Sharing is caring: combination of theories

2010

Abstract One of the main shortcomings of the traditional methods for combining theories is the complexity of guessing the arrangement of the variables shared by the individual theories. This paper presents a reformulation of the Nelson-Oppen method that takes into account explicit equality propagation and can ignore pairs of shared variables that the theories do not care about. We show the correctness of the new approach and present care functions for the theories of uninterpreted functions and the theory of arrays.

1996

Decision procedures are increasingly being employed for deciding or simplifying propositional combinations of ground equalities involving uninterpreted function symbols, linear arithmetic, arrays, and other theories. Two approaches for constructing decision procedures for combinations of ground theories were pioneered in the late seventies. In the approach of Nelson and Oppen, decision procedures for two disjoint theories are combined by introducing variables to name subterms and iteratively propagating any deduced equalities between variables from one theory to another. Shostak employs a different approach that works far more efficiently in practice. He uses an optimized implementation of the congruence closure procedure for ground equality over uninterpreted function symbols to combine theories that are canonizable and algebraically solvable. Many useful theories have these properties. Shostak's algorithm is subtle and complex and his description of this procedure is lacking in rigor. We present, for the first time, a careful development and clarification of Shostak's procedure that corrects several mistakes in Shostak's original presentation. Our analysis serves as a useful basis for the implementation, extension, and further optimization of Shostak's decision procedure.

1998

Annals of Mathematics …, 2009

… Margherita Ligure, Italy, April 8-10 …, 2002

1977

We restrict attention to the validity problem for unquantied dis- junctions of literals (possibly negated atomic formulae) over the do- main of integers, or what is just as good, the satisability problem for unquantied conjunctions. When = is the only predicate symbol and all function symbols are left uninterpreted, or when is the only pred- icate symbol (taking its standard

2006

Logic in Computer …, 2001

Electronic Notes in Theoretical Computer Science, 2007

Journal of Physics: Conference Series, 2020

A primary challenge of natural sciences in the new millennium is to cure the gap between metaphysics and empiricism-and puzzle out the obstacles to a unified theory and an understandable picture of reality. Antique science flourished via its strong philosophical impact but faded away due to the lack of supporting empirical science. The fast development of mathematical physics has led to the other opposite; theories are diversified, they are more like mathematical descriptions of observations; they provide precise predictions but lack a solid metaphysical basis and an understandable picture of reality. Anyway, modern science has increased our understanding of physics from elementary particles to cosmological structures and produced information that allows re-evaluation of the basis. By switching from an observer-oriented perspective to a system-oriented perspective, any local object is related to the rest of space and relativity appears as a direct consequence of the conservation of total energy in the system-without scarifying the absolute time and distance essential for human comprehension. Such a holistic approach has led to the Dynamic Universe theory (DU). After maturing for the last twenty-five years, DU produces precise, well-tested predictions for local and cosmological observables and an uncontradictory linkage to quantum mechanics.