## A formulae-as-types interpretation of subtractive logic (2004)

Venue: | Journal of Logic and Computation |

Citations: | 23 - 1 self |

### BibTeX

@ARTICLE{Crolard04aformulae-as-types,

author = {Tristan Crolard},

title = {A formulae-as-types interpretation of subtractive logic},

journal = {Journal of Logic and Computation},

year = {2004},

volume = {14},

pages = {2004}

}

### Years of Citing Articles

### OpenURL

### Abstract

We present a formulae-as-types interpretation of Subtractive Logic (i.e. bi-intuitionistic logic). This presentation is two-fold: we first define a very natural restriction of the λµ-calculus which is closed under reduction and whose type system is a constructive restriction of the Classical Natural Deduction. Then we extend this deduction system conservatively to Subtractive Logic. From a computational standpoint, the resulting calculus provides a type system for first-class coroutines (a restricted form of first-class continuations). Keywords: Curry-Howard isomorphism, Subtractive Logic, control operators, coroutines. 1

### Citations

686 |
Introduction to Metamathematics
- Kleene
- 1952
(Show Context)
Citation Context ... extension of natural deduction to sequents with several conclusions. As expected, this extension leads to classical logic. In order to stay in a constructive framework, several authors (for instance =-=[29]-=- p. 481 and also [3, 14, 16]) have suggested to restrict only the introduction rule of implication of LK to sequents with at most one conclusion. The same restriction can be applied to CND and by dual... |

321 |
λµ-calculus: an algorithmic interpretation of classic natural deduction
- Parigot
- 1992
(Show Context)
Citation Context ...horoughly this duality (but still in a categorical setting). On the other hand, Curien and Herbelin’s work is based on Parigot’s λµ-calculus and its type system, the Classical Natural Deduction (CND) =-=[36, 37]-=-. In order to complete the duality, they are led to extend CND with the subtraction (Walder does not consider the subtraction in [52]). However, this new connective is presented in a purely formal way... |

234 | A formulae-as-types notion of control
- Griffin
- 1990
(Show Context)
Citation Context ...equent in CND→∨∧− (resp. SND→∨∧−) which does not contain any occurrence of subtraction belongs to CND→∨∧ (resp. SND→∨∧). Computational interpretation TRISTAN CROLARD 3 Since Griffin’s pioneering work =-=[23]-=-, the extension of the well-known formulas-as-types paradigm to classical logic has been widely investigated for instance by Murthy [31], Barbanera and Berardi [2], Rehof and S⊘rensen [46], de Groote ... |

161 | The duality of computation
- Curien, Herbelin
(Show Context)
Citation Context ...ical logics. 1. Universit£ Paris XII ¤ 61, avenue du G£n£ral de Gaulle ¤ 94010 Cr£teil Cedex ¤ France 1s2 A Formulae-as-Types Interpretation of Subtractive Logic • Curien and Herbelin demonstrated in =-=[9]-=- a striking result: duality in classical logic exchanges call-by-value with call-by-name (see also Wadler’s recent paper [52]). Actually, a first attempt at a categorical continuation semantics was Fi... |

132 |
Contraction-free Sequent Calculi for Intuitionistic Logic
- Dyckhoff
- 1992
(Show Context)
Citation Context ... deduction to sequents with several conclusions. As expected, this extension leads to classical logic. In order to stay in a constructive framework, several authors (for instance [29] p. 481 and also =-=[3, 14, 16]-=-) have suggested to restrict only the introduction rule of implication of LK to sequents with at most one conclusion. The same restriction can be applied to CND and by duality it can be generalized to... |

93 | Typing first-class continuations in ML
- Duba, Harper, et al.
- 1987
(Show Context)
Citation Context ...cessfully used to implement Simula-like cooperative coroutines in Scheme [53, 18, 19]. This approach has been extended in the Standard ML of New Jersey (with the typed counterpart of Scheme’s call/cc =-=[15]-=-) to provide simple and elegant implementations of light-weight processes (or threads), where concurrency is obtained by having individual threads voluntarily suspend themselves [48, 42] (providing ti... |

