Model Theory. These are non-empty families I of partial isomorphisms between models M and N , closed under taking restrictions to smaller domains, and satisfying the usual Back-and-Forth properties for extension with objects on either side -- restricted to apply only to partial isomorphisms of size at most k . 'Invariance for k--partial isomorphism' means having the same truth value at tuples of objects in any two models that are connected by a partial isomorphism in such a set. The precise sense of this is spelt out in the following proof. 21 Proof (Outline.) k-variable formulas are preserved under partial isomorphism, by a simple induction. More precisely, one proves, for any assignment A and any partial isomorphism IÎI which is defined on the A-values for all variables x 1 , ..., x k , that M, A |= f iff N , IoA |= f . The crucial step in the induction is the quantifier case. Quantified variables are irrelevant to the assignment, so that the relevant partial isomorphism can be res...

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 proof-theory 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

Planning from second principles by reusing and modifying plans is one way of improving the efficiency of planning systems. In this paper, we study it in the general framework of deductive planning and develop a logical formalization of planning from second principles, which relies on a systematic decomposition of the planning process. Deductive inference processes with clearly defined semantics formalize each of the subtasks a second principles planner has to address. Plan modification, which comprises matching and adaptation tasks, is based on a deductive approach yielding provably correct modified plans. Description logics are introduced as query languages to plan libraries, which leads to a novel and efficient solution to the indexing problem in case-based reasoning. Apart from sequential plans, this approach enables a planner to reuse and modify complex plans containing control structures like conditionals and loops. 1 Introduction Planning from first principles generat...

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 well-known 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 right-regular grammars, are decidable by means of the filtration methods, by defining an extension of the Fischer-Ladner closure.

The paper presents a set-theoretic translation method for polymodal logics that reduces the derivability problem of a large class of propositional polymodal logics to the derivability problem of a very weak firstorder set theory\Omega\Gamma Unlike most existing translation methods, the one we proposed applies to any normal complete finitely-axiomatizable polymodal logic, regardless if it is first-order complete or if an explicit semantics is available for it. Moreover, the finite axiomatizability of\Omega makes it possible to implement mechanical proof search procedures via the deduction theorem or more specialized and efficient techniques. In the last part of the paper, we briefly discuss the application of set T -resolution to support automated derivability in (a suitable extension of) \Omega\Gamma This work has been supported by funds MURST 40% and 60%. The second author was supported by a grant from the Italian Consiglio Nazionale delle Ricerche (CNR). 1 Introduction The paper...

this paper we present a tableau-like proof system for multi-modal logics based on D'Agostino and Mondadori's classical refutation system KE [DM94]. The proposed system, that we call KEM , works for the logics S5A and S5P(n) which have been devised by Mayer and van der Hoek [MvH92] for formalizing the notions of actuality and preference. We shall also show how KEM works with the normal modal logics K45, D45, and S5 which are frequently used as bases for epistemic operators -- knowledge, belief (see, for example [Hoe93, Wan90]), and we shall briefly sketch how to combine knowledge and belief in a multi-agent setting through KEM modularity

In this paper we present a framework for developing modal extensions of logic programming, which are parametric with respect to the properties chosen for the modalities and which allow sequences of modalities of the form [t], where t is a term of the language, to occur in front of clauses, goals and clause heads. The properties of modalities are specified by a set A of inclusion axioms of the form [t 1 ] : : : [t n ]ff oe [s 1 ] : : : [s m ]ff. The language can deal with many of the wellknown modal systems and several examples are provided. Due to its features, it is particularly suitable for performing epistemic reasoning, defining parametric and nested modules, describing inheritance in a hierarchy of classes and reasoning about actions. A goal directed proof procedure of the language is presented, which is modular with respect to the properties of modalities. Moreover, we define a fixpoint semantics, by generalizing the standard construction for Horn clauses, which is used to prov...

We present an approach to a coherent program synthesis system which integrates a variety of interactively controlled and automated techniques from theorem proving and algorithm design at different levels of abstraction. Besides providing an overall view we summarize the individual research results achieved in the course of this development.

We investigate modal deduction through translation into standard logic and set theory. Derivability in the minimal modal logic is captured precisely by translation into a weak, computationally attractive set theory \Omega\Gamma This approach is shown equivalent to working with standard first-order translations of modal formulas in a theory of general frames. Next, deduction in a more powerful second-order logic of general frames is shown equivalent with set-theoretic derivability in an `admissible variant' of \Omega\Gamma Our methods are mainly model-theoretic and set-theoretic, and they admit extension to richer languages than that of basic modal logic. 1 Introduction We are interested in analyzing general deduction for modal formulae [4]. The standard systems used for this purpose are the so-called "minimal modal logic" K or, if one wants to work over general frames (as we do), the system K s obtained from K adding a substitution rule. We do not consider special purpose calculi for ...

. Skolemization is not an equivalence preserving transformation. For the purposes of refutational theorem proving it is suÆcient that Skolemization preserves satisability and unsatis- ability. Therefore there is sometimes some freedom in interpreting Skolem functions in a particular way. We show that in certain cases it is possible to exploit this freedom for simplifying formulae considerably. Examples for cases where this occurs systematically are the relational translation from modal logics to predicate logic and the relativization of rst-order logics with sorts. Key words: Skolemization, Refutational Theorem Proving 1. Introduction Refutational theorem proving has a certain degree of freedom which so far is not very often exploited. All kinds of transformations preserving satisability and unsatisability of the formulae to be refuted are allowed. Skolemization is a typical transformation which is not equivalence preserving but satisability and unsatis- ability preserving. Bu...