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The ProofTheory and Semantics of Intuitionistic Modal Logic
, 1994
"... Possible world semantics underlies many of the applications of modal logic in computer science and philosophy. The standard theory arises from interpreting the semantic definitions in the ordinary metatheory of informal classical mathematics. If, however, the same semantic definitions are interpret ..."
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Possible world semantics underlies many of the applications of modal logic in computer science and philosophy. The standard theory arises from interpreting the semantic definitions in the ordinary metatheory of informal classical mathematics. If, however, the same semantic definitions are interpreted in an intuitionistic metatheory then the induced modal logics no longer satisfy certain intuitionistically invalid principles. This thesis investigates the intuitionistic modal logics that arise in this way. Natural deduction systems for various intuitionistic modal logics are presented. From one point of view, these systems are selfjustifying in that a possible world interpretation of the modalities can be read off directly from the inference rules. A technical justification is given by the faithfulness of translations into intuitionistic firstorder logic. It is also established that, in many cases, the natural deduction systems induce wellknown intuitionistic modal logics, previously given by Hilbertstyle axiomatizations. The main benefit of the natural deduction systems over axiomatizations is their
A Tableau Calculus for Multimodal Logics and Some (Un)Decidability Results
 IN PROC. OF TABLEAUX98
, 1998
"... In this paper we present a prefixed analytic tableau calculus for a class of normal multimodal logics and we present some results about decidability and undecidability of this class. The class is characterized by axioms of the form [t 1 ] : : : [t n ]' oe [s1 ] : : : [sm ]', called incl ..."
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In this paper we present a prefixed analytic tableau calculus for a class of normal multimodal logics and we present some results about decidability and undecidability of this class. The class is characterized by axioms of the form [t 1 ] : : : [t n ]' oe [s1 ] : : : [sm ]', called inclusion axioms, where the t i 's and s j 's are constants. This class of logics, called grammar logics, was introduced for the first time by Farinas del Cerro and Penttonen to simulate the behaviour of grammars in modal logics, and includes some wellknown modal systems. The prefixed tableau method is used to prove the undecidability of modal systems based on unrestricted, context sensitive, and context free grammars. Moreover, we show that the class of modal logics, based on rightregular grammars, are decidable by means of the filtration methods, by defining an extension of the FischerLadner closure.
Algebraic logic, varieties of algebras, and algebraic varieties
, 1995
"... Abstract. The aim of the paper is discussion of connections between the three kinds of objects named in the title. In a sense, it is a survey of such connections; however, some new directions are also considered. This relates, especially, to sections 3, 4 and 5, where we consider a field that could ..."
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Cited by 14 (5 self)
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Abstract. The aim of the paper is discussion of connections between the three kinds of objects named in the title. In a sense, it is a survey of such connections; however, some new directions are also considered. This relates, especially, to sections 3, 4 and 5, where we consider a field that could be understood as an universal algebraic geometry. This geometry is parallel to universal algebra. In the monograph [51] algebraic logic was used for building up a model of a database. Later on, the structures arising there turned out to be useful for solving several problems from algebra. This is the position which the present paper is written from.
NonMonotonic Predicate Logics Syntax, Semantics, Completeness 1 Summary
"... To simplify our account, rewrite your favorite usual predicate logic, classical, modal, intuitionistic, as a logic of the set L of all statements. This is an inessential change. As Tarski once did, arrange it so that only statements occur in all axioms a and rules of inference I, all of which have t ..."
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To simplify our account, rewrite your favorite usual predicate logic, classical, modal, intuitionistic, as a logic of the set L of all statements. This is an inessential change. As Tarski once did, arrange it so that only statements occur in all axioms a and rules of inference I, all of which have the (monotonic) form ”from a,a ′,..., infer a ′ ′. Then a deductively closed set D is merely a subset D of L such that if a is an axiom, then a is in D; and for all such I, if a,a ′,... are in D, then c is in D. If A is a subset of L, call D grounded in A if every member of D has a finite derivation from A using the rules of inference I. Call E an extension of A if E is deductively closed and grounded in A. In the usual (monotonic) logics if E is an extension of A, then E is uniquely determined, the deductive closure of A via an ordinary inductive definition. We associate with each usual logic ”nonmonotonic prolongations ” based on additional rules of inference only, keeping the statements exactly the same
A Matrix Characterization of Validity for Multimodal Logics
"... abstract. We present a new matrix characterization of validity for a family of propositional multimodal logics with interacting modalities. Unlike previous matrix characterizations for modal logics, which could only cope with a few wellbehaved unimodal logics, our formulation is not based on prefix ..."
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abstract. We present a new matrix characterization of validity for a family of propositional multimodal logics with interacting modalities. Unlike previous matrix characterizations for modal logics, which could only cope with a few wellbehaved unimodal logics, our formulation is not based on prefixed tableaus in the style of Fitting, but has a more direct relationship with the semantics of the logics and their defining frame conditions. The resulting formalism uniformly and elegantly characterizes all 15 basic unimodal logics, as well as a number of multimodal logics with interacting modalities.
82 Il Milione: A Journey in the Computational Logic in Italy
"... Questo articolo presenta alcune delle attività condotte nel corso degli ultimi anni dal gruppo di ricerca guidato da Alberto Martelli. In particolare verrà presentato un percorso che comprende la specifica, lo sviluppo e la verifica di protocolli di interazione. Il filo conduttore è costituito dall’ ..."
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Questo articolo presenta alcune delle attività condotte nel corso degli ultimi anni dal gruppo di ricerca guidato da Alberto Martelli. In particolare verrà presentato un percorso che comprende la specifica, lo sviluppo e la verifica di protocolli di interazione. Il filo conduttore è costituito dall’uso di logiche multimodali e di formalismi dichiarativi e tecniche di ragionamento basati sulla logica computazionale. In this paper, we report some of the activities carried on in the last years by the research group leaded by Alberto Martelli. In particular, it presents a research line that encompasses the specification, the development and the verification of interaction protocols. The leading thread is given by the use of multimodal logics and of declarative formalisms and reasoning techniques, based on computational logic.