Formal language

About: Formal language is a research topic. Over the lifetime, 5763 publications have been published within this topic receiving 154114 citations.

Natural and formal infinities

Abstract: Concepts of infinity usually arise by reflecting on finite experiences and imagining them extended to the infinite. This paper will refer to such personal conception as natural infinities.Research has shown that individuals' natural conceptions of infinity are `labile and self-contradictory' (Fischbein et al.,1979, p. 31). The formal approach to mathematics in the twentieth century attempted to rationalize these inconsistencies by selecting a finite list of specific properties (or axioms) from which the conception of a formal infinity is built by formal deduction. By beginning with different properties of finite numbers, such as counting,ordering or arithmetic,different formal systems may be developed. Counting and ordering lead to cardinal and ordinal number theory and the properties of arithmetic lead to ordered fields that may contain infinite and infinitesimalquantities. Cardinal and ordinal numbers can be added and multiplied but not divided or subtracted. The operations of cardinals are commutative, but the operations of ordinals are not. Meanwhile an ordered field has a full system of arithmetic in which the reciprocals of infinite elements are infinitesimals. Thus, while natural concepts of infinity may contain built-in contradictions, there are several different kinds of formal infinity, each with its own coherent properties, yet each system having properties that differ from the others. The construction of both natural and formal infinities are products of human thought and so may be considered in terms of embodied cognition' (Lakoff and Nunez,2000). The viewpoint forwarded here, however, is that formal deduction focuses as far as possible on formal logic in preference to perceptual imagery, developing a network of formal properties that do not depend on specific embodiments. Indeed, I shall show that formal theory can lead to structure theorems, whose formal properties may then be re-interpreted as a more subtle form of embodied imagery. Not only can natural embodied theory inspire theorems to be proved formally, but formal theory can also feed back into human embodiment, now subtly enhanced by the underlying network of formal relationships.

Specification and verification of object-oriented programs using supertype abstraction

Abstract: We present a formal specification language and a formal verification logic for a simple object-oriented programming language. The language is applicative and statically typed, and supports subtyping and message-passing. The verification logic relies on a behavioral notion of subtyping that captures the intuition that a subtype behaves like its supertypes. We give a formal definition for legal subtype relations, based on the specified behavior of objects, and show that this definition is sufficient to ensure the soundness of the verification logic. The verification logic reflects the way programmers reason informally about object-oriented programs, in that it allows them to use static type information, which avoids the need to consider all possible run-time subtypes.

Some Notes on the Formal Properties of Bidirectional Optimality Theory

Abstract: In this paper, we discuss some formal properties of the model of bidirectional Optimality Theory that was developed in Blutner (2000). We investigate the conditions under which bidirectional optimization is a well-defined notion, and we give a conceptually simpler reformulation of Blutner's definition. In the second part of the paper, we show that bidirectional optimization can be modeled by means of finite state techniques. There we rely heavily on the related work of Frank and Satta (1998) about unidirectional optimization.

Supervision of Infinite Behavior of Discrete-Event Systems

Abstract: Some basic results of supervisory control theory are extended to the setting of $\omega$-languages, formal languages consisting of infinite strings. The extension permits the investigation of both liveness and safety issues in the control of discrete-event systems. A new controllability property appropriate to the infinitary setting ($\omega$-controllability) is defined; this language property captures in a natural way the limitations of available control actions. It is shown that every specification language contains a unique maximal $\omega$-controllable sublanguage, representing the least upper bound of the set of achievable closed-loop sublanguages. This supremal $\omega$-controllable sublanguage allows a simple formulation of necessary and sufficient conditions for the solvability of an infinitary supervisory control problem. The problems of effectively deciding solvability of the control problem and of effectively synthesizing appropriate supervisors are solved for the case where the plant is represented by a deterministic Buchi automaton and the specification of legal behavior by a deterministic Rabin automaton.