## Categorical and Kripke Semantics for Constructive S4 Modal Logic (2001)

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Venue: | In International Workshop on Computer Science Logic, CSL’01, L. Fribourg, Ed. Lecture Notes in Computer Science |

Citations: | 27 - 1 self |

### BibTeX

@INPROCEEDINGS{Alechina01categoricaland,

author = {Natasha Alechina and Michael Mendler and Valeria De Paiva and Eike Ritter},

title = {Categorical and Kripke Semantics for Constructive S4 Modal Logic},

booktitle = {In International Workshop on Computer Science Logic, CSL’01, L. Fribourg, Ed. Lecture Notes in Computer Science},

year = {2001},

pages = {292--307},

publisher = {Springer}

}

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### Abstract

We consider two systems of constructive modal logic which are computationally motivated. Their modalities admit several computational interpretations and are used to capture intensional features such as notions of computation, constraints, concurrency, etc. Both systems have so far been studied mainly from type-theoretic and category-theoretic perspectives, but Kripke models for similar systems were studied independently. Here we bring these threads together and prove duality results which show how to relate Kripke models to algebraic models and these in turn to the appropriate categorical models for these logics.

### Citations

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Citation Context ...ity, to model the quote mechanism of Lisp, is proposed by Goubault-Larrecq [GL96]. A ✸-style modality has been extensively used to distinguish a computation from its result in the λ-calculus: Moggi=-=’s [Mog91] in-=-fluential work on computational monads describes the computational λ-calculus, which corresponds to an intuitionistic modal type theory with a ✸-like modality (see [BBdP98]). Fairtlough and Mendler... |

232 |
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Citation Context ...ir associated modal operators. 2.1 Constructive S4 The first modal system, which we call Constructive S4 (CS4), is a version of the intuitionistic S4 first introduced by Prawitz in his 1965 monograph =-=[Pra65]. Th-=-e Hilbertstyle formulation of CS4 is obtained by extending IPL by a pair ✷,✸ of S4-like intuitionistic modalities satisfying the axioms and the necessitation rule listed in Figure 1. The normal ba... |

197 | A modal analysis of staged computation
- Davies, Pfenning
- 1996
(Show Context)
Citation Context ...tional logic (IPL), which are particularly interesting because of their various applications in computer science. To give the reader a taste for these applications, we list a few. Davies and Pfenning =-=[DP96] use -=-the ✷-modality to give a λ-calculus for computation in stages. The idea is that a term ✷t represents a delayed computation. Ghani et al. [GdPR98] investigate refinements of this calculus which ar... |

188 | Modal Logic - Chagrov, Zakharyaschev - 1997 |

167 | A judgmental reconstruction of modal logic
- Pfenning, Davies
(Show Context)
Citation Context ...the beginning of this section, though most applications so far focus on the ✷ modality. The formal role of ✸ and its interaction with ✷ has recently been studied systematically by Pfenning and D=-=avies [PD01]. ✷K : ✷(A → B) → (✷A → ✷B) ✸K : ✷(A → B) -=-→ (✸A → ✸B) ✷T : ✷A → A ✸T : A → ✸A ✷4 :✷A → ✷✷A ✸4 :✸✸A → ✸A Nec : If A is a theorem then ✷A is a theorem. Fig. 1. Hilbert-style system for Constructive S4 T... |

144 |
Introduction to Higher-Order Categorical Logic. Number 7
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- 1986
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Citation Context ...g A as hypothesis. Cartesian closed categories (with coproducts) are the categorical models for intuitionistic propositional logic. For a proper explanation the reader should consult Lambek and Scott =-=[LS85]-=-; Here we just outline the intuitions. Conjunction is modelled by cartesian products, a suitable generalisation of the products in Heyting algebras. The usual logical relationship between conjunction ... |

143 | Elements of Intuitionism - Dummett - 1977 |

125 | Primitive Recursion for Higherorder Abstract syntax
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Citation Context ...bstract machines. Similar ideas relating ✷ with staged evaluation and the distinction between run-time andscompile-time semantics have been developed by Moggi et.al. [BMTS99]. Despeyroux and Pfennin=-=g [DPS97] u-=-se a box modality to encode higher-order abstract syntax in theorem-provers like Elf and Isabelle. Still another use of the ✷ modality, to model the quote mechanism of Lisp, is proposed by Goubault-... |

