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Structural Cut Elimination  I. Intuitionistic and Classical Logic
 Information and Computation
, 2000
"... this paper we present new proofs of cut elimination for intuitionistic and classical sequent calculi and give their representations in the logical framework LF [HHP93] as implemented in the Elf system [Pfe91]. Multisets are avoided altogether in these proofs, and termination measures are replaced b ..."
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Cited by 53 (17 self)
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this paper we present new proofs of cut elimination for intuitionistic and classical sequent calculi and give their representations in the logical framework LF [HHP93] as implemented in the Elf system [Pfe91]. Multisets are avoided altogether in these proofs, and termination measures are replaced by three nested structural inductions. Parameters are treated as variables bound in derivations, thus naturally capturing occurrence conditions. A starting point for the proofs is Kleene's sequent system G 3 [Kle52], which we derive systematically from the point of view that a sequent calculus should be a calculus of proof search for natural deductions. It can easily be related to Gentzen's original and other sequent calculi. We augment G 3 with proof terms that are stable under weakening. These proof terms enable the structural induction and furthermore form the basis of the representation of the proof in LF. The most closely related work on cut elimination is MartinLo# f 's proof of admissibility [ML68]. In MartinLo# f 's system the cut rule incorporates aspects of both weakening and contraction which enables a structural induction argument closely related to ours. However, without the introduction of proof terms, the implicit weakening in the cut rule makes it difficult to implement this proof directly. Herbelin [Her95] restates this proof and proceeds by assigning proof terms only to restricted sequent calculi LJT and LKT which correspond more immediately to
Modality in Dialogue: Planning, Pragmatics and Computation
, 1998
"... Natural language generation (NLG) is first and foremost a reasoning task. In this reasoning, a system plans a communicative act that will signal key facts about the domain to the hearer. In generating action descriptions, this reasoning draws on characterizations both of the causal properties of the ..."
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Cited by 36 (9 self)
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Natural language generation (NLG) is first and foremost a reasoning task. In this reasoning, a system plans a communicative act that will signal key facts about the domain to the hearer. In generating action descriptions, this reasoning draws on characterizations both of the causal properties of the domain and the states of knowledge of the participants in the conversation. This dissertation shows how such characterizations can be specified declaratively and accessed efficiently in NLG. The heart of this dissertation is a study of logical statements about knowledge and action in modal logic. By investigating the prooftheory of modal logic from a logic programming point of view, I show how many kinds of modal statements can be seen as straightforward instructions for computationally manageable search, just as Prolog clauses can. These modal statements provide sufficient expressive resources for an NLG system to represent the effects of actions in the world or to model an addressee whose knowledge in some respects exceeds and in other respects falls short of its own. To illustrate the use of such statements, I describe how the SPUD sentence planner exploits a modal knowledge base to
Representing Communicative Intentions in Collaborative Conversational Agents
 IN AAAI FALL SYMPOSIUM ON INTENT INFERENCE FOR COLLABORATIVE TASKS
, 2001
"... This paper pursues a formal analogy between natural language dialogue and collaborative realworld action in general. The analogy depends on ..."
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Cited by 6 (2 self)
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This paper pursues a formal analogy between natural language dialogue and collaborative realworld action in general. The analogy depends on
Representing Scope in Intuitionistic Deductions
 THEORETICAL COMPUTER SCIENCE
, 1997
"... Intuitionistic proofs can be segmented into scopes which describe when assumptions can be used. In standard descriptions of intuitionistic logic, these scopes occupy contiguous regions of proofs. This leads to an explosion in the search space for automated deduction, because of the difficulty of pla ..."
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Cited by 4 (3 self)
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Intuitionistic proofs can be segmented into scopes which describe when assumptions can be used. In standard descriptions of intuitionistic logic, these scopes occupy contiguous regions of proofs. This leads to an explosion in the search space for automated deduction, because of the difficulty of planning to apply a rule inside a particular scoped region of the proof. This paper investigates an alternative representation which assigns scope explicitly to formulas, and which is inspired in part by semanticsbased translation methods for modal deduction. This calculus is simple and is justified by direct prooftheoretic arguments that transform proofs in the calculus so that scopes match standard descriptions. A Herbrand theorem, established straightforwardly, lifts this calculus to incorporate unification. The resulting system has no impermutabilities whatsoeverrules of inference may be used equivalently anywhere in the proof. Nevertheless, a natural specification describes how terms...
Firstorder MultiModal Deduction
"... This report aims to help provide such links by providing a set of extremely general results about firstorder multimodal deduction in terms of analytic tableaux and a prefix representation of possible worlds. We first provide sound and complete ground tableau and sequent inference systems, extendin ..."
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Cited by 1 (1 self)
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This report aims to help provide such links by providing a set of extremely general results about firstorder multimodal deduction in terms of analytic tableaux and a prefix representation of possible worlds. We first provide sound and complete ground tableau and sequent inference systems, extending and refining those presented in [Fitting and Mendelsohn, 1998] to the multimodal case. Then we show how to apply general prooftheoretic techniques to derive an equivalent calculus where Herbrand terms streamline proof search [Lincoln and Shankar, 1994]. Finally, we derive a lifted multimodal sequent inference system, which uses unification (or constraintsatisfaction) to resolve the values of variables, in the style of [Voronkov, 1996]. From one point of view, this report can be regarded as the multimodal generalization of the results presented for linear logic and firstorder modal logic in [Lincoln and Shankar, 1994, Fitting, 1996, Fitting and Mendelsohn, 1998]; alternatively, it can be seen as recasting into a modal setting the results of [Stone, 1999b], which investigates firstorder intuitionistic logic along similar lines. Formal modal logic goes back eighty years [Lewis, 1918, Lewis and Langford, 1932]. Yet according to McCarthy [McCarthy, 1997], for example, the modal logic literature still does not offer a formalism with the intensional expressive powerincluding fresh modalities defined ad hoc,and means to describe knowing what by concise and easily manipulated formulasthat is needed for knowledge representation in Artificial Intelligence. Moreover, typical results from the modal logic literature do not support the design of specialized modal inference mechanisms to solve particular knowledge representation tasks. The approach adopted here is a response t...