Constructive Logics Part I: A Tutorial on Proof Systems and Typed Lambda-Calculi

The purpose of this paper is to give an exposition of material dealing with constructive logic, typed λ-calculi, and linear logic. The emergence in the past ten years of a coherent field of research often named logic and computation has had two major (and related) effects: firstly, it has rocked vigorously the world of mathematical logic; secondly, it has created a new computer science discipline, which spans from what is traditionally called theory of computation, to programming language design. Remarkably, this new body of work relies heavily on some old concepts found in mathematical logic, like natural deduction, sequent calculus, and λ-calculus (but often viewed in a different light), and also on some newer concepts. Thus, it may be quite a challenge to become initiated to this new body of work (but the situation is improving, there are now some excellent texts on this subject matter). This paper attempts to provide a coherent and hopefully gentle initiation to this new body of work. We have attempted to cover the basic material on natural deduction, sequent calculus, and typed λ-calculus, but also to provide an introduction to Girard\u27s linear logic, one of the most exciting developments in logic these past five years. The first part of these notes gives an exposition of background material (with the exception of the Girard-translation of classical logic into intuitionistic logic, which is new). The second part is devoted to linear logic and proof nets

Proof Transformation with Built-in Equality Predicate

One of the main reasons why computer generated proofs are not widely accepted is often their complexity and incomprehensibility. Especially proofs of mathematical theorems with equations are normally presented in an inadequate and not intuitive way. This is even more of a problem for the presentation of inferences drawn by automated reasoning components in other AI systems. For first order logic, proof transformation procedures have been designed in order to structure proofs and state them in a formalism that is more familiar to human mathematicians. In this report we generalize these approaches, so that proofs involving equational reasoning can also be handled. To this end extended refutation graphs are introduced to represent combined resolution and paramodulation proofs. In the process of transforming these proofs into natural deduction proofs with equality, the inherent structure can also be extracted by exploiting topological properties of refutation graphs

