Most automated theorem provers suffer from the problem that they can produce proofs only in formalisms difficult to understand even for experienced mathematicians. Effort has been made to reconstruct natural deduction (ND) proofs from such machine generated proofs. Although the single steps in ND proofs are easy to understand, the entire proof is usually at a low level of abstraction, containing too many tedious steps. To obtain proofs similar to those found in mathematical textbooks, we propose a new formalism, called ND style proofs at the assertion level , where derivations are mostly justified by the application of a definition or a theorem. After characterizing the structure of compound ND proof segments allowing assertion level justification, we show that the same derivations can be achieved by domain-specific inference rules as well. Furthermore, these rules can be represented compactly in a tre structure. Finally, we describe a system called PROVERB , which substantially sh...

This paper describes the linguistic part of a system called PROVERB, which transforms, abstracts, and verbalizes machine-found proofs into formated texts. Linguistically, the architecture of PROVERB follows most application oriented systems, and is a pipe-lined control of three components. Its macroplanner linearizes a proof and plans mediating communicative acts by employing a combination of hierarchical planning and focus-guided navigation. The microplanner

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This paper outlines an implemented system called PROVERB that explains machine-found natural deduction proofs in natural language. Different from earlier works, we pursue a reconstructive approach. Based on the observation that natural deduction proofs are at a too low level of abstraction compared with proofs found in mathematical textbooks, we define first the concept of socalled assertion level inference rules. Derivations justified by these rules can intuitively be understood as the application of a definition or a theorem. Then an algorithm is introduced that abstracts machine-found ND proofs using the assertion level inference rules. Abstracted proofs are then verbalized into natural language by a presentation module. The most significant feature of the presentation module is that it combines standard hierarchical text planning and techniques that locally organize argumentative texts based on the derivation relation under the guidance of a focus mechanism. The behavior of the s...

This paper deals with the reference choices involved in the generation of argumentative text. Since a natual segmentation of discourse into attentional spaces is needed to carry out this task, this paper first proposes an architecture for natural language generation that combines hierarchical planning and focus-guided navigation, a work in its own right. While hierarchical planning spans out an attentional hierarchy of the discourse produced, local navigation fills details into the primitive discourse spaces. The usefulness of this architecture actually goes beyond the particular domain of application for which it is developed. A piece of argumentative text such as the proof of a mathematical theorem conveys a sequence of derivations. For each step of derivation, the premises derived in the previous context and the inference method (such as the application of a particular theorem or definition) must be made clear. Although not restricted to nominal phrases, our reference decisions are similar to those concerning nominal subsequent referring expressions. Based on the work of Reichmann, this paper presents a discourse theory that handles reference choices by taking into account both textual distance as well as the attentional hierarchy.