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Reconstructing Proofs at the Assertion Level
, 1994
"... 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 pr ..."
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Cited by 69 (9 self)
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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 domainspecific 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...
Proof Verbalization as an Application of NLG
 PROCEEDINGS OF THE 15TH INTERNATIONAL JOINT CONFERENCE ON ARTIFICIAL INTELLIGENCE (IJCAI
, 1997
"... This paper describes the linguistic part of a system called PROVERB, which transforms, abstracts, and verbalizes machinefound proofs into formated texts. Linguistically, the architecture of PROVERB follows most application oriented systems, and is a pipelined control of three components. Its ..."
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Cited by 45 (10 self)
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This paper describes the linguistic part of a system called PROVERB, which transforms, abstracts, and verbalizes machinefound proofs into formated texts. Linguistically, the architecture of PROVERB follows most application oriented systems, and is a pipelined control of three components. Its macroplanner linearizes a proof and plans mediating communicative acts by employing a combination of hierarchical planning and focusguided navigation. The microplanner
PROVERB  A System Explaining MachineFound Proofs
 IN PROC. OF 16TH ANNUAL CONFERENCE OF THE COGNITIVE SCIENCE SOCIETY
, 1994
"... This paper outlines an implemented system called PROVERB that explains machinefound 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 ..."
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Cited by 10 (3 self)
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This paper outlines an implemented system called PROVERB that explains machinefound 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 machinefound 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...
Planning Reference Choices for Argumentative Texts
"... 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 planni ..."
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Cited by 10 (2 self)
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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 focusguided 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.
Expressing References to Rules in Proof Presentations
"... Presenting machinegenerated proofs in natural language has deserved considerable attention over the past decade. Despite respectable progress achieved, the produced texts appear stilted in several places and are less coherent than comparable textbook presentations. One of the reasons for these shor ..."
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Cited by 4 (2 self)
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Presenting machinegenerated proofs in natural language has deserved considerable attention over the past decade. Despite respectable progress achieved, the produced texts appear stilted in several places and are less coherent than comparable textbook presentations. One of the reasons for these shortcomings is the low degree of variation in which references to the pieces of knowledge underlying a proof, theorems and axioms, are made. Addressing this deficit, we present a repertoire of descriptions of inference rules in proof presentations that indicate accurately how the underlying piece of generic knowledge is used in an involved inference. Generated expressions refer to conceptualizations of a rule's components, including partial instantiations, the role of assertions in inferencing, and focus emphasizing structural hints. Quite interestingly, some proof justifications are altered opportunistcally to enable the use of referring expressions that make the text fluent. The produced texts make reasoning lines much easier to follow, which is particularly valuable for texts serving didactic purposes.
Argumentation within . . .
, 2002
"... Deductive reasoning is an area related to argumentation where machinebased techniques, notably theorem proving, can contribute substantially to the formation of arguments. However, making use of the functionality of theorem provers for this issue is associated The problem of obtaining a natur ..."
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Deductive reasoning is an area related to argumentation where machinebased techniques, notably theorem proving, can contribute substantially to the formation of arguments. However, making use of the functionality of theorem provers for this issue is associated The problem of obtaining a natural language proof from a machinewith a number of difficulties and, as we will demonstrate, requires found proof can be divided into two subproblems: First, the proof considerable effort for obtaining reasonable results. Aiming at the is transformed from its original machineoriented formalism into a exploitation of machineoriented reasoning for humanadequate ar humanoriented calculus, which is much better suited for presentagumentation in a broader sense, we present our model for producing tion. Second, the transformed proof is verbalized in natural language. proof presentations from machineoriented inference structures. CaSince the lines of reasoning in machineoriented calculi are ofpabilities of the model include adaptation to humanadequate degrees ten unnatural and obscure, algorithms (see, e.g., [1, 18]) have been of granularitiy and explicitness in the underlying argumentation and developed to transform machinefound proofs into more natural forinteractive exploration of proofs. Enhancing capabilities in all these malisms, such as the natural deduction (ND) calculus [8]. ND inferrespects, even just those we have addressed so far, does not only imence steps consist of a small set of simple reasoning patterns, such as prove the interactive use of theorem provers, but they are essential in forallelimination leads) and implication eliminagredients
The Presentation of Proofs at the Assertion Level
"... Most automated theorem provers suffer from the problem that they can produce proofs only in formalisms difficult to understand even for experienced mathematicians. Efforts have been made to transform such machine generated proofs into natural deduction (ND) proofs. Although the single steps are now ..."
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Most automated theorem provers suffer from the problem that they can produce proofs only in formalisms difficult to understand even for experienced mathematicians. Efforts have been made to transform such machine generated proofs into natural deduction (ND) proofs. Although the single steps are now easy to understand, the entire proof is usually at a low level of abstraction, containing too many tedious steps. Therefore, it is not adequate as input to natural language generation systems. To overcome these problems, we propose a new intermediate representation, called ND style proofs at the assertion level . After illustrating the notion intuitively, we show that the assertion level steps can be justified by domainspecific inference rules, and that these rules can be represented compactly in a tree structure. Finally, we describe a procedure which substantially shortens ND proofs by abstracting them to the assertion level, and report our experience with further transformation into natu...
Extracting Text from Proofs
 In Typed Lambda Calculus and its Applications
, 1995
"... : In this paper, we propose a method for presenting formal proofs in an intelligible form. We describe a transducer from proof objects (terms in the Calculus of Constructions) to pseudo natural language that has been implemented for the Coq system Keywords: Proof Explanation, Natural Deduction, C ..."
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: In this paper, we propose a method for presenting formal proofs in an intelligible form. We describe a transducer from proof objects (terms in the Calculus of Constructions) to pseudo natural language that has been implemented for the Coq system Keywords: Proof Explanation, Natural Deduction, Calculus of Constructions. (R'esum'e : tsvp) Unite de recherche INRIA SophiaAntipolis 2004 route des Lucioles, BP 93, 06902 SOPHIAANTIPOLIS Cedex (France) Telephone : (33) 93 65 77 77  Telecopie : (33) 93 65 77 Extraction de Texte `a partir de Preuves R'esum'e : Ce papier pr'esente une m'ethode pour produire `a partir de preuves formelles une explication textuelle compr'ehensible. Nous d'ecrivons un traducteur d'un objet preuve (terme du Calcul des Constructions) vers un pseudo langage naturel qui a 'et'e implant'e dans le syst`eme Coq. Motscl'e : Explication de Preuves, D'eduction Naturelle, Calcul des Constructions. Extracting Text from Proofs 3 1 Introduction Almost all comput...