83 | Control categories and duality: on the categorical semantics of the lambda-mu calculus
- Selinger
- 2001
(Show Context)
Citation Context ...c exchanges call-by-value with call-by-name (see also Wadler’s recent paper [52]). Actually, a first attempt at a categorical continuation semantics was Filinski’s pioneering work [17]. Then Selinger =-=[50]-=- has investigated thoroughly this duality (but still in a categorical setting). On the other hand, Curien and Herbelin’s work is based on Parigot’s λµ-calculus and its type system, the Classical Natur... |

75 |
Proofs of strong normalization for second order classical natural deduction
- Parigot
- 1997
(Show Context)
Citation Context ...vine [30]. Wes4 A Formulae-as-Types Interpretation of Subtractive Logic shall consider here Parigot’s λµ-calculus mainly because it is confluent and strongly normalizing in the second order framework =-=[38]-=-. However, Parigot’s original CND is a second-order logic, in which ∨ , ∧ , ∃, ∃ 2 are definable from → , ∀, ∀ 2 . Since we are also interested in the subformula property, we shall restrict ourselves ... |

74 | Continuation-Based Multiprocessing
- Wand
(Show Context)
Citation Context ... used as a programming technique to simulate backtracking and coroutines. For instance, first-class continuations have been successfully used to implement Simula-like cooperative coroutines in Scheme =-=[53, 18, 19]-=-. This approach has been extended in the Standard ML of New Jersey (with the typed counterpart of Scheme’s call/cc [15]) to provide simple and elegant implementations of light-weight processes (or thr... |

60 | Adding threads to Standard ML
- Cooper, Morrisett
- 1990
(Show Context)
Citation Context ... concurrency is obtained by having individual threads voluntarily suspend themselves [48, 42] (providing time-sliced processes using pre-emptive scheduling requires additional run-time system support =-=[5, 47]-=-). The key point in these implementations is that control operators make it possible to switch between coroutine contexts, where the context of a coroutine is encoded as its continuation. The definiti... |

56 |
Mathematical Intuitionism: Introduction to Proof Theory, volume 67
- Dragalin
- 1988
(Show Context)
Citation Context ... deduction to sequents with several conclusions. As expected, this extension leads to classical logic. In order to stay in a constructive framework, several authors (for instance [29] p. 481 and also =-=[3, 14, 16]-=-) have suggested to restrict only the introduction rule of implication of LK to sequents with at most one conclusion. The same restriction can be applied to CND and by duality it can be generalized to... |

49 |
Classical proofs as programs
- Parigot
- 1993
(Show Context)
Citation Context ...horoughly this duality (but still in a categorical setting). On the other hand, Curien and Herbelin’s work is based on Parigot’s λµ-calculus and its type system, the Classical Natural Deduction (CND) =-=[36, 37]-=-. In order to complete the duality, they are led to extend CND with the subtraction (Walder does not consider the subtraction in [52]). However, this new connective is presented in a purely formal way... |

46 |
Call-by-value is dual to call-by-name
- Wadler
(Show Context)
Citation Context ...s Interpretation of Subtractive Logic • Curien and Herbelin demonstrated in [9] a striking result: duality in classical logic exchanges call-by-value with call-by-name (see also Wadler’s recent paper =-=[52]-=-). Actually, a first attempt at a categorical continuation semantics was Filinski’s pioneering work [17]. Then Selinger [50] has investigated thoroughly this duality (but still in a categorical settin... |

44 |
Classical Logic, Storage Operators and Second-Order lambda-Calculus
- Krivine
- 1994
(Show Context)
Citation Context ... the well-known formulas-as-types paradigm to classical logic has been widely investigated for instance by Murthy [31], Barbanera and Berardi [2], Rehof and S⊘rensen [46], de Groote [11, 12], Krivine =-=[30]-=-. Wes4 A Formulae-as-Types Interpretation of Subtractive Logic shall consider here Parigot’s λµ-calculus mainly because it is confluent and strongly normalizing in the second order framework [38]. How... |

