This paper generalizes many-sorted algebra (hereafter, MSA) to order-sorted algebra (hereafter, OSA) by allowing a partial ordering relation on the set of sorts. This supports abstract data types with multiple inheritance (in roughly the sense of object-oriented programming), several forms of polymorphism and overloading, partial operations (as total on equationally defined subsorts), exception handling, and an operational semantics based on term rewriting. We give the basic algebraic constructions for OSA, including quotient, image, product and term algebra, and we prove their basic properties, including Quotient, Homomorphism, and Initiality Theorems. The paper's major mathematical results include a notion of OSA deduction, a Completeness Theorem for it, and an OSA Birkhoff Variety Theorem. We also develop conditional OSA, including Initiality, Completeness, and McKinsey-Malcev Quasivariety Theorems, and we reduce OSA to (conditional) MSA, which allows lifting many known MSA results to OSA. Retracts, which intuitively are left inverses to subsort inclusions, provide relatively inexpensive run-time error handling. We show that it is safe to add retracts to any OSA signature, in the sense that it gives rise to a conservative extension. A final section compares and contrasts many different approaches to OSA. This paper also includes several examples demonstrating the flexibility and applicability of OSA, including some standard benchmarks like STACK and LIST, as well as a much more substantial example, the number hierarchy from the naturals up to the quaternions.

. This paper presents an overview of the development of the field of temporal and modal logic programming. We review temporal and modal logic programming languages under three headings: (1) languages based on interval logic, (2) languages based on temporal logic, and (3) languages based on (multi)modal logics. The overview includes most of the major results developed, and points out some of the similarities, and the differences, between languages and systems based on diverse temporal and modal logics. The paper concludes with a brief summary and discussion. Categories: Temporal and Modal Logic Programming. 1 Introduction In logic programming, a program is a set of Horn clauses representing our knowledge and assumptions about some problem. The semantics of logic programs as developed by van Emden and Kowalski [96] is based on the notion of the least (minimum) Herbrand model and its fixed-point characterization. As logic programming has been applied to a growing number of problem domai...

A growing body of work demonstrates that syntactic structure can evolve in populations of genetically identical agents. Traditional explanations for the emergence of syntactic structure employ an argument based on genetic evolution: syntactic structure is specified by an innate Language Acquisition Device (LAD). Knowledge of language is complex, yet the data available to the language learner is sparse. This incongruous situation, termed the "poverty of the stimulus", is accounted for by placing much of the specification of language in the LAD. The assumption is that the characteristic structure of language is somehow coded genetically. The effect of language evolution on the cultural substrate, in the absence of genetic change, is not addressed by this explanation. We show that the poverty of the stimulus introduces a pressure for compositional language structure when we consider language evolution resulting from iterated observational learning. We use a mathematical model to map the space of parameters that result in compositional syntax. Our hypothesis is that compositional syntax cannot be explained by understanding the LAD alone: compositionality is an emergent property of the dynamics resulting from sparse language exposure.

Although “now ” is expressed in SQL as CURRENT_TIMESTAMP within queries, this value cannot be stored in the database. However, this notion of an ever-increasing current-time value has been reflected in some temporal data models by inclusion of database-resident variables, such as “now”, “until-changed, ” “�, ” “@, ” and “–”. Time variables are very desirable, but their use also leads to a new type of database, consisting of tuples with variables, termed a variable database. This article proposes a framework for defining the semantics of the variable databases of the relational and temporal relational data models. A framework is presented because several reasonable meanings may be given to databases that use some of the specific temporal variables that have appeared in the literature. Using the framework, the article defines a useful semantics for such databases. Because situations occur where the existing time variables are inadequate, two new types of modeling entities that address these shortcomings, timestamps that we call now-relative and now-relative indeterminate, are introduced and defined within the framework. Moreover, the article provides a foundation, using algebraic

We present a compositional program logic for call-by-value imperative higherorder functions with general forms of aliasing, which can arise from the use of reference names as function parameters, return values, content of references and part of data structures. The program logic

