Jape is a program which supports the step-by-step interactive development of proofs in formal logics, in the style of proofs-on-paper. It is uncommitted to any particular logic and is customized by a description of a collection of inference rules and the syntax of judgements. It works purely at the surface syntactic level, as a person working on paper might. In that spirit it makes use of expli...

Proofs in standard logical calculi have a simple structure (mostly a sequence, tree or set of related formulas). Therefore, formal proofs are hard to understand or to present in an intelligible way. The Block Calculus for rst order logic introduced in this paper is a variant of natural deduction that has highly structured proofs. These proofs can be presented in many ways by hiding blocks of su...

A well-known problem with Hintikka-style logics of knowledge is that of logical omniscience. One knows too much. This breaks down into two subproblems: one knows all tautologies, and one’s knowledge is closed under consequence. A way of addressing the second of these is to move from knowledge simpliciter, to knowledge for a reason. Then, as consequences become ‘further away’ from one’s basic kn...

This paper describes a generalization of the LowerUnits algorithm [10] for the compression of propositional resolution proofs. The generalized algorithm, called LowerUnivalents, is able to lower not only units but also subproofs of non-unit clauses, provided that they satisfy some additional conditions. This new algorithm is particularly suited to be combined with the RecyclePivotsWithIntersect...

This document gives the formal proofs of the following results about the sums of two and four squares: 1. Any prime number p ≡ 1 mod 4 can be written as the sum of two squares. 2. (Lagrange) Any natural number can be written as the sum of four

Problem 11 is due to P. E. Howard.’ The results of Problems 12 and 13 were announced by Pincus. The independence result in Theorem 7.1 1 was proved by Levy [I9621 for ZFA and transferred to Z F by Pincus [ 1969,1973b1, whose is also the observation in Problem 15.’ Example 7.12 in Section 7.4 is due to Tarski. Conditions (S) and (D) were formulated by Mostowski [1945], who proved the independenc...

The prime number theorem, established by Hadamard and de la Vallée Poussin independently in 1896, asserts that the density of primes in the positive integers is asymptotic to 1/ ln x. Whereas their proofs made serious use of the methods of complex analysis, elementary proofs were provided by Selberg and Erdös in 1948. We describe a formally verified version of Selberg’s proof, obtained using th...

Code-based authentication schemes are very fast and efficient for implementation in small devices. The security of these schemes and the size of underlying coding matrices are often the main barriers to their use in practice. In this paper, we remove the security flaw of a classical codebased authentication scheme and modify it to achieve a secure code-based authentication scheme. We provide a ...

We propose a log signing scheme that enables (a) verification of the integrity of the whole log, and (b) presentation of any record, along with a compact proof that the record has not been altered since the log was signed, without leaking any information about the contents of other records in the log. We give a formal proof of the security of the proposed scheme, discuss practical consideration...