## Bidirectional Decision Procedures for the Intuitionistic Propositional Modal Logic IS4

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Citations: | 2 - 0 self |

### BibTeX

@MISC{Heilala_bidirectionaldecision,

author = {Samuli Heilala and Brigitte Pientka},

title = {Bidirectional Decision Procedures for the Intuitionistic Propositional Modal Logic IS4},

year = {}

}

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### Abstract

Abstract. We present a multi-context focused sequent calculus whose derivations are in bijective correspondence with normal natural deductions in the propositional fragment of the intuitionistic modal logic IS4. This calculus, suitable for the enumeration of normal proofs, is the starting point for the development of a sequent calculus-based bidirectional decision procedure for propositional IS4. In this system, relevant derived inference rules are constructed in a forward direction prior to proof search, while derivations constructed using these derived rules are searched over in a backward direction. We also present a variant which searches directly over normal natural deductions. Experimental results show that on most problems, the bidirectional prover is competitive with both conventional backward provers using loop-detection and inverse method provers, significantly outperforming them in a number of cases. 1

### Citations

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(Show Context)
Citation Context ... to choose NJIS4 as a foundation on which to develop proof search strategies for IS4. While Girard, Lafont, and Taylor suggest that we should think of natural deductions as the “true ‘proof’ objects” =-=[10]-=-, natural deduction systems have traditionally not seen much use as formalisms for proof search, mainly as a result of their lack of syntax-directedness. The first step towards being able to perform p... |

199 | A modal analysis of staged computation
- Davies, Pfenning
(Show Context)
Citation Context ... range of proof-theoretical properties of various logics of this kind. Recently, such logics have also found applications in hardware verification [7] and proposed type systems for staged computation =-=[4]-=- and distributed computing [14]. A logic frequently used in these settings is either the intuitionistic variant of the classical modal logic S4, which we will call IS4, or a logic that can be expresse... |

175 |
A new constructive logic: classical logic
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(Show Context)
Citation Context ... can be proved from assumptions ∆, Γ, A, focusing on the assumption A. If a sequent is focused on a formula A, then the only applicable rules are those with A as a principal formula. Following Girard =-=[9]-=-, we will call the position of the focused formula the stoup. As in the natural deduction formulations, contexts in MJIS4 are technically ordered lists, but the usual structural properties of weakenin... |

167 | A judgmental reconstruction of modal logic
- Pfenning, Davies
(Show Context)
Citation Context ...onstructive logics incorporating operators of necessity (✷) and possibility (✸). Fitch [8], Prawitz [17], Satre [19], and more recently Simpson [20], Bierman and de Paiva [2], and Pfenning and Davies =-=[16]-=- have investigated a broad range of proof-theoretical properties of various logics of this kind. Recently, such logics have also found applications in hardware verification [7] and proposed type syste... |

110 | The Proof Theory and Semantics of Intuitionistic Modal Logic
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- 1994
(Show Context)
Citation Context ... cases. 1 Introduction Intuitionistic modal logics are constructive logics incorporating operators of necessity (✷) and possibility (✸). Fitch [8], Prawitz [17], Satre [19], and more recently Simpson =-=[20]-=-, Bierman and de Paiva [2], and Pfenning and Davies [16] have investigated a broad range of proof-theoretical properties of various logics of this kind. Recently, such logics have also found applicati... |

67 |
Natural Deduction: A Proof Theoretical Study
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(Show Context)
Citation Context ...nificantly outperforming them in a number of cases. 1 Introduction Intuitionistic modal logics are constructive logics incorporating operators of necessity (✷) and possibility (✸). Fitch [7], Prawitz =-=[16]-=-, Satre [18], and more recently Simpson [19], Bierman and de Paiva [1], and Pfenning and Davies [15] have investigated a broad range of proof-theoretical properties of various logics of this kind. Rec... |

63 |
Contextual modal type theory
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(Show Context)
Citation Context ...IS4 in this paper, we believe that the techniques presented are general enough to find applications in other constructive logics, such as the contextual modal logic of Nanevski, Pfenning, and Pientka =-=[15]-=-. This paper is an extended version of [11]. In Sect. 2 we summarize the relevant background and introduce our core natural deduction formalism, while Sect. 3 presents a corresponding Hilbertstyle axi... |

