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Logic of subtyping
 Theoretical Computer Science
, 2005
"... We introduce new modal logical calculi that describe subtyping properties of Cartesian product and disjoint union type constructors as well as mutuallyrecursive types defined using those type constructors. Basic Logic of Subtyping S extends classical propositional logic by two new binary modalities ..."
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We introduce new modal logical calculi that describe subtyping properties of Cartesian product and disjoint union type constructors as well as mutuallyrecursive types defined using those type constructors. Basic Logic of Subtyping S extends classical propositional logic by two new binary modalities ⊗ and ⊕. An interpretation of S is a function that maps standard connectives into settheoretical operations (intersection, union, and complement) and modalities into Cartesian product and disjoint union type constructors. This allows S to capture many subtyping properties of the above type constructors. We also consider logics Sρ and S ω ρ that incorporate into S mutuallyrecursive types over arbitrary and wellfounded universes correspondingly. The main results are completeness of the above three logics with respect to appropriate type universes. In addition, we prove Cut elimination theorem for S and establish decidability of S and S ω ρ.
Combining Conjunction with Disjunction
 Proceedings of the 2nd Indian International Conference on Artificial Intelligence (IICAI 2005
, 2005
"... Abstract. In this paper we address some central problems of combination of logics through the study of a very simple but highly informative case, the combination of the logics of disjunction and conjunction. At first it seems that it would be very easy to combine such logics, but the following probl ..."
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Abstract. In this paper we address some central problems of combination of logics through the study of a very simple but highly informative case, the combination of the logics of disjunction and conjunction. At first it seems that it would be very easy to combine such logics, but the following problem arises: if we combine these logics in a straightforward way, distributivity holds. On the other hand, distributivity does not arise if we use the usual notion of extension between consequence relations. A detailed discussion about this phenomenon, as well as some possible solutions for it, are given. 1
Why sets?
 PILLARS OF COMPUTER SCIENCE: ESSAYS DEDICATED TO BORIS (BOAZ) TRAKHTENBROT ON THE OCCASION OF HIS 85TH BIRTHDAY, VOLUME 4800 OF LECTURE NOTES IN COMPUTER SCIENCE
, 2008
"... Sets play a key role in foundations of mathematics. Why? To what extent is it an accident of history? Imagine that you have a chance to talk to mathematicians from a faraway planet. Would their mathematics be setbased? What are the alternatives to the settheoretic foundation of mathematics? Besi ..."
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Sets play a key role in foundations of mathematics. Why? To what extent is it an accident of history? Imagine that you have a chance to talk to mathematicians from a faraway planet. Would their mathematics be setbased? What are the alternatives to the settheoretic foundation of mathematics? Besides, set theory seems to play a significant role in computer science; is there a good justification for that? We discuss these and some related issues.
Inheritance Reasoning and HeadDriven Phrase Structure Grammar
, 1990
"... Inheritance networks are a type of semantic network which represent both strict (classical implication) and defeasible (nonclassical) relationships among entities. We present an established approach to defeasible reasoning which defines inference in terms of the construction of paths through a netw ..."
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Inheritance networks are a type of semantic network which represent both strict (classical implication) and defeasible (nonclassical) relationships among entities. We present an established approach to defeasible reasoning which defines inference in terms of the construction of paths through a network. Much of literature on inheritance is concerned with specifying the most "intuitive" system of path construction. However, when considering a fundamental feature of these approachesthe status accorded to redundant linkswe find that topological considerations espoused in the literature are insufficient for determining the valid inferences of a network. This implies that the "intuitiveness" of a particular method depends upon the domain being represented. Though Touretzky has demonstrated that it is unsound in some cases, the pathpreference algorithm known as shortest path reasoning, is actually the most intuitive algorithm to use when reasoning about the inheritance network which r...
Recognizing plans with loops represented in a lexicalized grammar
 In The twenty fifth AAAI Conference on Artificial Intelligence (AAAI
, 2011
"... This paper extends existing plan recognition research to handle plans containing loops. We supply an encoding of plans with loops for recognition, based on techniques used to parse lexicalized grammars, and demonstrate its effectiveness empirically. To do this, the paper first shows how encoding p ..."
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This paper extends existing plan recognition research to handle plans containing loops. We supply an encoding of plans with loops for recognition, based on techniques used to parse lexicalized grammars, and demonstrate its effectiveness empirically. To do this, the paper first shows how encoding plan libraries as context free grammars permits the application of standard rewriting techniques to remove left recursion and productions, thereby enabling polynomial time parsing. However, these techniques alone fail to provide efficient algorithms for plan recognition. We show how the loophandling methods from formal grammars can be extended to the more general plan recognition problem and provide a method for encoding loops in an existing plan recognition system that scales linearly in the number of loop iterations.