37 | A simple calculus of exception handling
- Groote
- 1995
(Show Context)
Citation Context ..., the extension of the well-known formulas-as-types paradigm to classical logic has been widely investigated for instance by Murthy [31], Barbanera and Berardi [2], Rehof and S⊘rensen [46], de Groote =-=[11, 12]-=-, Krivine [30]. Wes4 A Formulae-as-Types Interpretation of Subtractive Logic shall consider here Parigot’s λµ-calculus mainly because it is confluent and strongly normalizing in the second order frame... |

37 |
Declarative continuations: an investigation of duality in programming language semantics
- Filinski
- 1989
(Show Context)
Citation Context ...ty in classical logic exchanges call-by-value with call-by-name (see also Wadler’s recent paper [52]). Actually, a first attempt at a categorical continuation semantics was Filinski’s pioneering work =-=[17]-=-. Then Selinger [50] has investigated thoroughly this duality (but still in a categorical setting). On the other hand, Curien and Herbelin’s work is based on Parigot’s λµ-calculus and its type system,... |

32 | Extracting constructive content from classical logic via control-like reductions
- Barbanera, Berardi
- 1993
(Show Context)
Citation Context ...D 3 Since Griffin’s pioneering work [23], the extension of the well-known formulas-as-types paradigm to classical logic has been widely investigated for instance by Murthy [31], Barbanera and Berardi =-=[2]-=-, Rehof and S⊘rensen [46], de Groote [11, 12], Krivine [30]. Wes4 A Formulae-as-Types Interpretation of Subtractive Logic shall consider here Parigot’s λµ-calculus mainly because it is confluent and s... |

30 | Minimal classical logic and control operators
- Ariola, Herbelin
- 2003
(Show Context)
Citation Context ...l with the negation, we add the propositional constant for falsum ⊥. We use the same name ε for occurrences of ⊥ in all sequents. Moreover, we do not represent ⊥ ε in the conclusions of sequents (see =-=[1]-=- for more details about the various treatments of ⊥ in CND). Now the elimination rule for ⊥ is just an instance of (WR) (take ⊥ ε as A α ): As usual, we also define ¬A as (A → ⊥). Derivation of the in... |

30 |
Obtaining Coroutines with Continuations
- Haynes, Friedman, et al.
- 1986
(Show Context)
Citation Context ... used as a programming technique to simulate backtracking and coroutines. For instance, first-class continuations have been successfully used to implement Simula-like cooperative coroutines in Scheme =-=[53, 18, 19]-=-. This approach has been extended in the Standard ML of New Jersey (with the typed counterpart of Scheme’s call/cc [15]) to provide simple and elegant implementations of light-weight processes (or thr... |

30 | Asynchronous signals in standard ML
- Reppy
- 1990
(Show Context)
Citation Context ... concurrency is obtained by having individual threads voluntarily suspend themselves [48, 42] (providing time-sliced processes using pre-emptive scheduling requires additional run-time system support =-=[5, 47]-=-). The key point in these implementations is that control operators make it possible to switch between coroutine contexts, where the context of a coroutine is encoded as its continuation. The definiti... |

23 |
Semi-boolean algebras and their applications to intuitionistic logic with dual operators
- Rauszer
- 1974
(Show Context)
Citation Context ...roduction Subtractive logic, also calledsbi-intuitionistic logic ¡ , is an extension of intuitionistic logic with a new connector, the subtraction (called pseudo-difference in Rauszer’s original work =-=[43, 44, 45]-=-), which is dual to implication. This duality has already been widely investigated from algebraic, relational, axiomatic and sequent perspectives by various authors [6, 8, 21, 43, 44, 45, 49]. In part... |

21 |
Strong normalization of classical natural deduction with disjunction. TLCA’01
- Groote
- 2001
(Show Context)
Citation Context ...rovides the subject reduction for SND→∨∧− (resp. SND→∨∧). Strong normalization and the subformula property Since our calculi are derived from de Groote’s λµ →∧∨⊥ , they enjoy the properties listed in =-=[12]-=-, namely: a) all connectives are taken as primitive; b) normal deductions satisfy the subformula property; c) the reduction relation is defined by means of local reduction steps; d) the reduction rela... |