A Possible and Necessary Consistency Proof

After Gödel's incompleteness theorems and the collapse of Hilbert's programme Gerhard Gentzen continued the quest for consistency proofs of Peano arithmetic. He considered a finitistic or constructive proof still possible and necessary for the foundations of mathematics. For a proof to be meaningful, the principles relied on should be considered more reliable than the doubtful elements of the theory concerned. He worked out a total of four proofs between 1934 and 1939. This thesis examines the consistency proofs for arithmetic by Gentzen from different angles. The consistency of Heyting arithmetic is shown both in a sequent calculus notation and in natural deduction. The former proof includes a cut elimination theorem for the calculus and a syntactical study of the purely arithmetical part of the system. The latter consistency proof in standard natural deduction has been an open problem since the publication of Gentzen's proofs. The solution to this problem for an intuitionistic calculus is based on a normalization proof by Howard. The proof is performed in the manner of Gentzen, by giving a reduction procedure for derivations of falsity. In contrast to Gentzen's proof, the procedure contains a vector assignment. The reduction reduces the first component of the vector and this component can be interpreted as an ordinal less than epsilon_0, thus ordering the derivations by complexity and proving termination of the process.De begränsningar av formella system som uppdagades av Gödels ofullständighetsteorem år 1931 innebär att Peanoaritmetikens konsistens endast kan bevisas med hjälp av fundamentala principer som inte kan formaliseras inom systemet. Trots att Hilberts finitistiska metoder inte kunde producera ett konsistensbevis, så fortsatte sökandet efter ett bevis med konstruktiva metoder. För att ett bevis skall vara meningsfullt borde principerna som används vara mera pålitliga än de element som betvivlas inom teorin. Avhandlingens titel hänvisar till ett citat av Gentzen då han motiverar behovet av konsistensbevis för första ordningens aritmetik. Gentzen själv producerade fyra konsistensbevis och analyserade hur väl dessa stämde överens med Hilberts program. Gentzen använde konstruktiva metoder i sina bevis, men det debatteras huruvida dessa metoder kan anses vara finitistiska. Det tredje och mest kända beviset presenterar en reduktion av härledningar av kontradiktioner. Med hjälp av transfinit induktion visas att reduktionsprocessen terminerar i en enkel härledning som konstateras vara omöjlig. Därför kan det inte finnas någon härledning av en kontradiktion. Avhandlingen undersöker och jämför Gentzens bevis från olika aspekter. Konsistensen av intuitionistisk Heytingaritmetik bevisas både i sekvenskalkyl och i naturlig deduktion. Det tidigare beviset är i Gentzens anda och innehåller ett snittelimineringsbevis för kalkylen och en syntaktisk studie av den aritmetiska delen av systemet. Det senare beviset påminner om ett normaliseringsbevis och visar terminering med hjälp av en vektortilldelning.Gödelin vuonna 1931 jullkaisemista epätäydellisyyslauseista seurausi rajoituksia formaalisille järjestelmille: Niiden mukaan Peano-aritmetiikan ristiriidattomuus voidaan todistaa ainoastaan periaatteilla, jotka eivät ole formalisoitavissa järjestelmän itsensä sisällä. Vaikka Hilbertin finitistisillä menetelmillä ei siksi pystytty tuottamaan konsistenssitodistusta, todistuksen etsiminen jatkui konstruktiivisillä menetelmillä. Jotta todistus olisi mielekäs, siinä käytettyjen periaatteiden oli oltava luotettavampia kuin teorian itsensä sisältämät periaatteet. Väitöskirjan otsikko viittaa Gentzenin kirjoitukseen, jossa hän perustelee ensimmäisen kertaluvun aritmetiikan konsistenssitodistuksen tarvetta. Gentzen itse laati neljä sellaista konsistenssitodistusta ja analysoi, missä määrin ne olivat yhdenmukaisia Hilbertin ohjelman kanssa. Gentzen käytti konstruktiivisia menetelmiä todistuksissaan ja on paljon väitelty kysymys, voidaanko näitä menetelmiä pitää finitistisinä. Kolmannessa ja tunnetuimassa Gentzenin todistuksessa esitetään ristiriitaisuuksien päättelyn reduktiomenetelmä. Transfiniittistä induktiota käyttämällä osoitetaan, että reduktioprosessi päättyy yksinkertaiseen päättelyyn, jollainen on erikseen todettu mahdottomaksi. Tämän vuoksi ristiriitaa ei voida päätellä. Väitöskirjassa selvitetään ja vertaillaan Gentzenin todistuksia eri näkökulmista. Intuitionistisen Heyting-aritmetiikan ristiriidattomuus osoitetaan sekä sekvenssikalkyylissä että luonnollisessa päättelyssä. Ensimmäinen todistus seuraa Gentzenin henkeä ja siinä sovelletaan ns. leikkaussäänön eliminointitodistusta sekä syntaktista analyysia järjestelmän aritmeettisesta osasta. Jälkimmäinen todistus muistuttaa luonnollisen päättelyn normalisointitodistusta ja näyttää reduktion päättymisen vektorimäärityksen avulla

A theory and its metatheory in FS 0

Feferman has proposed FS0, a theory of finitary inductive systems, as a framework theory suitable for various purposes, including reasoning both in and about encoded theories. I look here at how practical FS0 really is. I formalise of a sequent calculus presentation of classical propositional logic in FS0 and show this can be used for work in both the theory and the metatheory. the latter is illustrated with a discussion of a proof of Gentzen's Hauptsatz

Normalization by Completeness with Heyting Algebras

International audienceUsual normalization by evaluation techniques have a strong relationship with completeness with respect to Kripke structures. But Kripke structures is not the only semantics that ts intuitionistic logic: Heyting algebras are a more algebraic alternative.In this paper, we focus on this less investigated area: how completeness with respect to Heyting algebras generate a normalization algorithm for a natural deduction calculus, in the propositional fragment. Our main contributions is that we prove in a direct way completeness of natural deduction with respect to Heyting algebras, that the underlying algorithm natively deals with disjunction, that we formalized those proofs in Coq, and give an extracted algorithm