Author(s):  
Diego Castaño ◽  
Juan Manuel Cornejo
Keyword(s):  
Intuitionistic Logic ◽  
Sequent Calculus

scholarly journals Construcción de un sensor electroquímico para determinación de gases contaminantes en el aire (CO, CO2 y O3), empleado una tarjeta Arduino

2020 ◽  
pp. 1-5
Author(s):  
Maria Luisa Lozano-Camargo ◽  
Christian Hugo Rodríguez-Gómez ◽  
Laura Galicia-Luis ◽  
Fernando Talavera-Romero
Keyword(s):  
Air Pollution ◽  
Air Quality ◽  
Cardiovascular Diseases ◽  
Electrochemical Sensor ◽  
Fixed Points ◽  
Anthropogenic Activities ◽  
The State ◽  
Physical And Chemical

As humanity evolves, air pollution has increased due to the various anthropogenic activities that man carries out, alienating emissions of gases and polluting particles to the environment, seriously affecting the health of living beings and the planet, since these cause Irreversible physical and chemical alterations in the environment, becoming a major problem worldwide. In Mexico there are meteorological stations that measure air quality and they are only found at fixed points, however, they do not cover all areas of the State of Mexico, the deterioration of air quality brings with it an increase in respiratory and cardiovascular diseases ; That is why this project's main objective is to build a portable electrochemical sensor using an Arduino board, using customizable software capable of quantifying and analyzing three polluting gases CO, CO2 and O3, especially in the municipality of Chimalhuacán located in the area East of the State of México.

scholarly journals Normalization as a consequence of cut elimination

2009 ◽  
Vol 86 (100) ◽  
pp. 27-34
Author(s):  
Mirjana Borisavljevic
Keyword(s):  
Intuitionistic Logic ◽  
Natural Deduction ◽  
Cut Elimination ◽  
Deduction System ◽  
Normalization Theorem ◽  
Natural Deduction System

Pairs of systems, which consist of a system of sequents and a natural deduction system for some part of intuitionistic logic, are considered. For each of these pairs of systems the property that the normalization theorem is a consequence of the cut-elimination theorem is presented.

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