20 | Subtractive logic
- Crolard
(Show Context)
Citation Context ...auszer’s original work [43, 44, 45]), which is dual to implication. This duality has already been widely investigated from algebraic, relational, axiomatic and sequent perspectives by various authors =-=[6, 8, 21, 43, 44, 45, 49]-=-. In particular, a unified proof-theoretic approach can be found in Gor ¢ ’s recent paper [21], while a connection with category theory is presented in the author’s previous work [6, 8]. We present in... |

20 |
On the semantics of classical disjunction
- Pym, Ritter
(Show Context)
Citation Context ...here the proof normalization process includes permutative conversions [12]. Such an extension of CND with primitive conjunction and disjunction has already been investigated by Pym, Ritter and Wallen =-=[40, 41, 39]-=- and de Groote [12]. The computational interpretation of classical logic is usually given by a λ-calculus extended with some form of control (such as the famous call/cc of Scheme or the catch/throw me... |

19 | On the intuitionistic force of classical search
- Ritter, Pym, et al.
(Show Context)
Citation Context ...here the proof normalization process includes permutative conversions [12]. Such an extension of CND with primitive conjunction and disjunction has already been investigated by Pym, Ritter and Wallen =-=[40, 41, 39]-=- and de Groote [12]. The computational interpretation of classical logic is usually given by a λ-calculus extended with some form of control (such as the famous call/cc of Scheme or the catch/throw me... |

19 |
The λ∆-calculus
- Rehof, Sorensen
(Show Context)
Citation Context ...eering work [23], the extension of the well-known formulas-as-types paradigm to classical logic has been widely investigated for instance by Murthy [31], Barbanera and Berardi [2], Rehof and S⊘rensen =-=[46]-=-, de Groote [11, 12], Krivine [30]. Wes4 A Formulae-as-Types Interpretation of Subtractive Logic shall consider here Parigot’s λµ-calculus mainly because it is confluent and strongly normalizing in th... |

16 |
Semantic construction of intuitionistic logic
- Beth
- 1956
(Show Context)
Citation Context ... deduction to sequents with several conclusions. As expected, this extension leads to classical logic. In order to stay in a constructive framework, several authors (for instance [29] p. 481 and also =-=[3, 14, 16]-=-) have suggested to restrict only the introduction rule of implication of LK to sequents with at most one conclusion. The same restriction can be applied to CND and by duality it can be generalized to... |

15 |
Full intuitionistic linear logic (extended abstract
- Hyland, Paiva
- 1993
(Show Context)
Citation Context ...[34]. Brauner and de Paiva [4] have proposed a restriction of Classical Linear Logic (in order to obtain a sequent-style formulation of Full Intuitionistic Linear Logic defined by Hyland and de Paiva =-=[25]-=-) very akin to the one presented in this paper. In a recent work [13], de Paiva and Ritter also consider a Parigot-style linear λ-calculus for this logic which is based on Pym and Ritter’s λµν-calculu... |

15 | Concurrent programming in ML
- Ramsey
- 1990
(Show Context)
Citation Context ... Scheme’s call/cc [15]) to provide simple and elegant implementations of light-weight processes (or threads), where concurrency is obtained by having individual threads voluntarily suspend themselves =-=[48, 42]-=- (providing time-sliced processes using pre-emptive scheduling requires additional run-time system support [5, 47]). The key point in these implementations is that control operators make it possible t... |

15 |
An algebraic and Kripke-style approach to a certain extension of intuitionistic logic
- Rauszer
- 1980
(Show Context)
Citation Context ...roduction Subtractive logic, also calledsbi-intuitionistic logic ¡ , is an extension of intuitionistic logic with a new connector, the subtraction (called pseudo-difference in Rauszer’s original work =-=[43, 44, 45]-=-), which is dual to implication. This duality has already been widely investigated from algebraic, relational, axiomatic and sequent perspectives by various authors [6, 8, 21, 43, 44, 45, 49]. In part... |

14 |
A confluent lambda-calculus with a catch/throw mechanism
- Crolard
- 1999
(Show Context)
Citation Context ... its continuation. The definition of a catch/throw mechanism is straightforward in the λµ-calculus: just set catch α t ≡ µα[α]t and throw α t ≡ µδ[α]t where δ is a name which does not occur in t (see =-=[7]-=- for a study of the sublanguage obtained when we restrict the λµ-terms to these operators). Then a name α may be reified as the first-class continuation λx.throw α x. However, the type of such a λµ-te... |

