## Knowledge Archives in Theorema: A Logic-Internal Approach

Citations: | 1 - 1 self |

### BibTeX

@MISC{Rosenkranz_knowledgearchives,

author = {Camelia Rosenkranz and Bruno Buchberger and Tudor Jebelean},

title = {Knowledge Archives in Theorema: A Logic-Internal Approach},

year = {}

}

### OpenURL

### Abstract

Abstract. Archives are implemented as an extension of Theorema for representing mathematical repositories in a natural way. An archive can be conceived as one large formula in a language consisting of higher-order predicate logic together with a few constructs for structuring knowledge: attaching labels to subhierarchies, disambiguating symbols by the use of namespaces, importing symbols from other namespaces and specifying the domains of categories and functors as namespaces with variable operations. All these constructs are logic-internal in the sense that they have a natural translation to higher-order logic so that certain aspects of Mathematical Knowledge Management can be realized in the object logic itself. There are a variety of operations on archives, though in this paper we can only sketch a few of them: knowledge retrieval and theory exploration, merging and splitting, insertion and translation to predicate logic.

### Citations

62 |
Theorema: Towards computeraided mathematical theory exploration
- Buchberger, Craciun, et al.
(Show Context)
Citation Context ...-internal in the sense that there is a transparent translation to the underlying Theorema language. • As a part of Theorema, it emphasizes a natural style similar to that of the working mathematician =-=[13]-=-. The framework presented here should only be understood as a first step in the general direction outlined by these principles. The benefits of the logic-internal approach become apparent when one con... |

55 |
The Theorema project: A progress report
- Buchberger, Dupre, et al.
(Show Context)
Citation Context ...of archives is developed within the Theorema system, let us now describe some of its features that are particularly interesting from the viewpoint of MKM. For a general survey of Theorema we refer to =-=[13, 14]-=-. The overall goal of the Theorema system is to provide computer support for the working mathematician in his routine tasks, supporting the entire process of mathematical work. Some characteristic fea... |

51 | User interaction with the Matita proof assistant
- Asperti, Coen, et al.
- 2007
(Show Context)
Citation Context ...ignificant portions of structured mathematics. Most of these systems employ textual-based retrieval. A similar mechanism [4], based on HELM, is incorporated into the interactive theorem prover Matita =-=[3]-=-. On a somewhat different line, MathWebSearch [19] allows to search OMDoc documents up to α-conversion. Another idea, similar to archives in spirit but using a proof-theoretic approach to MKM, is pres... |

33 |
Algorithm Synthesis by Lazy Thinking: Examples and Implementation in Theorema. in: Fairouz Kamareddine (ed
- Buchberger, Craciun
(Show Context)
Citation Context ...ated by SearchFunction[arch,form], where the unknown functions in form are designated as for ArchiveMatch. As an example, consider the problem of synthesizing the merge-sort algorithm as described in =-=[12]-=-. At a certain stage one has to come up with two functions on tuples, f ? and g ? , such that the concatenation of f ? [X] and g ? [X] is a permuted version of X. Written in the formalism employed the... |

22 | Technical aspects of the digital library of mathematical functions
- Miller, Youssef
- 2003
(Show Context)
Citation Context ...of mathematics include HELM [15] and the mathematical databases formalized in OMDoc [18]. The ongoing efforts for reworking the theory of special functions into an online database in the DLMF project =-=[23]-=- are another instance of structuring a considerable portion of mathematics. Authoring tools like ActiveMath [22] also contain significant portions of structured mathematics. Most of these systems empl... |

20 | A content based mathematical search engine: Whelp
- Asperti, Guidi, et al.
- 2004
(Show Context)
Citation Context ...ble portion of mathematics. Authoring tools like ActiveMath [22] also contain significant portions of structured mathematics. Most of these systems employ textual-based retrieval. A similar mechanism =-=[4]-=-, based on HELM, is incorporated into the interactive theorem prover Matita [3]. On a somewhat different line, MathWebSearch [19] allows to search OMDoc documents up to α-conversion. Another idea, sim... |

18 | A search engine for mathematical formulae
- Kohlhase, Șucan
- 2006
(Show Context)
Citation Context ...st of these systems employ textual-based retrieval. A similar mechanism [4], based on HELM, is incorporated into the interactive theorem prover Matita [3]. On a somewhat different line, MathWebSearch =-=[19]-=- allows to search OMDoc documents up to α-conversion. Another idea, similar to archives in spirit but using a proof-theoretic approach to MKM, is presented in [1, 2]. The library of proofs, theorems a... |

