## A Multi-context Architecture for Formalizing Complex Reasoning (1995)

Venue: | INTERNATIONAL JOURNAL OF INTELLIGENT SYSTEMS |

Citations: | 8 - 0 self |

### BibTeX

@TECHREPORT{Giunchiglia95amulti-context,

author = {Enrico Giunchiglia and Paolo Traverso},

title = {A Multi-context Architecture for Formalizing Complex Reasoning},

institution = {INTERNATIONAL JOURNAL OF INTELLIGENT SYSTEMS},

year = {1995}

}

### Years of Citing Articles

### OpenURL

### Abstract

We propose multi-context systems (MC systems) as a formal framework for the specification of complex reasoning. MC systems provide the ability to structure the specification of "global" reasoning in terms of "local" reasoning sub-patterns. Each sub-pattern is modeled as a deduction in a context, formally defined as an axiomatic formal system. The global reasoning pattern is modeled as a concatenation of contextual deductions via bridge rules, i.e. inference rules that infer a fact in one context from facts asserted in other contexts. Besides the formal framework, in this paper we propose a three layer architecture designed to specify and automatize complex reasoning. At the first level we have object-level contexts (called s-contexts) for domain specifications. Problem solving principles and, more in general, meta-level knowledge about the application domain is specified in a distinct context, called Problem Solving Context (PSC). On top of s-contexts and PSC, we have a further context, called MT , where it is possible to specify strategies to control multi-context reasoning spanning through s-contexts and PSC. We show how GETFOL can be used as a computer tool for the implementation of MC systems and for the automatization of multi-context deductions.

### Citations

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Citation Context ...s to find some general conditions under which we have terminating strategies. A lot of work has been done on providing conditions for and on proving the termination of recursive definitions (e.g. see =-=[4]-=-). This issue is not discussed here as out of the goals of this paper. 30 be applied to global reasoning, consider a reasoning strategy that continuously switches reasoning among contexts, by repeated... |

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Citation Context ...other issue is whether the meta-theory has to be distinct or not from the object-level theories. Supposing the object-level domain is specified in a unique context, we can think about encoding (as in =-=[20, 23]-=-) or amalgamating (as in [3]) the object-level theory with its own meta-level specification. The same approach could, in principle, be accomplished even with multiple object-level theories: each of su... |

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Citation Context ...2.1 to be any first order language. Within the realm of first order theories, we further restrict to contexts being systems of Natural Deduction (ND systems) [28]. ND systems were first introduced in =-=[6] "to -=-set up a formal system in which deduction comes as close as possible to actual reasoning". This is the main motivation which makes them particularly attractive for the formalization of reasoning.... |

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Citation Context ...). Therefore, we choose L i in definition 2.1 to be any first order language. Within the realm of first order theories, we further restrict to contexts being systems of Natural Deduction (ND systems) =-=[28]. ND syste-=-ms were first introduced in [6] "to set up a formal system in which deduction comes as close as possible to actual reasoning". This is the main motivation which makes them particularly attra... |

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Citation Context ...r Systems. -- Pragmatical: "multi first order context systems" have shown to be expressive enough to represent all the forms of reasoning that we have considered so far (e.g. reasoning by ab=-=straction [18]-=-, meta-reasoning [14, 15, 19], reasoning about propositional attitudes [13], non monotonic reasoning [19], reasoning about the temporal evolution of physical systems [7], reasoning with dynamic assump... |

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Citation Context ... meta-sentence via reification of the proposition ff (e.g. 2("ff")). It can be shown that any modal theory is logically equivalent to a particular MC system where each context is a first ord=-=er theory [13]. In [33],-=- analogous equivalence results have been proven about the equivalence between default logic [29] and (first order) Bi-Modular Systems. -- Pragmatical: "multi first order context systems" hav... |

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Citation Context ...scribed in section 6). As a consequence, they can be executed. The execution mechanism of MT strategies is based on the simulation structure machinery and the attach mechanism first introduced in FOL =-=[36]-=- (see sections 6.1 and 6.2). This achieves obj-6: executability of reasoning strategies and automatization of reasoning. 6 MC systems in GETFOL MC systems have been implemented inside GETFOL [11]. GET... |

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Citation Context ...heory has to be distinct or not from the object-level theories. Supposing the object-level domain is specified in a unique context, we can think about encoding (as in [20, 23]) or amalgamating (as in =-=[3]-=-) the object-level theory with its own meta-level specification. The same approach could, in principle, be accomplished even with multiple object-level theories: each of such theories could contain a ... |

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Citation Context ...thin a small search space and a unique simple inference rule (e.g. resolution), proving a theorem automatically may turn out to be an hard or even unfeasible task for a general purpose theorem prover =-=[26]-=-. The basic idea underlying the Meta-Theory MT is to provide a framework where the user can specify its own strategies to guide the problem solving 10 activity to prove the desired facts in a MC syste... |