12 |
A constructive logic behind the catch and throw mechanism
- Nakano
- 1994
(Show Context)
Citation Context ...he dual restriction from the safe λµ →+×− -calculus just ensures that one cannot obtain full-fledged first-class continuations back from first-class coroutines. Related work Nakano, Kameyama and Sato =-=[33, 32, 34, 26, 27, 28]-=- have proposed various logical frameworks that are intended to provide a type system for a lexical variant of the catch/throw mechanism used in functional languages such as Lisp. Moreover, Nakano has ... |

12 | Proof-terms for classical and intuitionistic resolution
- Ritter, Pym, et al.
(Show Context)
Citation Context ...here the proof normalization process includes permutative conversions [12]. Such an extension of CND with primitive conjunction and disjunction has already been investigated by Pym, Ritter and Wallen =-=[40, 41, 39]-=- and de Groote [12]. The computational interpretation of classical logic is usually given by a λ-calculus extended with some form of control (such as the famous call/cc of Scheme or the catch/throw me... |

11 | The logical structures of the catch and throw mechanism
- Nakano
- 1995
(Show Context)
Citation Context ...he dual restriction from the safe λµ →+×− -calculus just ensures that one cannot obtain full-fledged first-class continuations back from first-class coroutines. Related work Nakano, Kameyama and Sato =-=[33, 32, 34, 26, 27, 28]-=- have proposed various logical frameworks that are intended to provide a type system for a lexical variant of the catch/throw mechanism used in functional languages such as Lisp. Moreover, Nakano has ... |

10 |
Extension de l’isomorphisme de CurryHoward au traitement des exceptions
- Crolard
- 1996
(Show Context)
Citation Context ...auszer’s original work [43, 44, 45]), which is dual to implication. This duality has already been widely investigated from algebraic, relational, axiomatic and sequent perspectives by various authors =-=[6, 8, 21, 43, 44, 45, 49]-=-. In particular, a unified proof-theoretic approach can be found in Gor ¢ ’s recent paper [21], while a connection with category theory is presented in the author’s previous work [6, 8]. We present in... |

10 | The non-deterministic catch and throw mechanism and its subject reduction property. Pages 61--72 of: Logic, language and computation
- Nakano
- 1994
(Show Context)
Citation Context ...he dual restriction from the safe λµ →+×− -calculus just ensures that one cannot obtain full-fledged first-class continuations back from first-class coroutines. Related work Nakano, Kameyama and Sato =-=[33, 32, 34, 26, 27, 28]-=- have proposed various logical frameworks that are intended to provide a type system for a lexical variant of the catch/throw mechanism used in functional languages such as Lisp. Moreover, Nakano has ... |

10 |
A formalization of the propositional calculus of H-B logic
- Rauszer
- 1974
(Show Context)
Citation Context ...roduction Subtractive logic, also calledsbi-intuitionistic logic ¡ , is an extension of intuitionistic logic with a new connector, the subtraction (called pseudo-difference in Rauszer’s original work =-=[43, 44, 45]-=-), which is dual to implication. This duality has already been widely investigated from algebraic, relational, axiomatic and sequent perspectives by various authors [6, 8, 21, 43, 44, 45, 49]. In part... |

7 | First-class synchronous operations
- Reppy
- 1995
(Show Context)
Citation Context ... Scheme’s call/cc [15]) to provide simple and elegant implementations of light-weight processes (or threads), where concurrency is obtained by having individual threads voluntarily suspend themselves =-=[48, 42]-=- (providing time-sliced processes using pre-emptive scheduling requires additional run-time system support [5, 47]). The key point in these implementations is that control operators make it possible t... |

6 | A formulation of linear logic based on dependency-relations
- Brauner, Paiva
- 1997
(Show Context)
Citation Context ...framework [38]. • We consider a type system ¥ la Curry, which allows us to rephrase the above restriction on pure (i.e. untyped) λµ-terms, and not only on typed terms as in [34]. Brauner and de Paiva =-=[4]-=- have proposed a restriction of Classical Linear Logic (in order to obtain a sequent-style formulation of Full Intuitionistic Linear Logic defined by Hyland and de Paiva [25]) very akin to the one pre... |

