Typed feature structures, definite equivalences, greatest model semantics, and nonmonotonicity

Typed feature logics have been employed as description languages in modern type-oriented grammar theories like HPSG and have laid the theoretical foundations for many implemented systems. However, recursivity pose severe problems and have been addressed through specialized powerdomain constructions which depend on the particular view of the logician. In this paper, we argue that definite equivalences introduced by Smolka can serve as the formal basis for arbitrarily formalized typed feature structures and typed feature-based grammars/lexicons, as employed in, e.g., TFS or TDL. The idea here is that type definitions in such systems can be transformed into an equivalent definite program, whereas the meaning of the definite program then is identified with the denotation of the type system. Now, models of a definite program P can be characterized by the set of ground atoms which are logical consequences of the definite program. These models are ordered by subset inclusion and, for reasons that will become clear, we propose the greatest model as the intended interpretation of P, or equivalent, as the denotation of the associated type system. Our transformational approach has also a great impact on nonmonotonically defined types, since under this interpretation, we can view the type hierarchy as a pure transport medium, allowing us to get rid of the transitivity of type information (inheritance), and yielding a perfectly monotonic definite program

Static analysis of functional languages

Static analysis is the name given to a number of compile time analysis techniques used to automatically generate information which can lead to improvements in the execution performance of function languages. This thesis provides an introduction to these techniques and their implementation. The abstract interpretation framework is an example of a technique used to extract information from a program by providing the program with an alternate semantics and evaluating this program over a non-standard domain. The elements of this domain represent certain properties of interest. This framework is examined in detail, as well as various extensions and variants of it. The use of binary logical relations and program logics as alternative formulations of the framework , and partial equivalence relations as an extension to it, are also looked at. The projection analysis framework determines how much of a sub-expression can be evaluated by examining the context in which the expression is to be evaluated, and provides an elegant method for finding particular types of information from data structures. This is also examined. The most costly operation in implementing an analysis is the computation of fixed points. Methods developed to make this process more efficient are looked at. This leads to the final chapter which highlights the dependencies and relationships between the different frameworks and their mathematical disciplines.KMBT_22

Dynamical systems via domains:Toward a unified foundation of symbolic and non-symbolic computation

Non-symbolic computation (as, e.g., in biological and artificial neural networks) is astonishingly good at learning and processing noisy real-world data. However, it lacks the kind of understanding we have of symbolic computation (as, e.g., specified by programming languages). Just like symbolic computation, also non-symbolic computation needs a semantics—or behavior description—to achieve structural understanding. Domain theory has provided this for symbolic computation, and this thesis is about extending it to non-symbolic computation. Symbolic and non-symbolic computation can be described in a unified framework as state-discrete and state-continuous dynamical systems, respectively. So we need a semantics for dynamical systems: assigning to a dynamical system a domain—i.e., a certain mathematical structure—describing the system’s behavior. In part 1 of the thesis, we provide this domain-theoretic semantics for the ‘symbolic’ state-discrete systems (i.e., labeled transition systems). And in part 2, we do this for the ‘non-symbolic’ state-continuous systems (known from ergodic theory). This is a proper semantics in that the constructions form functors (in the sense of category theory) and, once appropriately formulated, even adjunctions and, stronger yet, equivalences. In part 3, we explore how this semantics relates the two types of computation. It suggests that non-symbolic computation is the limit of symbolic computation (in the ‘profinite’ sense). Conversely, if the system’s behavior is fairly stable, it may be described as realizing symbolic computation (here the concepts of ergodicity and algorithmic randomness are useful). However, the underlying concept of stability is limited by a no-go result due to a novel interpretation of Fitch’s paradox. This also has implications for AI-safety and, more generally, suggests fruitful applications of philosophical tools in the non-symbolic computation of modern AI