109 | The Proof Theory and Semantics of Intuitionistic Modal Logic
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Citation Context ...ditions) and just how they should be used to interpret specifically the ✸ modality. The mainstream approach as exemplified by Ewald [Ewa86], Fischer-Servi [FS80], Plotkin and Stirling [PS86], Simpso=-=n [Sim94] is based -=-on the analogy of ✷ with ∀ and of ✸ with ∃-quantification over the modal accessibility R. Reading these quantifiers intuitionistically, relative to ≤, one arrives at the semantic interpretat... |

61 | Propositional lax logic
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- 1997
(Show Context)
Citation Context ...ics is a more recent development. In this paper we develop a suitable categorical semantics and associated λ-calculus for CS4 and PLL. It should be mentioned that the results for PLL are not new (see=-= [FM97]-=- for the Kripke and [BBdP98] for categorical semantics for PLL). Our contribution here is to show how PLL is related to CS4 and how these known results for PLL can be derived from those from CS4, or, ... |

56 | Computational types from a logical perspective
- Benton, Bierman, et al.
- 1998
(Show Context)
Citation Context ...opment. In this paper we develop a suitable categorical semantics and associated λ-calculus for CS4 and PLL. It should be mentioned that the results for PLL are not new (see [FM97] for the Kripke and=-= [BBdP98]-=- for categorical semantics for PLL). Our contribution here is to show how PLL is related to CS4 and how these known results for PLL can be derived from those from CS4, or, to put it the other way roun... |

54 |
Metamathematics of modal logic
- Goldblatt
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(Show Context)
Citation Context ...a + b) ✷a ≤ ✷✷a ✸✸a ≤ ✸a 1 ≤ ✷1 ✷a × ✸b ≤ ✸(✷a × b). Next, we identify the corresponding algebraic structure for PLL, which are also known, in a somewhat different axio=-=matisation, as “local algebras” [Gol76]: Definition 4. A PLL-modal algebra A = ��-=-�A, ≤, ×, +, ⇒, 0,✸〉 consists of a Heyting algebra 〈A, ≤, ×, +, ⇒, 0〉 with a unary operator ✸ on A, such that for every a, b ∈ A, ✸a ≤ ✸(a + b) a ≤ ✸a ✸✸a ≤ ✸a ... |

31 |
A framework for intuitionistic modal logics
- Plotkin, Stirling
- 1986
(Show Context)
Citation Context ...eract (frame conditions) and just how they should be used to interpret specifically the ✸ modality. The mainstream approach as exemplified by Ewald [Ewa86], Fischer-Servi [FS80], Plotkin and Stirlin=-=g [PS86], Simpson -=-[Sim94] is based on the analogy of ✷ with ∀ and of ✸ with ∃-quantification over the modal accessibility R. Reading these quantifiers intuitionistically, relative to ≤, one arrives at the sem... |

24 | Logical modalities and multi-stage programming
- Benaissa, Moggi, et al.
- 1999
(Show Context)
Citation Context ...h are suitable for the design of abstract machines. Similar ideas relating ✷ with staged evaluation and the distinction between run-time andscompile-time semantics have been developed by Moggi et.al=-=. [BMTS99]. -=-Despeyroux and Pfenning [DPS97] use a box modality to encode higher-order abstract syntax in theorem-provers like Elf and Isabelle. Still another use of the ✷ modality, to model the quote mechanism ... |

23 | Intuitionistic necessity revisited
- Bierman, Paiva
- 1996
(Show Context)
Citation Context ...eorem then ✷A is a theorem. Fig. 1. Hilbert-style system for Constructive S4 The natural deduction formulation of CS4 is subject to some controversy. We recall it in the style of Bierman and de Paiv=-=a [BdP96]. Th-=-e naive introduction rule for✷ (corresponding to the necessitation rule Nec) insists that all of the undischarged assumptions at the time of application are modal, i.e. they are all of the form ✷A... |