6 | Strong normalizability of the non-deterministic catch/throw calculi
- Kameyama, Sato
(Show Context)
Citation Context |

5 | A new formulation of the catch/throw mechanism
- Kameyama
- 1997
(Show Context)
Citation Context |

5 | A classical catch/throw calculus with tag abstraction and its strong normalizability. Pages 183--197
- Kameyama
- 1998
(Show Context)
Citation Context |

5 |
Classical proofs as programs: How, when, and why
- Murthy
- 1991
(Show Context)
Citation Context ...nterpretation TRISTAN CROLARD 3 Since Griffin’s pioneering work [23], the extension of the well-known formulas-as-types paradigm to classical logic has been widely investigated for instance by Murthy =-=[31]-=-, Barbanera and Berardi [2], Rehof and S⊘rensen [46], de Groote [11, 12], Krivine [30]. Wes4 A Formulae-as-Types Interpretation of Subtractive Logic shall consider here Parigot’s λµ-calculus mainly be... |

3 | Extending intuitionistic logic with subtraction. Unpublished note. Available at http://consequently.org/writing/extendingj
- Restall
- 1977
(Show Context)
Citation Context ...auszer’s original work [43, 44, 45]), which is dual to implication. This duality has already been widely investigated from algebraic, relational, axiomatic and sequent perspectives by various authors =-=[6, 8, 21, 43, 44, 45, 49]-=-. In particular, a unified proof-theoretic approach can be found in Gor ¢ ’s recent paper [21], while a connection with category theory is presented in the author’s previous work [6, 8]. We present in... |

3 |
Gentzen Collected work
- Szabo
- 1969
(Show Context)
Citation Context ...disjunction and subtraction are replaced by (LI ∨), (LE ∨ 1 ), (LE ∨ 2 ), and (LI −), (LE −). 3 Constructive restrictions of CND It is well-known that if we restrict the classical sequent calculus LK =-=[51]-=- to sequents with at most one conclusion we obtain the intuitionistic sequent calculus LJ [51]. As for natural deduction, it was originally presented for sequents having one conclusion and formalTRIS... |

2 |
Continuations and coroutines: An exercise in metaprogramming
- Friedman, Haynes, et al.
- 1984
(Show Context)
Citation Context ... used as a programming technique to simulate backtracking and coroutines. For instance, first-class continuations have been successfully used to implement Simula-like cooperative coroutines in Scheme =-=[53, 18, 19]-=-. This approach has been extended in the Standard ML of New Jersey (with the typed counterpart of Scheme’s call/cc [15]) to provide simple and elegant implementations of light-weight processes (or thr... |

2 |
Model extension theorem and Graigs interpolation theorem for intermediate predicate logics
- Ono
- 1983
(Show Context)
Citation Context ...ic logic. However, in the first-order framework, subtractive logic is no longer conservative over intuitionistic logic but over Constant Domain Logic (CDL). This logic as been studied for instance in =-=[20, 22, 35]-=- and [45]. Although CDL is not conservative over intuitionistic logic (in the firstorder framework), it is important to note that CDL is still a constructive logic (i.e. both disjunction and existence... |

1 |
On the relation between the lambda-calculus and the syntactic theory of sequential control
- Groote
- 1994
(Show Context)
Citation Context ...Γ ⊢ ∆; A set-context α t: Γ ⊢ ∆, A α ; B (WR) Γ ⊢ ∆, A α ; A get-context α t: Γ ⊢ ∆; A (CR) Remark 4.2. In order to deal with the negation, we add a macro-definition called abort (following de Groote =-=[10]-=-) defined as set-context ε t where ε is a free name (the name of ⊥ given in section 2.1) considered as a constant (see [1] for more details). Here is the typing rule for abort: t: Γ ⊢∆; ⊥ abort t: Γ ⊢... |

1 | A Parigot-style linear lambda-calculus for full intuitionistic linear logic. Submitted - Paiva, Ritter - 2003 |

1 |
Dual intuitionistic logic revisited
- Gor�
(Show Context)
Citation Context |