Citations: A Proof-Theoretical Study - Prawitz (ResearchIndex)

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Dag Prawitz. Natural Deduction, A Proof-Theoretical Study.Almquist and Wiksell, 1965.

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Constructive Logics. Part I: A Tutorial on Proof Systems and.. - Gallier (2003) (26 citations) (Correct)

....to motivate the introduction of various concepts by showing that they are indispensable to achieve certain natural goals. For pedagogical reasons, it seems that it is best to begin with proof systems in natural deduction style (originally due to Gentzen [8] and thoroughly investigated by Prawitz [23] in the sixties) This way, it is fairly natural to introduce the distinction between intuitionistic and classical logic. By adopting a description of natural deduction in terms of judgements, as opposed to the tagged trees used by Gentzen and Prawitz, we are also led quite naturally to the ....

....calculus. For an in depth study of constructivism in mathematics, we highly recommend Troelstra and van Dalen [32] 2 Natural Deduction, Simply Typed Calculus We rst consider a syntactic variant of the natural deduction system for implicational propositions due to Gentzen [8] and Prawitz [23]. In the natural deduction system of Gentzen and Prawitz, a deduction consists in deriving a proposition from a nite number of packets of assumptions, using some prede ned inference rules. Technically, packets are multisets of propositions. During the course of a deduction, certain packets of ....

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D. Prawitz. Natural deduction, a proof-theoretical study. Almquist & Wiksell, Stockholm, 1965.

On Zucker's isomorphism for LJ and its extension to Pure Type.. - Joachimski (2003) (Correct)

.... quite appropriate (yet not ecient) for proof search purposes, it somehow resists the otherwise beautiful correspondence between sequent calculus and natural deduction and leads to technical diculties when comparing the notions of cut elimination on the one side and normalization on the other (see [Pra65,Zuc74,Pot77]) From an intuitive natural deduction perspective, the left sided rules of sequent calculus work on the leaves of derivations the assumptions , while the right sided rules, just as all natural deduction rules, pertain the derivation root only. From the more formal (Curry Howard) viewpoint of ....

Dag Prawitz. Natural deduction, a proof-theoretical study. Almquist and Wiksell, Stockholm, 1965.

Minimal Classical Logic and Control Operators - Ariola, Herbelin (2003) (Correct)

....terms. We show the correspondence of top with a new theory of control, C top. The calculus C top is interesting in its own right, since it extends Felleisen s theory of control ( C) FH92] The study of C top leads to the development of a re nement of Prawitz s natural deduction [Pra65] in which one can distinguish between aborting a computation and throwing to a continuation (aborting corresponds to throwing to the top level continuation) This logic provides a solution to the mismatch between the operational and proof theoretical interpretation of Felleisen s C reduction ....

....deduction. A RAA A RAA A Activate ; cA RAA RAA A RAAc RAA ; A RAA B RAA A B RAA A B RAA A RAA B Fig. 5. Natural Deduction with RAAc Minimal Prawitz Classical Logic. Prawitz de nes classical logic as minimal logic plus the Reductio Ad Absurdum rule (RAA) [Pra65]: from ; A deduce A. This rule implies EFQ (as DN implies EFQ) and hence yields full classical logic. In here we are interested in exploring the possibility of de ning minimal classical logic from minimal logic and RAA but without deriving EFQ. Equivalently, we would like to devise a ....

D. Prawitz. Natural Deduction, a Proof-Theoretical Study. Almquist and Wiksell, Stockholm, 1965.

On the Geometry of Intuitionistic S4 Proofs - Goubault-Larrecq, Goubault (2002) (Correct)

.... = Convertibility = Equality of Maps The = signs are exact, except possibly for the Programs=Augmented Simplicial Maps one (we only get definable augmented simplicial maps) In particular, it is well known that equality of proofs, as defined by the symmetric closure of detour, or cut elimination [47], is exactly convertibility of terms (programs) We shall in addition show that two (definable) augmented simplicial maps are equal if and only if their defining terms are convertible, i.e. equal as proofs (bottom right = sign) This will be Theorem 72 and Corollary 73, an S4 variant of ....

