Abstract
We propose a new semantics for rewrite systems based on interpreting rewrite rules as inequations between terms in an ordered algebra. In particular, we show that the algebra of normal forms in a terminating system is a uniquely minimal covering of the term algebra. In the non-terminating case, the existence of this minimal covering is established in the completion of an ordered algebra formed by rewriting sequences. We thus generalize the properties of normal forms for non-terminating systems to this minimal covering. These include the existence of normal forms for arbitrary rewrite systems, and their uniqueness far confluent systems, in which case the algebra of normal forms is isomorphic to the canonical quotient algebra associated with the rules when seen as equations. This extends the benefits of algebraic semantics to systems with non-deterministic and non-terminating computations. We first study properties of abstract orders, and then instantiate these to term rewriting systems.

Abstract
This paper revisits the idea of slicing programs based on their axiomatic semantics, rather than using criteria based on control/data dependencies. We show how the forward propagation of preconditions and the backward propagation of postconditions can be combined in a new slicing algorithm that is more precise than the existing specification-based algorithms. The algorithm is based on (a) a precise test for removable statements, and (b) the construction of a slice graph, a program control flow graph extended with semantic labels and additional edges that "short-circuit" removable commands. It improves on previous approaches in two aspects: it does not fail to identify removable commands; and it produces the smallest possible slice that can be obtained (in a sense that will be made precise). Iteration is handled through the use of loop invariants and variants to ensure termination. The paper also discusses in detail applications of these forms of slicing, including the elimination of (conditionally) unreachable and dead code, and compares them to other related notions.

Abstract
This paper revisits the idea of slicing programs based on their axiomatic semantics, rather than using criteria based on control/data dependencies. We show how the forward propagation of preconditions and the backward propagation of post conditions can be combined in a new slicing algorithm that is more precise than the existing specification-based algorithms. The algorithm is based on (i) a precise test for removable statements, and (ii) the construction of a slice graph, a program control flow graph extended with semantic labels. It improves on previous approaches in two aspects: it does not fail to identify removable commands; and it produces the smallest possible slice that can be obtained (in a sense that will be made precise). The paper also reviews in detail, through examples, the ideas behind the use of preconditions and post conditions for slicing programs. © 2010 IEEE.