## A Resolution Theorem Prover for Intuitionistic Logic (1996)

Venue: | Proceedings of CADE-13 |

Citations: | 43 - 4 self |

### BibTeX

@INPROCEEDINGS{Tammet96aresolution,

author = {Tanel Tammet},

title = {A Resolution Theorem Prover for Intuitionistic Logic},

booktitle = {Proceedings of CADE-13},

year = {1996},

pages = {2--16},

publisher = {Springer-Verlag LNCS}

}

### Years of Citing Articles

### OpenURL

### Abstract

We use the general scheme of building resolution calculi (also called the inverse method) originating from S.Maslov and G.Mints to design and implement a resolution theorem prover for intuitionistic logic. A number of search strategies is introduced and proved complete. The resolution method is shown to be a decision procedure for a new syntactically described decidable class of intuitionistic logic. We compare the search strategies suitable for the resolution method with strategies suitable for the tableau method. The performance of our prover is compared with the performance of a tableau prover for intuitionistic logic presented in [17].

### Citations

966 |
A machine-oriented logic based on the resolution principle
- Robinson
- 1965
(Show Context)
Citation Context ...n the following we will refer to the method shortly as the resolution method. Ordinarily the name "resolution method" (coined by J.Robinson) denotes the proof search method presented by J.Ro=-=binson in [16]. In our c-=-ontext the usage of "resolution" is justified by the fact that Robinson's resolution method can be seen as a special case of the generic resolution. The forward-chaining method which systema... |

280 |
R.: Symbolic Logic and Mechanical Theorem Proving
- Chang, Lee
- 1987
(Show Context)
Citation Context ...of analogous help for program synthesis and type theory. 2 Terminology For the basic terminology of resolution (term, atom, literal, clause, substitution, most general unifier (denoted mgu)) see e.g. =-=[6, 4]-=-. Let us fix the terminology concerning Gentzen-type systems (sequent calculus). In the sequent \Gamma ` \Delta the left-hand side \Gamma and right-hand side \Delta are sometimes called antecedent and... |

109 | Resolution in Type Theory
- Andrews
- 1971
(Show Context)
Citation Context ...22] is limited to propositional calculus. In recent years we observe a renewed interest in proof search for intuitionistic logic, originating mostly from research in intuitionistic type theories (see =-=[1]-=- and [2] for early research in automating type theories). Although type theories are essentially higher-order, there exist useful fragments of type theories which can be directly encoded in first orde... |

46 |
Resolution Methods for the Decision Problem
- Fermuller, Tammet, et al.
- 1993
(Show Context)
Citation Context ...of analogous help for program synthesis and type theory. 2 Terminology For the basic terminology of resolution (term, atom, literal, clause, substitution, most general unifier (denoted mgu)) see e.g. =-=[6, 4]-=-. Let us fix the terminology concerning Gentzen-type systems (sequent calculus). In the sequent \Gamma ` \Delta the left-hand side \Gamma and right-hand side \Delta are sometimes called antecedent and... |

42 | Proof search in the intuitionistic sequent calculus
- Shankar
- 1992
(Show Context)
Citation Context ...a positive subformula of F . Thus all the derivable clauses satisfy conditions of the lemma 4. Thus the number of derivable clauses is finite. 2 For example, consider the following four formulas from =-=[18]-=-. Each of these is provable classically, but not intuitionistically. For each of these our resolution prover exhausts the search space in ca 0.016 seconds, keeping 0--2 of the derived clauses, and the... |

31 |
Gentzen-type systems and resolution rule, part II: predicate logic
- Mints
- 1990
(Show Context)
Citation Context ...theorem proving has been concentrated on classical logic. Relatively few papers are devoted on proof search in intuitionistic logic. The following is an incomplete list of such papers: [22],[7], [3], =-=[11]-=-, [23], [15], [17], [18], [5], [13]. Despite the fact that several intuitionistic theorem provers have been implemented (see [5], [18], [3]), only very few published papers describe the actual impleme... |

26 |
An intuitionistic predicate logic theorem prover
- Sahlin, Franzen, et al.
- 1992
(Show Context)
Citation Context ...ble for the resolution method with strategies suitable for the tableau method. The performance of our prover is compared with the performance of a tableau prover for intuitionistic logic presented in =-=[17]-=-. 1 Introduction Intuitionistic logic is interesting since intuitionistic proofs contain more information than the corresponding classical proofs: each intuitionistic proof corresponds to a computable... |

24 | Proof strategies in linear logic
- Tammet
- 1994
(Show Context)
Citation Context ...ly deepening depth-first search, where each iteration has a certain bound on a length of branches. Experiments for both the intuitionistic tableau (see [17]) and the linear propositional tableau (see =-=[19]-=-) have showed the good bound to be a number of contractions in a branch. The chief gains of efficiency for the tableau provers are obtained by restricting the calculus in order to obtain a low branchi... |

23 |
Theorem proving in non-standard logics based on the inverse method
- Voronkov
- 1992
(Show Context)
Citation Context ... is possible to allow only a single rule to be applied to the clause and not to consider any other rule applications. A general scheme of reduction strategies for the resolution method is proposed in =-=[21]-=-. The reduction strategy for linear logic developed independently in [19] was of crucial importance for efficiency of the linear resolution prover in [19]. Definition 5 We say that a clause \Gamma is ... |

