Databases and information systems are often hard to use because they do not explicitly attempt to cooperate with their users. Direct answers to database and knowledge base queries may not always be the best answers. Instead, an answer with extra or alternative information may be more useful and less misleading to a user. This paper surveys foundational work that has been done toward endowing intelligent information systems with the ability to exhibit cooperative behavior. Grice's maxims of cooperative conversation, which provided a starting point for the field of cooperative answering, are presented along with relevant work in natural language dialogue systems, database query answering systems, and logic programming and deductive databases. The paper gives a detailed account of cooperative techniques that have been developed for considering users' beliefs and expecations, presuppositions, and misconceptions. Also, work in intensional answering and generalizing queries and answers is co...

A logic for specification and verification is derived from the axioms of Zermelo-Fraenkel set theory. The proofs are performed using the proof assistant Isabelle. Isabelle is generic, supporting several different logics. Isabelle has the flexibility to adapt to variants of set theory. Its higher-order syntax supports the definition of new binding operators. Unknowns in subgoals can be instantiated incrementally. The paper describes the derivation of rules for descriptions, relations and functions, and discusses interactive proofs of Cantor’s Theorem, the Composition of Homomorphisms challenge [9], and Ramsey’s Theorem [5]. A generic proof assistant can stand up against provers dedicated to particular logics. Key words. Isabelle, set theory, generic theorem proving, Ramsey’s Theorem,

It is generally accepted that in principle it’s possible to formalize completely almost all of present-day mathematics. The practicability of actually doing so is widely doubted, as is the value of the result. But in the computer age we believe that such formalization is possible and desirable. In contrast to the QED Manifesto however, we do not offer polemics in support of such a project. We merely try to place the formalization of mathematics in its historical perspective, as well as looking at existing praxis and identifying what we regard as the most interesting issues, theoretical and practical.

Most general purpose proof assistants support versions of typed higher order logic. Experience has shown that these logics are capable of representing most of the mathematical models needed in Computer Science. However, perhaps there exist applications where ZF-style set theory is more natural, or even necessary. Examples may include Scott's classical inverse-limit construction of a model of the untyped - calculus (D1 ) and the semantics of parts of the Z specification notation. This paper

Set theory is the standard foundation for mathematics, but the majority of general purpose mechanised proof assistants support versions of type theory (higher order logic). Examples include Alf, Automath, Coq, EHDM, HOL, IMPS, LAMBDA, LEGO, Nuprl, PVS and Veritas. For many applications type theory works well and provides, for specification, the benefits of type-checking that are well-known in programming. However, there are areas where types get in the way or seem unmotivated. Furthermore, most people with a scientific or engineering background already know set theory, whereas type theory may appear inaccessable and so be an obstacle to the uptake of proof assistants based on it. This paper describes some experiments (using HOL) in combining set theory and type theory; the aim is to get the best of both worlds in a single system. Three approaches have been tried, all based on an axiomatically specified type V of ZF-like sets: (i) HOL is used without any additions besides V; (ii) an emb...

. The majority of general purpose mechanised proof assistants support versions of typed higher order logic, even though set theory is the standard foundation for mathematics. For many applications higher order logic works well and provides, for specification, the benefits of type-checking that are well-known in programming. However, there are areas where types get in the way or seem unmotivated. Furthermore, most people with a scientific or engineering background already know set theory, but not higher order logic. This paper discusses some approaches to getting the best of both worlds: the expressiveness and standardness of set theory with the efficient treatment of functions provided by typed higher order logic. 1 Introduction Higher order logic is a successful and popular formalism for computer assisted reasoning. Proof systems based on higher order logic include ALF [18], Automath [20], Coq [9], EHDM [19], HOL [13], IMPS [10], LAMBDA [11], LEGO [17], Nuprl [6], PVS [22]...

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Databases and information systems are often hard to use because they do not explicitly attempt to cooperate with their users. Direct answers to database and knowledge base queries may not always be the best answers. Instead, an answer with extra or alternative information may be more useful and less misleading to a user. This paper surveys foundational work that has been done toward endowing intelligent information systems with the ability to exhibit cooperative behavior. Grice's maxims of cooperative conversation, which provided a starting point for the field of cooperative answering, are presented along with relevant work in natural language dialogue systems, database query answering systems, and logic programming and deductive databases. The paper gives a detailed account of cooperative techniques that have been developed for considering users' beliefs and expecations, presuppositions, and misconceptions. Also, work in intensional answering and generalizing queries and answers is c...