21 | Intuitionistic modal logics as fragments of classical bimodal logics. Logics at work, Essays in honour of Helena Rasiowa
- Wolter, Zakharyaschev
- 1999
(Show Context)
Citation Context ...alities corresponding to R). This means that our variant of S4 does not fall directly in the scope of Wolter and Zakharyaschev’s analysis of intuitionistic modal logics as classical bimodal logics i=-=n [WZ97] s-=-ince they assume ✸ to be a normal modality. However, analogous techniques could probably be used to give a new proof of decidability and finite modal property of CS4 and PLL. Theorem 1. CS4 is sound... |

20 |
Intuitionistic tense and modal logic
- Ewald
- 1986
(Show Context)
Citation Context ...R, it is not so clear how these relations should interact (frame conditions) and just how they should be used to interpret specifically the ✸ modality. The mainstream approach as exemplified by Ewal=-=d [Ewa86], Fische-=-r-Servi [FS80], Plotkin and Stirling [PS86], Simpson [Sim94] is based on the analogy of ✷ with ∀ and of ✸ with ∃-quantification over the modal accessibility R. Reading these quantifiers intuit... |

20 |
Constructive modal logics I
- Wijesekera
- 1990
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Citation Context ...ssitation rule listed in Figure 1. The normal basis of CS4, i.e., consisting only of axioms ✷K and ✸K plus the axiom ¬✸⊥ (which we reject, see below) has been introduced 1 and motivated by Wi=-=jesekera [Wij90]-=- as a predecessor to constructive concurrent dynamic logic. The practical importance of CS4 as a type system for functional programming is evident from the literature, e.g. as cited in the beginning o... |

17 | An incompleteness theorem in modal logic - Thomason - 1974 |

15 |
Monad as modality
- Kobayashi
- 1997
(Show Context)
Citation Context ... special characteristics: namely we want our monad to be strong with respect to the ✷ operator, i.e. we assume a natural transformation stA,B:✷A ×✸B −→ ✸(✷A × B) satisfying the conditi=-=ons detailed in [Kob97]. The stre-=-ngth is needed to model the explicit substitution in the ✸E-rule. Definition 5. A CS4-category consists of a cartesian closed category C with coproducts, a monoidal comonad (✷,δ,ɛ,m−,−,m1) w... |

14 | An incomplete logic containing S4 - Fine - 1974 |

12 | First-order lax logic as a framework for constraint logic programming - Fairtlough, Mendler, et al. - 1997 |

12 | Grothendieck topology as geometric modality - Goldblatt - 1981 |

12 |
A Modal Logic for Handling Behavioural Constraints in Formal Hardware Verification
- Mendler
- 1993
(Show Context)
Citation Context ...xiomatic presentations, such as Computational Logic [BBdP98] or Propositional Lax Logic (PLL) [FM97]. Henceforth we shall call it PLL. The natural deduction system contains the following rules for ✸=-= ([Men93]): . B (✸I-=-) ✸B [A] . . . . (✸E) ✸A ✸B ✸B PLL also has a colourful history. As a modal logic it was invented in the forties by Curry [Cur57] (who seems to have dropped it again because of its wild prop... |

10 | Topological duality for intuitionistic modal algebras
- Hilken
(Show Context)
Citation Context ...e accessible through R ∗ . This too, expresses negative information, though of a second-order nature. A quite different, but still second-order representation of CK models has been proposed by Hilke=-=n [Hil96]. As w-=-e have shown, however, the representation for CS4 can be done in a first-order fashion. Our constructive S4 models satisfy the inclusion R;≤ ⊆≤;R, a frame condition that is typically assumed in ... |

8 |
Benthem, Modal logic and classical logic, Bibliopolis
- van
- 1983
(Show Context)
Citation Context ...positional variable p and be closed under intersection, union and operations which give the set of worlds satisfying ✷ϕ (✸ϕ) from the set of worlds satisfying ϕ. (For more background, see for e=-=xample [Ben83]-=-.) 2 More precisely, a general frame; see the discussion below.sHere, we will be somewhat sloppy and identify elements of the algebra with logical formulae straightaway. We assume that some subset P o... |

8 |
Explicit Substitutions for Constructive Necessity, volume Automata, Languages and Programming of Lecture
- Ghani, Paiva, et al.
(Show Context)
Citation Context ...se applications, we list a few. Davies and Pfenning [DP96] use the ✷-modality to give a λ-calculus for computation in stages. The idea is that a term ✷t represents a delayed computation. Ghani et=-= al. [GdPR98] i-=-nvestigate refinements of this calculus which are suitable for the design of abstract machines. Similar ideas relating ✷ with staged evaluation and the distinction between run-time andscompile-time ... |

