16,762 research outputs found
A HOL basis for reasoning about functional programs
Domain theory is the mathematical theory underlying denotational semantics. This thesis presents a formalization of domain theory in the Higher Order Logic (HOL) theorem proving system along with a mechanization of proof functions and other tools to support reasoning about the denotations of functional programs. By providing a fixed point operator for functions on certain domains which have a special undefined (bottom) element, this extension of HOL supports the definition of recursive functions which are not also primitive recursive. Thus, it provides an approach to the long-standing and important problem of defining non-primitive recursive functions in the HOL system. Our philosophy is that there must be a direct correspondence between elements of complete partial orders (domains) and elements of HOL types, in order to allow the reuse of higher order logic and proof infrastructure already available in the HOL system. Hence, we are able to mix domain theoretic reasoning with reasoning in the set theoretic HOL world to advantage, exploiting HOL types and tools directly. Moreover, by mixing domain and set theoretic reasoning, we are able to eliminate almost all reasoning about the bottom element of complete partial orders that makes the LCF theorem prover, which supports a first order logic of domain theory, difficult and tedious to use. A thorough comparison with LCF is provided. The advantages of combining the best of the domain and set theoretic worlds in the same system are demonstrated in a larger example, showing the correctness of a unification algorithm. A major part of the proof is conducted in the set theoretic setting of higher order logic, and only at a late stage of the proof domain theory is introduced to give a recursive definition of the algorithm, which is not primitive recursive. Furthermore, a total well-founded recursive unification function can be defined easily in pure HOL by proving that the unification algorithm (defined in domain theory) always terminates; this proof is conducted by a non-trivial well-founded induction. In such applications, where non-primitive recursive HOL functions are defined via domain theory and a proof of termination, domain theory constructs only appear temporarily
The complex geomety of a domain related to -synthesis
We describe the basic complex geometry and function theory of the {\em
pentablock} , which is the bounded domain in given
by where denotes
the open unit ball in the space of complex matrices. We prove
several characterizations of the domain. We describe its distinguished boundary
and exhibit a -parameter group of automorphisms of . We show
that is intimately connected with the problem of -synthesis
for a certain cost function on the space of matrices defined
in connection with robust stabilization by control engineers. We demonstrate
connections between the function theories of and . We
show that is polynomially convex and starlike.Comment: 36 pages, 2 figures. This version contains corrections of some
inaccuracies and an expanded argument for Proposition 12.
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