The Sigma-Semantics: A Comprehensive Semantics for Functional Programs

A comprehensive semantics for functional programs is presented, which generalizes the well-known call-by-value and call-by-name semantics. By permitting a separate choice between call-by value and call-by-name for every argument position of every function and parameterizing the semantics by this choice we abstract from the parameter-passing mechanism. Thus common and distinguishing features of all instances of the sigma-semantics, especially call-by-value and call-by-name semantics, are highlighted. Furthermore, a property can be validated for all instances of the sigma-semantics by a single proof. This is employed for proving the equivalence of the given denotational (fixed-point based) and two operational (reduction based) definitions of the sigma-semantics. We present and apply means for very simple proofs of equivalence with the denotational sigma-semantics for a large class of reduction-based sigma-semantics. Our basis are simple first-order constructor-based functional programs with patterns

The Sigma-Semantics: A Comprehensive Semantics for Functional Programs

A comprehensive semantics for functional programs is presented, which generalizes the well-known call-by-value and call-by-name semantics. By permitting a separate choice between call-by value and call-by-name for every argument position of every function and parameterizing the semantics by this choice we abstract from the parameter-passing mechanism. Thus common and distinguishing features of all instances of the sigma-semantics, especially call-by-value and call-by-name semantics, are highlighted. Furthermore, a property can be validated for all instances of the sigma-semantics by a single proof. This is employed for proving the equivalence of the given denotational (fixed-point based) and two operational (reduction based) definitions of the sigma-semantics. We present and apply means for very simple proofs of equivalence with the denotational sigma-semantics for a large class of reduction-based sigma-semantics. Our basis are simple first-order constructor-based functional programs with patterns

Annotated Type Systems for Program Analysis

In this Ph.D. thesis, we study four program analyses. Three of them are specified by annotated type systems and the last one by abstract interpretation.We present a combined strictness and totality analysis. We are specifying the analysis as an annotated type system. The type system allows conjunctions of annotated types, but only at the top-level. The analysis is somewhat more powerful than the strictness analysis by Kuo and Mishra due to the conjunctions and in that we also consider totality. The analysis is shown sound with respect to a natural-style operational semantics. The analysis is not immediately extendable to full conjunction.The second analysis is also a combined strictness and totality analysis, however with ``full´´ conjunction. Soundness of the analysis is shown with respect to a denotational semantics. The analysis is more powerful than the strictness analyses by Jensen and Benton in that it in addition to strictness considers totality. So far we have only specified the analyses, however in order for the analyses to be practically useful we need an algorithm for inferring the annotated types. We construct an algorithm for the second analysis using the lazy type approach by Hankin and Le Métayer. The reason for choosing the second analysis from the thesis is that the approach is not applicable to the first analysis.The third analysis we study is a binding time analysis. We take the analysis specified by Nielson and Nielson and we construct a more efficient algorithm than the one proposed by Nielson and Nielson. The algorithm collects constraints in a structural manner like the type inference algorithm by Damas. Afterwards the minimal solution to the set of constraints is found.The last analysis in the thesis is specified by abstract interpretation. Hunt shows that projection based analyses are subsumed by PER (partial equivalence relation) based analyses using abstract interpretation. The PERs used by Hunt are strict, i.e. bottom is related to bottom. Here we lift this restriction by requiring the PERs to be uniform, in the sense that they treat all the integers equally. By allowing non-strict PERs we get three properties on the integers, corresponding to the three annotations used in the first and second analysis in the thesis

Proof-Theoretic Methods for Analysis of Functional Programs

We investigate how, in a natural deduction setting, we can specify concisely a wide variety of tasks that manipulate programs as data objects. This study will provide us with a better understanding of various kinds of manipulations of programs and also an operational understanding of numerous features and properties of a rich functional programming language. We present a technique, inspired by structural operational semantics and natural semantics, for specifying properties of, or operations on, programs. Specifications of this sort are presented as sets of inference rules and are encoded as clauses in a higher-order, intuitionistic meta-logic. Program properties are then proved by constructing proofs in this meta-logic. We argue the following points regarding these specifications and their proofs: (i) the specifications are clear and concise and they provide intuitive descriptions of the properties being described; (ii) a wide variety of program analysis tools can be specified in a single unified framework, and thus we can investigate and understand the relationship between various tools; (iii) proof theory provides a well-established and formal setting in which to examine meta-theoretic properties of these specifications; and (iv) the meta-logic we use can be implemented naturally in an extended logic programming language and thus we can produce experimental implementations of the specifications. We expect that our efforts will provide new perspectives and insights for many program manipulation tasks

Selective Strictness and Parametricity in Structural Operational Semantics, Inequationally

Parametric polymorphism constrains the behavior of pure functional pro-grams in a way that allows the derivation of interesting theorems about them solely from their types, i.e., virtually for free. The formal background of such ‘free theorems’ is well developed for extensions of the Girard-Reynolds polymorphic lambda calculus by algebraic datatypes and general recursion, provided the resulting calculus is endowed with either a purely strict or a purely nonstrict semantics. But modern functional languages like Clean and Haskell, while using nonstrict evaluation by default, also provide means to enforce strict evaluation of subcomputations at will. The resulting selective strictness gives the advanced programmer explicit control over evaluation order, but is not without semantic consequences: it breaks standard parametricity results. This paper develops an operational semantics for a core calculus supporting all the language features emphasized above. Its main achievement is the characterization of observational approximation with respect to this operational semantics via a carefully constructed logical relation. This establishes the formal basis for new parametricity results, as illustrated by several example applications, including the first complete correctness proof for short cut fusion in the presence of selective strictness. The focus on observational approximation, rather than equivalence, allows a finer-grained analysis of computational behavior in the presence of selective strictness than would be possible with observational equivalence alone

Abstract interpretation

Abstract. Abstract interpretation has been widely used for verifying properties of computer systems. Here, we present a way to extend this framework to the case of probabilistic systems. The probabilistic abstraction framework that we propose allows us to systematically lift any classical analysis or verification method to the probabilistic setting by separating in the program semantics the probabilistic behavior from the (non-)deterministic behavior. This separation provides new insights for designing novel probabilistic static analyses and verification methods. We define the concrete probabilistic semantics and propose different ways to abstract them. We provide examples illustrating the expressiveness and effectiveness of our approach.

The Correctness of an Optimized Code Generation

For a functional programming language with a lazy standard semantics, we define a strictness analysis by means of abstract interpretation. Using the information from the strictness analysis we are able to define a code generation which avoids delaying the evaluation of the argument to an application, provided that the corresponding function is strict.To show the correctness of the code generation, we adopt the framework of logical relations and define a layer of predicates which finally will ensure that the code generation is correct with respect to the standard semantics