80 | A Comparison of Languages which Operationalise and Formalise KADS Models of Expertise. The Knowledge Engineering Review
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Citation Context ... Reflection principles formalize some of the interactions between modules. Logic based specifications of modular architectures are also provided by some languages which formalize the KADS methodology =-=[5]-=-. For instance, in (ML) 2 [1, 2], axioms are organized into a number of logical sub-theories which are then combined by using meta-theoretic operators. All of these systems and approaches share with M... |

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Citation Context ...ons for this are the same explained for PSC plus four further facts. First, a meta-theory distinguished from PSC can describe reasoning within PSC and avoid selfreferential problems (see for instance =-=[27, 31]-=-). Second, a distinguished meta-theory can explicitly represent the interactions (through bridge rules) among PSC and s-contexts. Third, PSC language is not rich enough to represent deductions. In PSC... |

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34 |
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Citation Context ...s of modular architectures, where "local" reasoning sub-patterns are specified as logical deductions, and interaction among modules are formalized (mainly by the use of reflection principles=-=). MILORD [21]-=- (and its descendant MILORD-II [30]) provides hierarchical structures of modules, where, within each module, an object-level logical theory and a meta-level first order theory interact through a reifi... |

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Citation Context ...ons for this are the same explained for PSC plus four further facts. First, a meta-theory distinguished from PSC can describe reasoning within PSC and avoid selfreferential problems (see for instance =-=[27, 31]-=-). Second, a distinguished meta-theory can explicitly represent the interactions (through bridge rules) among PSC and s-contexts. Third, PSC language is not rich enough to represent deductions. In PSC... |

25 |
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Citation Context ...ning within and among the other contexts (PSC and s-contexts). The proposed architecture has been implemented within GETFOL, an interactive reasoning system that supports the "multi-context parad=-=igm" [11]-=-. The paper is structured as follows. MC systems are described in section 2. The three level architecture is presented in section 3. The problem solving context PSC and the meta-theory MT are describe... |

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(Show Context)
Citation Context ...ical: "multi first order context systems" have shown to be expressive enough to represent all the forms of reasoning that we have considered so far (e.g. reasoning by abstraction [18], meta-=-=reasoning [14, 15, 19]-=-, reasoning about propositional attitudes [13], non monotonic reasoning [19], reasoning about the temporal evolution of physical systems [7], reasoning with dynamic assumptions [9]). Therefore, we cho... |

23 | A metatheory of a mechanized object theory
- Giunchiglia, Traverso
- 1996
(Show Context)
Citation Context ...ification of reasoning strategies. A first version of the meta-theory MT and of its goals has been introduced in [15]. Its formal definition and the proofs of its formal properties have been given in =-=[16]-=-. Some extensions of MT to represent reasoning strategies are described in [17]. In all these papers, MT has been defined as a meta-theory of a single object-level theory. In this paper, we take these... |

22 | R.: A Multi-Context Monotonic Axiomatizations of Inessential Non-Monotonicity
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Citation Context ...ical: "multi first order context systems" have shown to be expressive enough to represent all the forms of reasoning that we have considered so far (e.g. reasoning by abstraction [18], meta-=-=reasoning [14, 15, 19]-=-, reasoning about propositional attitudes [13], non monotonic reasoning [19], reasoning about the temporal evolution of physical systems [7], reasoning with dynamic assumptions [9]). Therefore, we cho... |

19 | Multi-context Systems as a Specification framework for Complex Reasoning Systems
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Citation Context ...-reasoning [14, 15, 19], reasoning about propositional attitudes [13], non monotonic reasoning [19], reasoning about the temporal evolution of physical systems [7], reasoning with dynamic assumptions =-=[9]-=-). Therefore, we choose L i in definition 2.1 to be any first order language. Within the realm of first order theories, we further restrict to contexts being systems of Natural Deduction (ND systems) ... |

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17 |
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Citation Context ...ovides hierarchical structures of modules, where, within each module, an object-level logical theory and a meta-level first order theory interact through a reification/reflection mechanism. In DESIRE =-=[24, 32]-=-, inference processes are described by means of modules which consist of propositional logical theories, where the models of the theories are represented explicitly in the system. Reflection principle... |

14 | Hierarchical Meta-Logics: Intuitions, Proof Theory and Semantics
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Citation Context ...ical: "multi first order context systems" have shown to be expressive enough to represent all the forms of reasoning that we have considered so far (e.g. reasoning by abstraction [18], meta-=-=reasoning [14, 15, 19]-=-, reasoning about propositional attitudes [13], non monotonic reasoning [19], reasoning about the temporal evolution of physical systems [7], reasoning with dynamic assumptions [9]). Therefore, we cho... |

13 | Multilanguage systems
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- 1991
(Show Context)
Citation Context ... the proposed architecture are implemented in GETFOL. Finally, in sections 7 and 8 we have the related work and the conclusions. 2 MC systems MC systems have been motivated and formally introduced in =-=[10, 12]-=-. Further investigations and applications have been given in [8]. They are also called multilanguage systems (ML systems) to emphasize the fact that they allow for the definition of multiple languages... |

