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An introduction to substructural logics
, 2000
"... Abstract: This is a history of relevant and substructural logics, written for the Handbook of the History and Philosophy of Logic, edited by Dov Gabbay and John Woods. 1 1 ..."
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Cited by 138 (16 self)
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Abstract: This is a history of relevant and substructural logics, written for the Handbook of the History and Philosophy of Logic, edited by Dov Gabbay and John Woods. 1 1
The Family of Stable Models
, 1993
"... The family of all stable models for a logic program has a surprisingly simple overall structure, once two naturally occurring orderings are made explicit. In a socalled knowledge ordering based on degree of definedness, every logic program P has a smallest stable model, s k P it is the well ..."
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Cited by 54 (4 self)
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The family of all stable models for a logic program has a surprisingly simple overall structure, once two naturally occurring orderings are made explicit. In a socalled knowledge ordering based on degree of definedness, every logic program P has a smallest stable model, s k P it is the wellfounded model. There is also a dual largest stable model, S k P , which has not been considered before. There is another ordering based on degree of truth. Taking the meet and the join, in the truth ordering, of the two extreme stable models s k P and S k P just mentioned, yields the alternating fixed points of [29], denoted s t P and S t P here. From s t P and S t P in turn, s k P and S k P can be produced again, using the meet and join of the knowledge ordering. All stable models are bounded by these four valuations. Further, the methods of proof apply not just to logic programs considered classically, but to logic programs over any bilattice meeting certain co...
A Nonstandard Approach to the Logical Omniscience Problem
 Artificial Intelligence
, 1990
"... We introduce a new approach to dealing with the wellknown logical omniscience problem in epistemic logic. Instead of taking possible worlds where each world is a model of classical propositional logic, we take possible worlds which are models of a nonstandard propositional logic we call NPL, which ..."
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Cited by 50 (4 self)
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We introduce a new approach to dealing with the wellknown logical omniscience problem in epistemic logic. Instead of taking possible worlds where each world is a model of classical propositional logic, we take possible worlds which are models of a nonstandard propositional logic we call NPL, which is somewhat related to relevance logic. This approach gives new insights into the logic of implicit and explicit'belief considered by Levesque and Lakemeyer. In particular, we show that in a precise sense agents in the structures considered by Levesque and Lakemeyer are perfect reasoners in NPL. 1
Possible Worlds and Resources: The Semantics of BI
 THEORETICAL COMPUTER SCIENCE
, 2003
"... The logic of bunched implications, BI, is a substructural system which freely combines an additive (intuitionistic) and a multiplicative (linear) implication via bunches (contexts with two combining operations, one which admits Weakening and Contraction and one which does not). BI may be seen to a ..."
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Cited by 46 (17 self)
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The logic of bunched implications, BI, is a substructural system which freely combines an additive (intuitionistic) and a multiplicative (linear) implication via bunches (contexts with two combining operations, one which admits Weakening and Contraction and one which does not). BI may be seen to arise from two main perspectives. On the one hand, from prooftheoretic or categorical concerns and, on the other, from a possibleworlds semantics based on preordered (commutative) monoids. This semantics may be motivated from a basic model of the notion of resource. We explain BI's prooftheoretic, categorical and semantic origins. We discuss in detail the question of completeness, explaining the essential distinction between BI with and without ? (the unit of _). We give an extensive discussion of BI as a semantically based logic of resources, giving concrete models based on Petri nets, ambients, computer memory, logic programming, and money.
PartialGaggles Applied to Logics with Restricted Structural Rules
 In Peter SchroederHeister and Kosta Dosen, editors, Substructural Logics
, 1991
"... Law of Residuation (in their jth place) when f and g are contrapositives (with respect to their jth place) and S(f; a 1 ; : : : ; a j ; : : : ; a n ; b) iff S(g; a 1 ; : : : ; b; : : : ; a n ; a j ). (5) Two operators f , g 2 OP are relatives when they satisfy the Abstract Law of Residuation in ..."
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Cited by 40 (1 self)
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Law of Residuation (in their jth place) when f and g are contrapositives (with respect to their jth place) and S(f; a 1 ; : : : ; a j ; : : : ; a n ; b) iff S(g; a 1 ; : : : ; b; : : : ; a n ; a j ). (5) Two operators f , g 2 OP are relatives when they satisfy the Abstract Law of Residuation in some position. (6) The family of operations OP is founded when there is a distinguished operator f 2 OP (the head) such that any other operator g 2 OP is a relative of f . Definition. A partialgaggle is a tonoid T = (X; ; OP), in which OP is a founded family. As examples, consider a p.o. residuated groupoid, with OP chosen to be any of the following families of operations (ffi is the head of the families of which it is a member): fffig, fffi; /g, fffi; !g, fffi; /;!g, f/g, f!g. Note that f!;/g does not formally fall under our definition since the trace of one is not directly the contrapositive of the trace of the other, even though the trace of each is a contrapositive of the trace of f...