11 | Theory Exploration Versus Theorem Proving - Buchberger |

11 |
The Mathematica Book, WolframMedia Inc
- Wolfram
- 1999
(Show Context)
Citation Context ...lename Algebra.nb in the home directory, one uses the command: archive = LoadArchive["Algebra.nb"] By this call, the box structure of the notebook "Algebra.nb" is parsed into a Mathematica expression =-=[30]-=-, subsequently stored in the variable archive. The underlying expression language is that of Theorema [14], extended by the language features presented in the preceding sections.20 Camelia Rosenkranz... |

10 | Theory exploration with theorema
- Buchberger
(Show Context)
Citation Context ... existing textbooks, articles, etc. In practice, both methods should be combined for optimal results. In the current implementation, theory exploration is provided as an experimental feature based on =-=[7]-=- and [13, §2]. We distinguish two types of exploration: • New propositions are searched by InventPropositions[arch,lib,sym]. • New concepts are created by InventConcepts[arch,lib,sym,new]. Here arch i... |

9 | An Environment for Building Mathematical Knowledge Libraries
- Piroi, Buchberger
(Show Context)
Citation Context ...e are discarded. One way to cope with this problem is to employ logic-external means for supplying the missing structure in the form of metadata [5], semantic annotations [20, 29], the label tools of =-=[25, 24]-=- or content dictionaries within OMDoc [18]. In this paper we would like to point out an alternative that has a very natural flavor: By a slight extension of the object language (the Theorema language,... |

9 | Mathematical Knowledge Management in Mizar
- Rudnicki, Trybulec
- 2001
(Show Context)
Citation Context ...informal mathematical practice are discarded. One way to cope with this problem is to employ logic-external means for supplying the missing structure in the form of metadata [5], semantic annotations =-=[20, 29]-=-, the label tools of [25, 24] or content dictionaries within OMDoc [18]. In this paper we would like to point out an alternative that has a very natural flavor: By a slight extension of the object lan... |

8 | Methods for Access and Retrieval of Mathematical Content in ActiveMath
- Libbrecht, Melis
- 2006
(Show Context)
Citation Context ...informal mathematical practice are discarded. One way to cope with this problem is to employ logic-external means for supplying the missing structure in the form of metadata [5], semantic annotations =-=[20, 29]-=-, the label tools of [25, 24] or content dictionaries within OMDoc [18]. In this paper we would like to point out an alternative that has a very natural flavor: By a slight extension of the object lan... |

8 |
Tools for Using Automated Provers in Mathematical Theory Exploration
- Piroi
- 2004
(Show Context)
Citation Context ...e are discarded. One way to cope with this problem is to employ logic-external means for supplying the missing structure in the form of metadata [5], semantic annotations [20, 29], the label tools of =-=[25, 24]-=- or content dictionaries within OMDoc [18]. In this paper we would like to point out an alternative that has a very natural flavor: By a slight extension of the object language (the Theorema language,... |

7 | A Proof-Theoretic Approach to Mathematical Knowledge Management
- Aboul-Hosn
- 2007
(Show Context)
Citation Context ...mewhat different line, MathWebSearch [19] allows to search OMDoc documents up to α-conversion. Another idea, similar to archives in spirit but using a proof-theoretic approach to MKM, is presented in =-=[1, 2]-=-. The library of proofs, theorems and tactics are integrated in the underlying proof logic, where scoping and tactics are represented in a typed λ-calculus. But the point of emphasis there seems to be... |

6 |
Algorithm Retrieval: Concept Clarification and Case Study in Theorema
- Buchberger
- 2003
(Show Context)
Citation Context ...rting archive arch2 into arch1 at the label position lab, one may use the command InsertArchive[arch1,arch2,lab]. 5.5. Retrieval in Archives We regard retrieval as an integral part of theorem proving =-=[9]-=-, typically employing a drastically reduced set of inference rules (“symbolic computation proving”) for obtaining the formula needed. Textual search is usually not a sufficient solution since this wil... |

6 |
Towards the Automated Synthesis of Groebner Bases Algorithm”, RACSAM, vol Falta
- Buchberger
(Show Context)
Citation Context ...versal variables in the assumptions of arch and for the metavariables in form.Knowledge Archives in Theorema 25 Consider the following example occurring in a synthesis of the Gröbner basis algorithm =-=[10]-=-. Omitting restriction predicates like is-power-product and is-polynomial for the sake of simplicity, the crucial point in the synthesis requires proving ∀ p ∀ g,h (lp[g] | p ∧ lp[h] | p ⇒ F ? [g, h] ... |

5 |
and the semantic MathWeb
- Asperti, Padovani, et al.
- 2001
(Show Context)
Citation Context ...rchical layers present in informal mathematical practice are discarded. One way to cope with this problem is to employ logic-external means for supplying the missing structure in the form of metadata =-=[5]-=-, semantic annotations [20, 29], the label tools of [25, 24] or content dictionaries within OMDoc [18]. In this paper we would like to point out an alternative that has a very natural flavor: By a sli... |

