Abstract. The practice of first-order logic is replete with meta-level concepts. Most notably there are meta-variables ranging over formulae, variables, and terms, and properties of syntax such as alpha-equivalence, capture-avoiding substitution and assumptions about freshness of variables with respect to metavariables. We present one-and-a-halfth-order logic, in which these concepts are made explicit. We exhibit both sequent and algebraic specifications of one-and-a-halfth-order logic derivability, show them equivalent, show that the derivations satisfy cut-elimination, and prove correctness of an interpretation of first-order logic within it. We discuss the technicalities in a wider context as a case-study for nominal algebra, as a logic in its own right, as an algebraisation of logic, as an example of how other systems might be treated, and also as a theoretical foundation

One of the main characteristics of knowledge representation systems based on the description of concepts is the clear distinction between terminological and assertional knowledge. Although this characteristic leads to several computational and representational advantages, it usually limits the expressive power of the system. For this reason, some attempts have been done, allowing for a limited form of amalgamation between the two components and a more complex interaction between them. In particular, one of these attempts is based on letting the individuals to be referenced in the concept expressions. This is generally performed by admitting a constructor for building a concept from a set of enumerated individuals. In this paper we investigate on the consequences of introducing constructors of this type in the concept description language. We also provide a complete reasoning procedure to deal with these constructors and we obtain some complexity results on it. 1 Introduction The ide...

We study proof systems for reasoning about logical consequences and refinement of structured specifications, based on similar systems proposed earlier in the literature [ST 88, Wir 91]. Following Goguen and Burstall, the notion of an underlying logical system over which we build specifications is formalized as an institution and extended to a more general notion, called (D, T )-institution. We show that under simple assumptions (essentially: amalgamation and interpolation) the proposed proof systems are sound and complete. The completeness proofs are inspired by proofs due to M. V. Cengarle (see [Cen 94]) for specifications in first-order logic and the logical systems for reasoning about them. We then propose a methodology for reusing proof systems built over institutions rich enough to satisfy the properties required for the completeness results for specifications built over poorer institutions where these properties need not hold.

Reasoning about specifications is one of the fundamental activities in the process of formal program development. This ranges from proving the consequences of a specification, during the prototyping or testing phase for a requirements speci cation, to proving the correctness of refinements (or implementations) of specifications. The main proof techniques for algebraic specifications have their origin in equational Horn logic and term rewriting. These proof methods have been well studied in the case of nonstructured speci cations (see Chapters 9 and 10). For large systems of specifications built using the structuring operators of speci cation languages, relatively few proof techniques have been developed yet; for such proof systems, see [SB83, HST94, Wir91, Far92, Cen94, HWB97]. In this chapter we focus on proof systems designed particularly for modular specifications. The aim is to concentrate on the structuring concepts, while abstracting as much as possible from the par...

The process of rational inquiry can be defined as the evolution of the beliefs of a rational agent as a consequence of its internal inference procedures and its interaction with the environment. These beliefs can be modelled in a formal way using belief logics. The possible worlds model and its associated Kripke semantics provide an intuitive semantics for these logics, but they commit us to model agents that are logically omniscient and perfect reasoners. These problems can be avoided with a syntactic view of possible worlds, defining them as arbitrary sets of sentences in a propositional belief logic. In this article this syntactic view of possible worlds is taken, and a dynamic analysis of the beliefs of the agent is suggested in order to model the process of rational inquiry in which the agent is permanently engaged. 1 INTRODUCTION The aim of this work 1 is to model the process of rational inquiry, i.e. the (rationally controlled) transformation of the beliefs of an intelligent...