On the role of logic in information retrieval
 Information Processing and Management
, 1998
"... What is that makes a “good ” logical model of IR? What are the guidelines that we should follow when we want to build one, and how much can we depart from these guidelines and still claim to have a logical model of IR? We have been motivated to write this note from our dissatisfaction with the fact ..."
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Cited by 34 (4 self)
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What is that makes a “good ” logical model of IR? What are the guidelines that we should follow when we want to build one, and how much can we depart from these guidelines and still claim to have a logical model of IR? We have been motivated to write this note from our dissatisfaction with the fact that there seem to be many competing, incompatible views of what a logical model of IR should consist of; we think some of these views are misleading. 1 Information Retrieval and modelling In recent years, researchers in Information Retrieval (IR) have devoted an increasing amount of work to the design of models of IR, i.e. of theoretical descriptions of the IR task that could serve both as specifications for building running systems, and as theoretical tools for abstractly investigating the relative effectiveness of systems built along their guidelines. Modelling is fundamentally an activity of abstraction. A model is a description of a system that concentrates on the most important, architectural features of the system, and leaves out details that are believed not to be
A Logical View of Composition
 THEORETICAL COMPUTER SCIENCE
, 1993
"... We define two logics of safety specifications for reactive systems. The logics provide a setting for the study of composition rules. The two logics arise naturally from extant specification approaches; one of the logics is intuitionistic, while the other one is linear. ..."
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Cited by 34 (8 self)
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We define two logics of safety specifications for reactive systems. The logics provide a setting for the study of composition rules. The two logics arise naturally from extant specification approaches; one of the logics is intuitionistic, while the other one is linear.
On Bunched Typing
, 2002
"... We study a typing scheme derived from a semantic situation where a single category possesses several closed structures, corresponding to dierent varieties of function type. In this scheme typing contexts are trees built from two (or more) binary combining operations, or in short, bunches. Bunched ..."
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Cited by 33 (2 self)
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We study a typing scheme derived from a semantic situation where a single category possesses several closed structures, corresponding to dierent varieties of function type. In this scheme typing contexts are trees built from two (or more) binary combining operations, or in short, bunches. Bunched typing and its logical counterpart, bunched implications, have arisen in joint work of the author and David Pym. The present paper gives a basic account of the type system, and then focusses on concrete models that illustrate how it may be understood in terms of resource access and sharing. The most
Entailed Ranking Arguments
, 2002
"... An ‘elementary ranking condition ’ (ERC) embodies the kind of restrictions imposed by a comparison between a desired optimum and a single competitor. All entailments between elementary ranking conditions can be ascertained through three simple formal rules; one of them introduces a method of argumen ..."
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Cited by 31 (3 self)
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An ‘elementary ranking condition ’ (ERC) embodies the kind of restrictions imposed by a comparison between a desired optimum and a single competitor. All entailments between elementary ranking conditions can be ascertained through three simple formal rules; one of them introduces a method of argument combination, fusion, shown to have the same sense as in relevance logic. Fusion is also central to detecting inconsistency in a set of ERCs; inconsistency and entailment are closely related here, much as in ordinary logic. Fusion therefore plays a key role in the definition of Recursive Constraint Demotion (RCD: Tesar & Smolensky 1994, 1998). When ERCs are hierarchized by the ranking of the constraints that crucially evaluate them, their entailment and fusional relations are seen to correlate with aspects of ranking structure. RCD and the Minimal Stratified Hierarchy it produces also figure prominently in an efficient procedure for calculating entailments. Harmonic bounding, both simple and collective, leads to the existence of entailment relations, and removal of entailment dependencies from a set of ERCs eliminates harmonic bounding in its underlying candidate set. The logic of entailment in OT is seen to be the implicationnegation fragment of RM (Sobociski 1952, Parks 1972) and the logic of OT in general is shown by a semantical argument to be precisely RM itself. When the logic is extended from ERCs to constraints, it allows for a direct representation of the notion of a strict domination hierarchy using only the connectives of the logic; various ranking restrictions are shown to follow when logical relations exist between constraints.