61 | Propositional lax logic
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(Show Context)
Citation Context ...nd Pfenning and Davies [16] have investigated a broad range of proof-theoretical properties of various logics of this kind. Recently, such logics have also found applications in hardware verification =-=[7]-=- and proposed type systems for staged computation [4] and distributed computing [14]. A logic frequently used in these settings is either the intuitionistic variant of the classical modal logic S4, wh... |

41 | The Focused Inverse Method for Linear Logic
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(Show Context)
Citation Context ...t nature of how such a generalization affects proof search is yet to be explored. The reconciliation of forward and backward proof search has recently also been investigated by Chaudhuri and Pfenning =-=[3]-=-, who, in the context of linear logic, propose a focusing inverse method prover incorporating derived rules constructed in a backward way and searched over in a forward direction, precisely opposite t... |

41 | Cut-elimination and a permutation-free sequent calculus for intuitionistic logic
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- 1998
(Show Context)
Citation Context ...e LJIS4, presented in Sect. 4, as a basis for proof search in a backward direction, its refl rule is a source of inconvenient nondeterminism. We will instead follow the approach of Dyckhoff and Pinto =-=[6]-=- and construct a 25s∆; Γ ⊲ A → A init (A is atomic) ∆; Γ ⊲ ⊥ → A ⊥L ∆; Γ1, A, Γ2 ⊲ A → C ∆; Γ1, A, Γ2 → C ∆; Γ, A1 → A2 ∆; Γ → A1 ⊃ A2 ⊃R ∆; Γ → A1 ∆; Γ → A2 ∆; Γ → A1 ∧ A2 ∆; Γ → Aj ∆; Γ → A1 ∨ A2 ∨R... |

39 | Proof Search Issues in Some Non-Classical Logics
- Howe
- 1998
(Show Context)
Citation Context ...of enumeration and theorem proving in lax logic [13] and, coming closer to our work, has presented a backward decision procedure for the fragment of propositional IS4 without the possibility modality =-=[12]-=-. His system performs loop-detection using a history mechanism, but is encumbered by a large number of rules and related provisos (21 axioms and inference rules). It would only grow with the addition ... |

28 | Categorical and Kripke Semantics for Constructive S4 Modal Logic
- Alechina, Mendler, et al.
- 2001
(Show Context)
Citation Context ...several proposals for Hilbert-style axiomatic formulations of constructive versions of S4 exist in the literature, we will use a standard system motivated by applications discussed by Alechina et al. =-=[1]-=- and Pfenning and Davies [16]. In it, a standard axiomatization of intuitionistic propositional logic (see for instance [21]) is augmented by the six modal axioms K✷ : ✷(A1 ⊃ A2) ⊃ ✷A1 ⊃ ✷A2 T✷ : ✷A ⊃... |

24 | On an intuitionistic modal logic
- Bierman, Paiva
(Show Context)
Citation Context ...itionistic modal logics are constructive logics incorporating operators of necessity (✷) and possibility (✸). Fitch [8], Prawitz [17], Satre [19], and more recently Simpson [20], Bierman and de Paiva =-=[2]-=-, and Pfenning and Davies [16] have investigated a broad range of proof-theoretical properties of various logics of this kind. Recently, such logics have also found applications in hardware verificati... |

24 | Modal logic as a basis for distributed computation
- Moody
- 2003
(Show Context)
Citation Context ...operties of various logics of this kind. Recently, such logics have also found applications in hardware verification [7] and proposed type systems for staged computation [4] and distributed computing =-=[14]-=-. A logic frequently used in these settings is either the intuitionistic variant of the classical modal logic S4, which we will call IS4, or a logic that can be expressed through IS4, such as Fairtlou... |

16 | The iltp problem library for intuitionistic logic
- Raths, Otten, et al.
- 2007
(Show Context)
Citation Context ...evaluate the performance of our bidirectional approach, we put together a benchmark set of 50 formulas for IS4, mostly problems from Raths et al.’s Intuitionistic Logic Theorem Proving (ILTP) library =-=[18]-=- to which we introduced modalities. Our full benchmark set is given in Appendix A. We implemented three IS4 decision procedures in SML: (1) an MJIS4-based backward prover with a conventional history m... |