20 | Resolution calculus for the first order linear logic - Mints - 1993 |

20 |
The Axioms of Constructive Geometry
- Plato
- 1995
(Show Context)
Citation Context ...time without Horn strategy or hyperresolution 0.36 seconds, clauses kept: 227. Resolution time with hyperresolution 0.04 seconds, clauses kept: 10. 7.3 Constructive Geometry of von Plato In the paper =-=[14]-=- Jan von Plato presents a first order axiomatization of constructive elementary geometry and proves a number of theorems in geometry. We will consider the axiomatization of geometry of apartness and c... |

19 |
Automating higher order logics
- Andrews, Longini-Cohen, et al.
- 1984
(Show Context)
Citation Context ...imited to propositional calculus. In recent years we observe a renewed interest in proof search for intuitionistic logic, originating mostly from research in intuitionistic type theories (see [1] and =-=[2]-=- for early research in automating type theories). Although type theories are essentially higher-order, there exist useful fragments of type theories which can be directly encoded in first order intuit... |

18 |
Automated Proof Search in Non-Classical Logics
- WALLEN
- 1990
(Show Context)
Citation Context ...m proving has been concentrated on classical logic. Relatively few papers are devoted on proof search in intuitionistic logic. The following is an incomplete list of such papers: [22],[7], [3], [11], =-=[23]-=-, [15], [17], [18], [5], [13]. Despite the fact that several intuitionistic theorem provers have been implemented (see [5], [18], [3]), only very few published papers describe the actual implementatio... |

17 |
Resolution strategies for the intuitionistic logic, (conf proc
- Mints
- 1994
(Show Context)
Citation Context ...ed on classical logic. Relatively few papers are devoted on proof search in intuitionistic logic. The following is an incomplete list of such papers: [22],[7], [3], [11], [23], [15], [17], [18], [5], =-=[13]-=-. Despite the fact that several intuitionistic theorem provers have been implemented (see [5], [18], [3]), only very few published papers describe the actual implementation of an automated theorem pro... |

14 |
Some applications of Gentzenâ€™s proof theory in automated deduction
- Beeson
- 1991
(Show Context)
Citation Context ...ated theorem proving has been concentrated on classical logic. Relatively few papers are devoted on proof search in intuitionistic logic. The following is an incomplete list of such papers: [22],[7], =-=[3]-=-, [11], [23], [15], [17], [18], [5], [13]. Despite the fact that several intuitionistic theorem provers have been implemented (see [5], [18], [3]), only very few published papers describe the actual i... |

14 |
2.0 Users Guide
- OTTER
(Show Context)
Citation Context ... same result as the unoptimized versions. 5.9 The Main Loop of the Prover Our resolution prover uses the given-clause algorithm which is common for resolution provers for classical logic, e.g. OTTER: =-=[10]-=-. Two main lists are maintained: sos (set of support) is the list of clauses yet to be considered, usable is the list of active clauses. All the axioms are initially put to the sos list, and the usabl... |

9 |
What is the inverse method
- Lifschitz
- 1989
(Show Context)
Citation Context ...ences of 8. 4 The Generic Resolution Method The generic resolution method (also called the "inverse method") originally developed by S.Maslov and G.Mints (see e.g. [9] (contains unification =-=already), [8]-=- and [11]) is a forward-chaining proof search method. The sequent calculus rules are applied top-tobottom (again, observe that premisses are written above the line and the conclusion below the line). ... |

8 |
Investigations into proof-search in a system of first-order dependent function types
- Pym, Wallen
- 1990
(Show Context)
Citation Context ...ing has been concentrated on classical logic. Relatively few papers are devoted on proof search in intuitionistic logic. The following is an incomplete list of such papers: [22],[7], [3], [11], [23], =-=[15]-=-, [17], [18], [5], [13]. Despite the fact that several intuitionistic theorem provers have been implemented (see [5], [18], [3]), only very few published papers describe the actual implementation of a... |

7 | Optimized encodings of fragments of type theory in first-order logic
- Tammet, Smith
- 1998
(Show Context)
Citation Context ...ssentially higher-order, there exist useful fragments of type theories which can be directly encoded in first order intuitionistic logic, with no additional axioms or axiom schemes required, see e.g. =-=[20]-=-. In fragments like these the problems of proof search in type theory translate directly into problems of proof search in first order intuitionistic logic. It is not realistic to expect that an automa... |

2 |
Resolution for Intuitionistic Logic. Paper presented at ISMIS '87
- Fitting
- 1987
(Show Context)
Citation Context ...automated theorem proving has been concentrated on classical logic. Relatively few papers are devoted on proof search in intuitionistic logic. The following is an incomplete list of such papers: [22],=-=[7]-=-, [3], [11], [23], [15], [17], [18], [5], [13]. Despite the fact that several intuitionistic theorem provers have been implemented (see [5], [18], [3]), only very few published papers describe the act... |

2 |
S.Ju.Maslov An inverse method of establishing deducibility in the classical predicate calculus
- Dokl
- 1964
(Show Context)
Citation Context ...mulas not containing negative occurrences of 8. 4 The Generic Resolution Method The generic resolution method (also called the "inverse method") originally developed by S.Maslov and G.Mints =-=(see e.g. [9]-=- (contains unification already), [8] and [11]) is a forward-chaining proof search method. The sequent calculus rules are applied top-tobottom (again, observe that premisses are written above the line ... |

1 | Contraction-free sequent calculi for intuitionistic logic - Dyckoff - 1992 |