13 |
editors. Formal Specification of Complex Reasoning Systems
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Citation Context ... where the information specified is not sufficient to solve the goal and we need to test the effects of some further (plausible) assumption. An extensive use of this kind of reasoning can be found in =-=[34]. As a par-=-ticular case of such a kind of reasoning, we may state contradictory facts about an s-context c. In this case, the s-context c turns out to be "inconsistent". Consider for instance the follo... |

11 |
A bi-modular approach to nonmonotonic reasoning
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(Show Context)
Citation Context ...tence via reification of the proposition ff (e.g. 2("ff")). It can be shown that any modal theory is logically equivalent to a particular MC system where each context is a first order theory=-= [13]. In [33], analogou-=-s equivalence results have been proven about the equivalence between default logic [29] and (first order) Bi-Modular Systems. -- Pragmatical: "multi first order context systems" have shown t... |

8 | A KADS/(ML) model of a scheduling task
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(Show Context)
Citation Context ...ize some of the interactions between modules. Logic based specifications of modular architectures are also provided by some languages which formalize the KADS methodology [5]. For instance, in (ML) 2 =-=[1, 2]-=-, axioms are organized into a number of logical sub-theories which are then combined by using meta-theoretic operators. All of these systems and approaches share with MS the ability to support formal ... |

8 | Specifying simple scheduling tasks in a reflective and modular architecture
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(Show Context)
Citation Context ... c and the injectivity of the mkcxt function) it is possible to have :TH("w"; "c") if w is not provable in c (17) as part of the set of inference rules of PSC. A similar approach i=-=s used in MILORD-II [30]-=-. Finally, we want to point out that also role limiting methods can be provided in PSC 6 . For instance, consider axiom PSC1. This axiom does not limit the role of the object-level implication denoted... |

7 |
A formalization of knowledge-level models for knowledge acquisition
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(Show Context)
Citation Context ...ize some of the interactions between modules. Logic based specifications of modular architectures are also provided by some languages which formalize the KADS methodology [5]. For instance, in (ML) 2 =-=[1, 2]-=-, axioms are organized into a number of logical sub-theories which are then combined by using meta-theoretic operators. All of these systems and approaches share with MS the ability to support formal ... |

5 |
Formalized Metatheory in Type Theory
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(Show Context)
Citation Context ...other issue is whether the meta-theory has to be distinct or not from the object-level theories. Supposing the object-level domain is specified in a unique context, we can think about encoding (as in =-=[20, 23]-=-) or amalgamating (as in [3]) the object-level theory with its own meta-level specification. The same approach could, in principle, be accomplished even with multiple object-level theories: each of su... |

4 |
Towards a logical treatment of qualitative reasoning
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(Show Context)
Citation Context ...e.g. reasoning by abstraction [18], meta-reasoning [14, 15, 19], reasoning about propositional attitudes [13], non monotonic reasoning [19], reasoning about the temporal evolution of physical systems =-=[7]-=-, reasoning with dynamic assumptions [9]). Therefore, we choose L i in definition 2.1 to be any first order language. Within the realm of first order theories, we further restrict to contexts being sy... |

2 |
Sistemi Multi-Contestuali: Teoria ed Applicazioni
- Giunchiglia
- 1993
(Show Context)
Citation Context ...tions 7 and 8 we have the related work and the conclusions. 2 MC systems MC systems have been motivated and formally introduced in [10, 12]. Further investigations and applications have been given in =-=[8]-=-. They are also called multilanguage systems (ML systems) to emphasize the fact that they allow for the definition of multiple languages, each language associated with a context. Here we show how the ... |

2 |
A Compositional Architecture for Simple Design Formally Specified in DESIRE
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- 1993
(Show Context)
Citation Context ...ovides hierarchical structures of modules, where, within each module, an object-level logical theory and a meta-level first order theory interact through a reification/reflection mechanism. In DESIRE =-=[24, 32]-=-, inference processes are described by means of modules which consist of propositional logical theories, where the models of the theories are represented explicitly in the system. Reflection principle... |

2 |
Improving the efficiency of Metalevel Reasoning
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Citation Context ...l axioms and thus they do not need to be actually implemented within the GETFOL code. The former solution gives some efficiency advantages, since it allows us to avoid any form of meta-level overhead =-=[35], but requ-=-ires some programming. On the other hand, meta-level axiomatic specifications provide important advantages. The GETFOL user can specify and "simulate" rules as axioms in the first order lang... |

1 | Multi-context systems as a tool to model temporal evolution
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- 1993
(Show Context)
Citation Context ...e context c (i.e. ` MS hw; ci). These two requirements are achieved via the two bridge rules, called Reflection Down and Reflection Up [13], given below: 1 "w" might be the quotation mark na=-=me (as in [25]) or the structural descr-=-iptive name (as in [7]) of the formula w. The only requirement on naming is that if "w 1 " and "w 2 " are identical constants, then w 1 and w 2 are identical formulas. 9 hTH("... |