....no normal proof (no rule can lead to a proof of false) So false has no proof, otherwise any proof r of false could be simplified to a normal proof of false, which does not exist. Hilbert systems as the one above are not really suited to the task, and we shall instead use natural deduction systems [47] in Section 3.3. 3.2. The Curry Howard Correspondence Note that there is another reading of the logic. Consider any formula as being a set: F D G will denote the set of all total functions from the set F to the set G. Then proofs are inhabitants of these sets: interpret the one step proof (1) as ....

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Dag Prawitz, Natural deduction, a proof-theoretical study, Almqvist and Wiskell, Stockholm, 1965.

On the Geometry of Intuitionistic S4 Proofs - Goubault-Larrecq, Goubault (2001) (Correct)

.... = Convertibility = Equality of Maps The = signs are exact, except possibly for the Programs=Augmented Simplicial Maps one (we only get de nable augmented simplicial maps) In particular, it is well known that equality of proofs, as de ned by the symmetric closure of detour, or cut elimination [44], is exactly convertibility of terms (programs) We shall in addition show that two (de nable) augmented simplicial maps are equal if and only if their de ning terms are convertible, i.e. equal as proofs (bottom right = sign) This will be Theorem 69 and Corollary 70, an S4 variant of Friedman s ....

....no normal proof (no rule can lead to a proof of false) So false has no proof, otherwise any proof of false could be simpli ed to a normal proof of false, which does not exist. Hilbert systems as the one above are not really suited to the task, and we shall instead use natural deduction systems [44] in Section 3.3. 3.2. The Curry Howard Correspondence Note that there is another reading of the logic. Consider any formula as being a set: F G will denote the set of all total functions from the set F to the set G. Then proofs are inhabitants of these sets: interpret the one step proof (1) as ....

[Article contains additional citation context not shown here]

Dag Prawitz, Natural deduction, a proof-theoretical study, Almqvist and Wiskell, Stockholm, 1965.

Towards a Logic for Reasoning about Logic Programs.. - Momigliano, Ornaghi (Correct)

....Introduction We give a proof theoretic analysis of logic programs transformations, viewed as operations on proof trees in the sense of [3, 4, 9, 10] We present a logic for reasoning about (equivalence preserving) transformations of logic programs. Our framework is natural deduction a la Prawitz [12], although a totally analogous presentation can be given in terms of sequent calculi with definitional reflection [14] We are not going to fully detail in this paper the systems we are using, though we have shown elsewhere [9] and it is beginning to be common knowledge, see [3, 4] that, w.r.t. ....

....the proof theoretical reconstruction of Partial Deduction. Section 5 targets Unfold Fold transformations. Finally, a brief Conclusion winds up the paper. 2 Background on Proof Trees First we recall some basic definitions: we assume some familiarity with standard notions of natural deduction [12] and logic programming [1] We use a variant of the Axiom Application Rules introduced in [9] We consider the Clause Application Rule ca, for applying axioms of the form B A 1 ; An . Using the formalism of natural deduction, ca can be expressed as follows: A 1 Delta Delta Delta ....

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Prawitz, D.: Natural deduction. A Proof-Theoretical Study. Almquist Wiksell (1965)

What is a Theory? - Dowek (Correct)

....uence is not a sucient condition for cut elimination. Computation rules are not the only alternative to axioms. Another one is to add non logical deduction rules to predicate logic either taking an introduction and elimination rule for the abstraction symbol in various formulations of set theory [15, 2, 10, 1, 3, 9] or interpreting logic programs or de nitions as deduction rules [11, 16, 17, 13] or in a more general setting [14] Non logical deduction rules and computation rules have some similarities, but we believe that computation rules have some advantages. For instance, non logical deduction rules may ....

D. Prawitz. Natural deduction, a proof-theoretical study. Almqvist & Wiksell, 1965.

Interpolation for Natural Deduction With Generalized Eliminations - Matthes (2001) (1 citation) (Correct)

.... : E) r (s ; x :t ) I) inj 1; r : inj 2; r : E) r (x :s ; y :t ) I) hr ; s i : E) r (x :y :t ) 1 I) IN1 : 1 (0 E) r 0 : 2 Note that elimination always [Pra65] has been given in the generalized style and that elimination is the same as tensor elimination in linear logic 1 . The essential new ingredient (by von Plato) is the generalized elimination. In calculus, we would have ( E) r s : E) r L : and r R : ....