5 |
Groebner rings and modules
- Buchberger
- 2001
(Show Context)
Citation Context ...axioms characterizing them; we call the corresponding model class a category. Note that in this paperKnowledge Archives in Theorema 5 the notions of categories and functors are meant in the sense of =-=[11, 8]-=- and not necessarily in the sense of Eilenberg and MacLane (although there will often be a natural correspondence). We will show later how categories can be defined succintly in the language of archiv... |

5 |
Groebner bases in Theorema using functors
- Buchberger
- 2008
(Show Context)
Citation Context ...iew them essentially as propositional constants. Likewise namespaces can be given a natural interpretation within the logic, somewhat similar to the domains of categories and functors in the sense of =-=[11]-=-. Let us briefly review some other systems operating on structured knowledge repositories; see also [26]. The largest organized library of mathematics available isKnowledge Archives in Theorema 3 the... |

4 |
2006, OMDoc: An Open Markup Format for
- Kohlhase
(Show Context)
Citation Context ...blem is to employ logic-external means for supplying the missing structure in the form of metadata [5], semantic annotations [20, 29], the label tools of [25, 24] or content dictionaries within OMDoc =-=[18]-=-. In this paper we would like to point out an alternative that has a very natural flavor: By a slight extension of the object language (the Theorema language, a version of predicate logic), this probl... |

3 | A proof-theoretic approach to hierarchical math library organization
- Aboul-Hosn, Andersen
- 2005
(Show Context)
Citation Context ...mewhat different line, MathWebSearch [19] allows to search OMDoc documents up to α-conversion. Another idea, similar to archives in spirit but using a proof-theoretic approach to MKM, is presented in =-=[1, 2]-=-. The library of proofs, theorems and tactics are integrated in the underlying proof logic, where scoping and tactics are represented in a typed λ-calculus. But the point of emphasis there seems to be... |

3 | Knowledge Management of Formal Mathematics and Interactive Proving
- Coen
- 2004
(Show Context)
Citation Context ...edge repositories; see also [26]. The largest organized library of mathematics available isKnowledge Archives in Theorema 3 the MML library of Mizar [29]. Other libraries of mathematics include HELM =-=[15]-=- and the mathematical databases formalized in OMDoc [18]. The ongoing efforts for reworking the theory of special functions into an online database in the DLMF project [23] are another instance of str... |

3 | Towards Practical Reflection for Formal Mathematics, extended abstract
- Giese, Buchberger
- 2007
(Show Context)
Citation Context ...culus. But the point of emphasis there seems to be more on the migration between object and metalevels in mathematics. Similar ideas have also been studied in Theorema under the heading of reflection =-=[17]-=-. The structure of the paper is as follows. We start by summarizing various MKM-related features of Theorema (Section 2). The structure of archives is described in detail within the next two sections.... |

2 |
Lazy Thinking Algorithm Synthesis in Grŏebner Bases Theory. Research Institute for Symbolic Computation (RISC
- Crăciun
- 2008
(Show Context)
Citation Context ...ynthesizing the merge-sort algorithm as in the scenario “Logical Algorithm Retrieval = Algorithm Invention” of [9]. But a full integration of the synthesis method described in [12] and implemented in =-=[16]-=-, is beyond the scope of the current implementation. Using labels, theory exploration can also be carried out in a single archive big-archive. In this case, arch and lib are labels referring to suitab... |

2 | Mathematical Journals as Reasoning Agents: Literature Review
- Piroi, Buchberger, et al.
- 2008
(Show Context)
Citation Context ...on within the logic, somewhat similar to the domains of categories and functors in the sense of [11]. Let us briefly review some other systems operating on structured knowledge repositories; see also =-=[26]-=-. The largest organized library of mathematics available isKnowledge Archives in Theorema 3 the MML library of Mizar [29]. Other libraries of mathematics include HELM [15] and the mathematical databa... |

2 |
Organisational Tools for MKM in Theorema
- Piroi, Buchberger, et al.
- 2007
(Show Context)
Citation Context ...uples",FilterType → All]). 6. Applicative higher-order pattern matching: Search all formulae in BN:Tuples using ≍ left of an equality by FormulaLookup[ ≍ = ,"BN:Tuples"]. For more details we refer to =-=[24, 27]-=-. The language of archives is a continuation of the features (1)–(6), with labels now being logic-internal. Moreover, it provides namespaces as a suitable construct for structuring the constants occur... |

1 | Retrieval and Structuring of Large Mathematical Knowledge Bases: A Theorema Approach - Rosenkranz |