On modal logics of partial recursive functions
 Studia Logica
"... The classical propositional logic is known to be sound and complete with respect to the set semantics that interprets connectives as set operations. The paper extends propositional language by a new binary modality that corresponds to partial recursive function type constructor under the above inter ..."
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The classical propositional logic is known to be sound and complete with respect to the set semantics that interprets connectives as set operations. The paper extends propositional language by a new binary modality that corresponds to partial recursive function type constructor under the above interpretation. The cases of deterministic and nondeterministic functions are considered and for both of them semantically complete modal logics are described and decidability of these logics is established.
Variants of the Basic Calculus of Constructions
, 2004
"... In this paper, a number of different versions of the basic calculus of constructions that have appeared in the literature are compared and the exact relationships between them are determined. The biggest differences between versions are those between the original version of Coquand and the version i ..."
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In this paper, a number of different versions of the basic calculus of constructions that have appeared in the literature are compared and the exact relationships between them are determined. The biggest differences between versions are those between the original version of Coquand and the version in early papers on the subject by Seldin. None of these results is very deep, but it seems useful to collect them in one place.
Transfer of Sequent Calculus Strategies to Resolution for S4
 IN PROOF THEORY OF MODAL LOGIC, STUDIES IN PURE AND APPLIED LOGIC 2
, 1996
"... This paper illustrates for the propositional S4 a general scheme of transferring strategies of proof search in a cutfree Gentzentype system into a resolution type system, preserving the structure of derivations. This is a direct extension of the method introduced by Maslov for classical predica ..."
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This paper illustrates for the propositional S4 a general scheme of transferring strategies of proof search in a cutfree Gentzentype system into a resolution type system, preserving the structure of derivations. This is a direct extension of the method introduced by Maslov for classical predicate logic.
On Lukasiewicz's fourvalued modal logic
, 2000
"... . # Lukasiewicz's fourvalued modal logic is surveyed and analyzed, together with # Lukasiewicz's motivations to develop it. A faithful interpretation of it into classical (nonmodal) twovalued logic is presented, and some consequences are drawn concerning its classification and its algeb ..."
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. # Lukasiewicz's fourvalued modal logic is surveyed and analyzed, together with # Lukasiewicz's motivations to develop it. A faithful interpretation of it into classical (nonmodal) twovalued logic is presented, and some consequences are drawn concerning its classification and its algebraic behaviour. Some counterintuitive aspects of this logic are discussed under the light of the presented results, # Lukasiewicz's own texts, and related literature. 1 Introduction The Polish philosopher and logician Jan # Lukasiewicz (Lwow, 1878  Dublin, 1956) is one of the fathers of modern manyvalued logic, and some of the systems he introduced are presently a topic of deep investigation. In particular his infinitelyvalued logic belongs to the core systems of mathematical fuzzy logic as a logic of comparative truth, see [5, 15, 14, 16]. However, it must be stressed here that # Lukasiewicz's logical work bears also a special relationship to modal logic. Actually, modal notions were part of #...
Basic Relevant Theories for Combinators at Levels I and II Koushik Pal
, 2005
"... Abstract: The system B+ is the minimal positive relevant logic. B+ is trivially extended to B+T on adding a greatest truth (Church constant) T. If we leave ∨ out of the formation apparatus, we get the fragment B∧T. It is known that the set of all B∧T theories provides a good model for the combinator ..."
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Abstract: The system B+ is the minimal positive relevant logic. B+ is trivially extended to B+T on adding a greatest truth (Church constant) T. If we leave ∨ out of the formation apparatus, we get the fragment B∧T. It is known that the set of all B∧T theories provides a good model for the combinators CL at LevelI, which is the theory level. Restoring ∨ to get back B+T was not previously fruitful at LevelI, because the set of all B+T theories is not a model of CL. It was to be expected from semantic completeness arguments for relevant logics that basic combinator laws would hold when restricted to prime B+T theories. Overcoming some previous difficulties, we show that this is the case, at Level I. But this does not form a model for CL. This paper also looks for corresponding results at LevelII, where we deal with sets of theories that we call propositions. We adapt work by Ghilezan to note that at LevelII also there is a model of CL in B∧T propositions. However, the corresponding result for B+T propositions extends smoothly to LevelII only in part. Specifically, only