We describe a proof by a computer program of the unsolvability of the halting problem. The halting problem is posed in a constructive, formal language. The computational paradigm formalized is Pure LISP, not Turing machines. The machine was led to the proof by the authors, who suggested certain function definitions and stated certain intermediate lemmas. The machine checked that every suggested definition was admissible and the machine proved the main theorem and every lemma. We believe this is the first instance (...) of a machine checking that a given problem is not solvable by machine. (shrink)

The development of mathematics toward greater precision has led, as is well known, to the formalization of large tracts of it, so that one can prove any theorem using nothing but a few mechanical rules. -- Godel [11].

We describe an extension to our quantifier-free computational logic to provide the expressive power and convenience of bounded quantifiers and partial functions. By quantifier we mean a formal construct which introduces a bound or indicial variable whose scope is some subexpression of the quantifier expression. A familiar quantifier is the Σ operator which sums the values of an expression over some range of values on the bound variable. Our method is to represent expressions of the logic as objects in the (...) logic, to define an interpreter for such expressions as a function in the logic, and then define quantifiers as "mapping functions." The novelty of our approach lies in the formalization of the interpreter and its interaction with the underlying logic. Our method has several advantages over other formal systems that provide quantifiers and partial functions in a logical setting. The most important advantage is that proofs not involving quantification or partial recursive functions are not complicated by such notions as "capturing," "bottom," or "continuity." Naturally enough, our formalization of the partial functions is nonconstructive. The theorem prover for the logic has been modified to support these new features. We describe the modifications. The system has proved many theorems that could not previously be stated in our logic. Among them are. (shrink)

We present an initial and elementary investigation of the formal specification and mechanical verification of programs that interact with environments. We describe a formal, mechanically produced proof that a simple, real time control program keeps a vehicle on a straightline course in a variable crosswind. To formalize the specification we define a mathematical function which models the interaction of the program and its environment. We then state and prove two theorems about this function: the simulated vehicle never gets farther than (...) three units away from the intended course and homes to the course if the wind ever remains steady for at least four sampling intervals. (shrink)

Meta-level reasoning was added to the Boyer-Moore theorem prover by 1979. We have recently re-implemented and, we think, slightly improved our approach to meta-theoretic reasoning in the Acl2 theorem prover. However, there is lots of room for much more substantial improvement. Our talk will consist of 3 parts: review, recent results, and general remarks.

How are the properties of computer programs proved? We discuss three approaches in this article: inductive invariants, functional semantics, and explicit semantics. Because the first approach has received by far the most attention, it has produced the most impressive results to date. However, the field is now moving away from the inductive invariant approach.