. This work belongs to a family of research efforts, called microtheories and aimed at describing the static meaning of all lexical categories in several languages in the framework of the MikroKosmos project on computational semantics. The latter also involves other static microtheories describing world knowledge and syntax-semantics mapping as well as dynamic microtheories connected with the actual process of text analysis. This paper describes our approach to determining and representing adjectival meaning, compares it with the body of knowledge on adjectives in literature and presents a detailed, practically tested methodology and heuristics for the acquisition of lexical entries for adjectives. The work was based on the set of over 6,000 English and about 1,500 Spanish adjectives obtained from task-oriented corpora. Introduction The topic of this paper is the information about adjectival meaning which should be included in a computational lexicon. Thus, we concentrate on...

The modern computational lexical semantics reached a point in its development when it has become necessary to define the premises and goals of each of its several trends. This paper proposes ten choices in terms of which these premises and goals can be discussed. It is argued that the central questions include the use of lexical rules for generating word senses; the role of syntax, pragmatics, and formal semantics in the specification of lexical meaning; the use of a world model, or ontology, as the organizing principle for lexical-semantic descriptions; the use of rules with limited scope; the relation between static and dynamic resources; the commitment to descriptive coverage; the trade-off between generalization and idiosyncracy; and, finally, the adherence to the "supply-side" (method-oriented) or "demand-side" (task-oriented) ideology of research. The discussion is inspired by, but not limited to, the comparison between the generative lexicon approach and the ontologi...

this paper we discuss the analysis of expressions such as John's team, John's brother, John's favorite movie, Mary's favorite chair, Mary's former mansion. Before introducing the concrete problems, we briefly describe our theoretical perspective. Our theoretical concern is the integration of formal semantics and lexical semantics, especially but not exclusively in the traditions of Montague Grammar and the Moscow School (Apresjan (1994), Mel'uk (1982), Paducheva (1996)), respectively. We have proposed (Borschev and Partee (in press)) to modify the Moscow school approach and represent lexical information in the form of sets of meaning postulates, which may or may not exhaust the meaning of the given lexical item. We believe this use of meaning postulates is consistent with actual Moscow school practice, and it makes it possible to integrate lexical semantics with the compositional "semantics of syntax" given by formal semantics. If the formal semantic interpretation of a sentence is given as a formula of intensional logic in which lexical items are primitives, and lexical semantics as a set of meaning postulates for these lexical items, then their integration can be seen as the drawing of entailments from these sources. This approach is in principle extendable to the integration of semantic interpretation with contextual and other information as well. So we semantically represent a sentence or a text as a theory consisting of different sorts of formulas, i.e. different sorts of axioms and their entailments. By "theory" here, we do not mean the metalevel linguistic theory, but the set of axioms from various sources plus the consequences that can be drawn from these axioms, which together constitute the interpretation of such a sentence in a given context. Such a theory (s...

Even though residuation is at the core of Categorial Grammar [11], it is not always immediate to realize how standard logic systems like Multi-modal Categorial Type Logics (MCTL) [17] actually embody this property. In this paper we focus on the basic system NL [12] and its extension with unary modalities NL(3) [16], and we spell things out by means of Display Calculi (DC) [3, 10]. The use of structural operators in DC permits a sharp distinction between the core properties we want to impose on the logic system and the way these properties are projected into the logic operators. We will show how we can obtain Lambek residuated triple n, = and of binary operators, and how the operators 3 and 2 introduced by Moortgat in [16] are indeed their unary counterpart.

Two recent lines of database research. proceeding independently, have been concerned with providing a richer, more intuitive view of information at the user level. Historical database research has focused on ways to provide users with a view of information anchored and evolving in the temporal dimension. Object-oriented database research focuses on encapsulating both the structure and the behavior of the objects that users intend to model. In this paper we explore how these two lines of research might be brought together. providing to the user the representation and management of objects in tzme. I.