7 |
Modal operators on Heyting algebras
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Citation Context ...he case that” by interpreting this in various ways to mean “at all nearby points” [Gol81,Gol93]. The algebraic properties of such operators (on complete Heyting algebras) have been explored by M=-=acnab [Mac81], who ca-=-lls them “modal operators”. In this paper we show how PLL can be naturally seen as a special CS4 theory or CS4 algebra in the sense that it can be obtained from CS4 by adding the axiom A → ✷A.... |

5 |
Semantics for a class of intuitionistic modal calculi
- Fischer-Servi
- 1980
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Citation Context ...ow these relations should interact (frame conditions) and just how they should be used to interpret specifically the ✸ modality. The mainstream approach as exemplified by Ewald [Ewa86], Fischer-Serv=-=i [FS80], Plotki-=-n and Stirling [PS86], Simpson [Sim94] is based on the analogy of ✷ with ∀ and of ✸ with ∃-quantification over the modal accessibility R. Reading these quantifiers intuitionistically, relative... |

5 | Logical foundations of eval/quote mechanisms, and the modal logic S4
- Goubault-Larrecq
- 1996
(Show Context)
Citation Context ...ms in the functional programming language based on these terms. For a discussion of the necessity modal operator ✷ and its interpretation as the ‘eval/quote’ operator in Lisp the reader is refer=-=ed to [GL96]-=-. In this sense both Kripke semantics and categorical semantics, presented here for CS4 and PLL, should be seen as two complementary elaborations of the algebraic semantics. They are both intensional ... |

5 | Characterising combinational timing analyses in intuitionistic modal logic,” The Logic - Mendler - 2000 |

4 |
A Theory of Formal Deducibility, volume 6
- Curry
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(Show Context)
Citation Context ...natural deduction system contains the following rules for ✸ ([Men93]): . B (✸I) ✸B [A] . . . . (✸E) ✸A ✸B ✸B PLL also has a colourful history. As a modal logic it was invented in the for=-=ties by Curry [Cur57] (-=-who seems to have dropped it again because of its wild properties) and independently rediscovered in the nineties by Benton et al. and Fairtlough and Mendler, who used the symbol ○ for the modality,... |

4 | Mathematics of Modality. CSLI Lecture Notes No - Goldblatt - 1993 |

2 |
Metamathematics of modal logic”, Reports on
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(Show Context)
Citation Context ...a + b) ✷a ≤ ✷✷a ✸✸a ≤ ✸a 1 ≤ ✷1 ✷a × ✸b ≤ ✸(✷a × b). Next, we identify the corresponding algebraic structure for PLL, which are also known, in a somewhat different axiomatisation, as “local algebras” =-=[Gol76]-=-: Definition 4. A PLL-modal algebra A = 〈A, ≤, ×, +, ⇒, 0, ✸〉 consists of a Heyting algebra 〈A, ≤, ×, +, ⇒, 0〉 with a unary operator ✸ on A, such that for every a, b ∈ A, ✸a ≤ ✸(a + b) a ≤ ✸a ✸✸a ≤ ✸a... |

1 |
Explicit Substitutions for Constructive
- Ghani, Paiva, et al.
(Show Context)
Citation Context ...se applications, we list a few. Davies and Pfenning [DP96] use the ✷-modality to give a λ-calculus for computation in stages. The idea is that a term ✷t represents a delayed computation. Ghani et al. =-=[GdPR98]-=- investigate refinements of this calculus which are suitable for the design of abstract machines. Similar ideas relating ✷ with staged evaluation and the distinction between run-time andcompile-time ... |

1 |
Logical foundations of eval/quote mechanisms, and the modal logic S4
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- 1998
(Show Context)
Citation Context ...ms in the functional programming language based on these terms. For a discussion of the necessity modal operator ✷ and its interpretation as the ‘eval/quote’ operator in Lisp the reader is refered to =-=[GL96]-=-. In this sense both Kripke semantics and categorical semantics, presented here for CS4 and PLL, should be seen as two complementary elaborations of the algebraic semantics. They are both intensional ... |