13 | The inverse method
- Degtyarev, Voronkov
- 2001
(Show Context)
Citation Context ...left-rule applications, the stoup formula is dropped from the stoup into one 35sof the contexts by an application of ch1 or ch2. In a focused forward calculus used as the basis for the inverse method =-=[5]-=-, we would proceed in a similar way, but it is not clear which context a stoup formula should be dropped into. To address this uncertainty, we refine the idea of focusing and develop the system MJF IS... |

12 |
Logic programming with focussing proofs in linear logic
- Andreoli
- 1992
(Show Context)
Citation Context ...d prover incorporating big-step rules constructed in a backward way and searched over in a forward direction, opposite to our approach. Unlike their work, which builds on Andreoli’s focusing property =-=[1]-=-, we extend ideas by Dyckhoff and Pinto [6] and develop a focusing calculus which directly gives rise to a bidirectional natural deduction proof search procedure In the future, we plan to explore exte... |

5 |
Intuitionistic modal logic with quantifiers
- Fitch
- 1948
(Show Context)
Citation Context ... provers, significantly outperforming them in a number of cases. 1 Introduction Intuitionistic modal logics are constructive logics incorporating operators of necessity (✷) and possibility (✸). Fitch =-=[8]-=-, Prawitz [17], Satre [19], and more recently Simpson [20], Bierman and de Paiva [2], and Pfenning and Davies [16] have investigated a broad range of proof-theoretical properties of various logics of ... |

4 | Proof search in lax logic
- Howe
(Show Context)
Citation Context ...tionship between IS4 and lax logic). In this light, it is surprising that proof search in IS4 has not received more attention. Howe has investigated proof enumeration and theorem proving in lax logic =-=[13]-=- and, coming closer to our work, has presented a backward decision procedure for the fragment of propositional IS4 without the possibility modality [12]. His system performs loop-detection using a his... |

3 | A permutation-free calculus for lax logic
- Howe
- 1998
(Show Context)
Citation Context ...al forms that we are interested in during proof search, an unrestricted variant NJIS4 can be obtained by dropping the arrow annotations and the rule ↑↓. Using an approach analogous to that of Howe in =-=[10]-=-, the two systems can be shown to be equivalent in terms of provability.∆; Γ, A, Γ ′ hyp1 ⊢ A ↓ ∆; Γ ⊢ Aj ↑ ∆; Γ, A1 ⊢ A2 ↑ ∆; Γ ⊢ A1 ⊃ A2 ↑ ⊃I ∆; Γ ⊢ A1 ∨ A2 ↑ ∨Ij ∆; Γ ⊢ A ↑ ∆; Γ ⊢ ✸A ↑ ✸I ∆; Γ ⊢ A... |

2 |
Natural deduction rules for modal logics
- Satre
- 1972
(Show Context)
Citation Context ...utperforming them in a number of cases. 1 Introduction Intuitionistic modal logics are constructive logics incorporating operators of necessity (✷) and possibility (✸). Fitch [8], Prawitz [17], Satre =-=[19]-=-, and more recently Simpson [20], Bierman and de Paiva [2], and Pfenning and Davies [16] have investigated a broad range of proof-theoretical properties of various logics of this kind. Recently, such ... |

2 |
Basic Proof Theory, Second Edition. Cambridge University Press, 2000. Benchmark Formulas Our 50 benchmark formulas are shown below. The equivalence operator ≡, used in some of the formulas, is defined notationally
- Troelstra, Schwichtenberg
(Show Context)
Citation Context ...use a standard system motivated by applications discussed by Alechina et al. [1] and Pfenning and Davies [16]. In it, a standard axiomatization of intuitionistic propositional logic (see for instance =-=[21]-=-) is augmented by the six modal axioms K✷ : ✷(A1 ⊃ A2) ⊃ ✷A1 ⊃ ✷A2 T✷ : ✷A ⊃ A T✸ : A ⊃ ✸A K✸ : ✷(A1 ⊃ A2) ⊃ ✸A1 ⊃ ✸A2 4✷ : ✷A ⊃ ✷✷A 4✸ : ✸✸A ⊃ ✸A and the necessitation rule “If A is a theorem, then ✷... |