On Spatial Conjunction as Second-Order Logic

Spatial conjunction is a powerful construct for reasoning about dynamically allocated data structures, as well as concurrent, distributed and mobile computation. While researchers have identified many uses of spatial conjunction, its precise expressive power compared to traditional logical constructs was not previously known. In this paper we establish the expressive power of spatial conjunction. We construct an embedding from first-order logic with spatial conjunction into second-order logic, and more surprisingly, an embedding from full second order logic into first-order logic with spatial conjunction. These embeddings show that the satisfiability of formulas in first-order logic with spatial conjunction is equivalent to the satisfiability of formulas in second-order logic. These results explain the great expressive power of spatial conjunction and can be used to show that adding unrestricted spatial conjunction to a decidable logic leads to an undecidable logic. As one example, we show that adding unrestricted spatial conjunction to two-variable logic leads to undecidability. On the side of decidability, the embedding into second-order logic immediately implies the decidability of first-order logic with a form of spatial conjunction over trees. The embedding into spatial conjunction also has useful consequences: because a restricted form of spatial conjunction in two-variable logic preserves decidability, we obtain that a correspondingly restricted form of second-order quantification in two-variable logic is decidable. The resulting language generalizes the first-order theory of boolean algebra over sets and is useful in reasoning about the contents of data structures in object-oriented languages.Comment: 16 page

Equivalence-Checking on Infinite-State Systems: Techniques and Results

The paper presents a selection of recently developed and/or used techniques for equivalence-checking on infinite-state systems, and an up-to-date overview of existing results (as of September 2004)

Computational Processes and Incompleteness

We introduce a formal definition of Wolfram's notion of computational process based on cellular automata, a physics-like model of computation. There is a natural classification of these processes into decidable, intermediate and complete. It is shown that in the context of standard finite injury priority arguments one cannot establish the existence of an intermediate computational process

Strong Turing Degrees for Additive BSS RAM's

For the additive real BSS machines using only constants 0 and 1 and order tests we consider the corresponding Turing reducibility and characterize some semi-decidable decision problems over the reals. In order to refine, step-by-step, a linear hierarchy of Turing degrees with respect to this model, we define several halting problems for classes of additive machines with different abilities and construct further suitable decision problems. In the construction we use methods of the classical recursion theory as well as techniques for proving bounds resulting from algebraic properties. In this way we extend a known hierarchy of problems below the halting problem for the additive machines using only equality tests and we present a further subhierarchy of semi-decidable problems between the halting problems for the additive machines using only equality tests and using order tests, respectively