Dag Prawitz. Natural Deduction. A Proof-Theoretical Study. Almquist and Wiksell, 1965.

Labelled Natural Deduction for Interval Logics - Rasmussen (2001) (1 citation) (Correct)

....this, some proofs are still tedious to perform as the way to the informal pen and paper level of abstraction seems fairly long. The present paper is an attempt to narrow this gap. It is inspired by work on Labelled Natural Deduction (LND) 15, 3] which combines classical natural deduction [11, 14] with labelled deductive systems [5] The LND formalism has shown its worth for traditional modal logics [1, 2] The rest of this paper is organized as follows: In Section 2 we consider propositional logics with a binary modality. These can be seen as the propositional basis for SIL and ITL. We ....

....below w i : i and w i : i is not the minor premise of a E or E rule. A track of order 0 ends in the root of ; a track of order n 1 ends in the minor premise of an E rule with major premise belonging to a track of order n. The above de nition of a track is an extension of that of [11] (we use the terminology of [14] for propositional logic to L AF . The key observation is that the structure of the rules I and E is similar to that of I and E, respectively (disregarding judgments concerning the accessibility relation R) Proposition 4. Let w 1 : 1 ; w 2 : 2 ; ....

D. Prawitz. Natural Deduction. A Proof-Theoretical Study. Almquist & Wiksell, 1965.

Simple Proof of the Completeness Theorem for Second Order.. - Nour, Raffalli (Correct)

.... G for (F G) G F ) We write F [x : t] for the rst order substitution of a term. We write F [X n : Y n ] for the second order substitution of a variable. We write F [X n : x 1 : x n G] for the second order substitution of a formula. We will use natural deduction [5, 6] both for second and rst order logic, and we will write n k F with k 2 fi; cg (for intuitionistic or classical logic) and n 2 f1; 2g (for rst or second order) We have the following lemma: Lemma 2.3 If n k A then, for every substitution , n k A[ De nition 2.4 (coding) We ....

Dag Prawitz. Natural Deduction, A proof-theoretical Study. Almqvist and Wiksells, Stockholm, 1965.

Parigot's Second Order λμ-Calculus and Inductive Types - Matthes (2001) (Correct)

....strong normalization. Hence, also the call by value and call by name formulations in [2] which are possible with ordinary substitution do not suce. Second order calculus is strongly normalizing [13] 1 It amounts to a calculus notation for classical natural deduction in the style of Prawitz [16]. 4 4 Extension of F by Iteration on Stabilization If : is provable, then is called stable. We consider an extension F ] of system F by a least stable supertype ] for any type . This is expressed as follows: We add a type constant for falsity 2 (and set : and for ....

Dag Prawitz. Natural Deduction. A Proof-Theoretical Study. Almquist and Wiksell, 1965.

Deforestation for Higher-Order Functional Programs - Marlow (1995) (11 citations) (Correct)

....a way that the resulting proof contains no uses of the cut rule. Furthermore, this can be done automatically and on any proof containing cuts. Gentzen, however, had no similar theorem for natural deduction. The strong normalisation property of natural deduction proofs was discovered by Prawitz [Pra65] some 30 years later. The Curry Howard isomorphism firmly bonds the mathematics of logic to the science of programming languages. Curry [CF58] noticed the relationship between natural deduction and combinatory logic, and Howard [How80] showed that there is an equivalent one toone correspondence ....

D. Prawitz. Natural Deduction, A Proof-Theoretical Study. Almquist & Wisell, Stockholm, 1965.

Explicit Substitutions and Reducibility - Herbelin (2001) (1 citation) (Correct)

.... with substitution made explicit can be found in the introduction of [10] Cut rule and explicit substitution Along the lines of the Curry Howard correspondence, the proofs of implicational Gentzen s natural deduction [13] are terms and Prawitz process of normalization for natural deduction [26] is reduction. Explicitation of substitution can be transfered to natural deduction and it happens substitution is nothing but a cut rule. The cut rule is central in this other kind of formal systems: Gentzen s sequent calculi LJ and LK [13] In contrast with the prominent r ole of reduction ....

D. Prawitz, Natural Deduction, a Proof-Theoretical Study, Almqvist & Wiksell (1965).

A Local System for Linear Logic - Stra�burger (2002) (5 citations) (Correct)