We introduce FMG (Fraenkel-Mostowski Generalised) set theory, a generalisation of FM set theory which allows binding of infinitely many names instead of just finitely many names. We apply this generalisation to show how three presentations of syntax — de Bruijn indices, FM sets, and name-carrying syntax — have a relation generalising to all sets and not only sets of syntax trees. We also give syntaxfree accounts of Barendregt representatives, scope extrusion, and other phenomena associated to α-equivalence. Our presentation uses a novel presentation based not on a theory but on a concrete model U.

of actions of the object G on the object X, in the sense of the theory of semi-direct products in V. We investigate the representability of the functor Act(−,X) in the case where V is locally presentable, with finite limits commuting with filtered colimits. This contains all categories of models of a semi-abelian theory in a Grothendieck topos, thus in particular all semi-abelian varieties of universal algebra. For such categories, we prove first that the representability of Act(−,X) reduces to the preservation of binary coproducts. Next we give both a very simple necessary condition and a very simple sufficient condition, in terms of amalgamation properties, for the preservation of binary coproducts by the functor Act(−,X) in a general semi-abelian category. Finally, we exhibit the precise form of the more involved “if and only if ” amalgamation property corresponding to the representability of actions: this condition is in particular related to a new notion of “normalization of a morphism”. We provide also a wide supply of algebraic examples and counter-examples, giving in particular evidence of the relevance of the object representing Act(−,X), when it turns out to exist. 1. Actions and split exact sequences A semi-abelian category is a Barr-exact, Bourn-protomodular, finitely complete and finitely cocomplete category with a zero object 0. The existence of finite limits and a zero object implies that Bourn-protomodularity is equivalent to, and so can be replaced with, the following split version of the short five lemma: K K given a commutative diagram of “kernels of split epimorphisms” qisi =1Q, ki = Ker qi, i =1, 2 k1 k2

The main aim of this paper is to present the completeness proof of a formal system for reasoning about logical consequences of structured specifications. The system is based on the proof rules for structural specifications build in an arbitrary institution as presented in [ST 88]. The proof of its completeness is inspired by the proof due to M. V. Cengarle (see [Cen 94]) for specifications in firstorder logic and the logical system for reasoning about them presented also in [Wir 91]. 1 Introduction In a number of papers on algebraic specifications (see [Cen 94, Far 92, ST 88, SST 92, Tar 86, Wir 91]) the main goal was to build: ffl a flexible enough specification formalism which allows one to cope with various problems of software engineering; ffl a sound and complete logic for reasoning about such specifications. We follow these goals when the first and a part of the second aim (soundness) is achieved by using structured specifications and the logical system built over an arbitrary...

The construction of rational agents is one of the goals that has been pursued in Artificial Intelligence (AI). In most of the architectures that have been proposed for this kind of agents, its behaviour is guided by its set of beliefs. In our work, rational agents are those systems that are permanently engaged in the process of rational inquiry; thus, their beliefs keep evolving in time, as a consequence of their internal inference procedures and their interaction with the environment. Both AI researchers and philosophers are interested in having a formal model of this process, and this is the main topic in our work. Beliefs have been formally modelled in the last decades using doxastic logics. The possible worlds model and its associated Kripke semantics provide an intuitive semantics for these logics, but they seem to commit us to model agents that are logically omniscient and perfect reasoners. We avoid these problems by replacing possible worlds by conceivable situations, which ar...

This paper concerns the representation of introspective belief and knowledge in multi-agent systems. An introspective agent is an agent that has the ability to refer to itself and reason about its own beliefs. It is well-known that representing introspective beliefs is theoretically very problematic. An agent which is given strong introspective abilities is most likely to have inconsistent beliefs, since it can use introspection to express self-referential beliefs that are paradoxical in the same way as the classical paradoxes of self-reference. In multi-agent systems these paradoxical beliefs can even be expressed as beliefs about the correctness and completeness of other agents' beliefs, i.e., even without the presence of explicit introspection. In this paper we explore the maximal sets of introspective beliefs that an agent can consistently obtain and retain when situated in a dynamic environment, and when treating beliefs "syntactically" (that is, formalizing beliefs as axioms of first-order predicate logic rather than using modal formalisms). We generalize some previous results by Perlis [1985] and des Rivieres &