As probabilistic computations play an increasing role in solving various problems, researchers have designed probabilistic languages to facilitate their modeling. Most of the existing probabilistic languages, however, focus only on discrete distributions, and there has been little effort to develop probabilistic languages whose expressive power is beyond discrete distributions. This dissertation presents a probabilistic language, called PTP (ProbabilisTic Programming), which supports all kinds of probability distributions.

Building on a judgmental formulation of lax logic, we propose a modal language which can be used as a framework for practical programming languages with e#ects. Its characteristic feature is a syntactic distinction between terms and expressions, where terms denote values and expressions denote computations. We distinguish between control e#ects and world e#ects, and allow control e#ects only in terms and world e#ects only in expressions. Therefore the distinction between values and computations is made only with respect to world e#ects. We give an explanation of the type system and the operational semantics from a modal logic perspective. We also introduce a term construct similar to Haskell's runST construct and augment the type system to ensure its safety.

Despite their invaluable contribution to the programming language community, monads as a foundation for the study of effects have three problems: they make it difficult to combine effects; they enforce sequentialization of computations by the syntax; they prohibit effect-free evaluations from invoking e#ectful computations. Building on the judgmental formulation and the possible worlds interpretation of modal logic, we propose a logical analysis of effects based upon the view monads are not identified with effects. Our analysis leads to a language called # # which distinguishes between control e#ects and world e#ects, enforces sequentialization of computations only by the semantics, and logically explains the invocation of computations from evaluations. # # also serves as a unified framework for studying Haskell and ML, which have traditionally been studied separately.

In the interest of designing a recursive module extension to ML that is as simple and general as possible, we propose a novel type system for general recursion over effectful expressions. The presence of effects seems to necessitate a backpatching semantics for recursion based on Scheme's. Our type system ensures statically that recursion is well-founded (that the body of a recursive expression will evaluate without attempting to access the undefined recursive variable), which avoids some unnecessary run-time costs associated with backpatching. To ensure well-founded recursion in the presence of multiple recursive variables and separate compilation, we track the usage of individual recursive variables, represented statically by "names". So that our type system may eventually be integrated smoothly into ML's, reasoning involving names is only required inside code that uses our recursive construct and does not need to infect existing ML code.