Atomic formula

Abstract: A term is an individual constant or variable or an n-adic function letter followed by n terms. An atomic formula is an n-adic predicate letter followed by n terms. A literal is an atomic formula or the negation thereof. A clause is a set of literals and is thought of as representing the universally-quantified disjunction of its members. It will sometimes be notationally convenient1 to distinguish between the empty clause □, viewed as a clause, and ‘other’ empty sets such as the empty set of clauses, even though all these empty sets are the same set-theoretic object o. A ground clause (term, literal) is one with no variables. A clause C’ (literal, term) is an instance of another clause C (literal, term) if there is a uniform replacement of the variables in C by terms that transform C into C’.

Abstract: An automatic analysis method for first-order logic with sets and relations is described. A first-order formula is translated to a quantifier-free boolean formula, which has a model when the original formula has a model within a given scope (that is, involving no more than some finite number of atoms). Because the satisfiable formulas that occur in practice tend to have small models, a small scope usually suffices and the analysis is efficient.The paper presents a simple logic and gives a compositional translation scheme. It also reports briefly on experience using the Alloy Analyzer, a tool that implements the scheme.

Abstract: Most branches of logic may be studied by means of two different (although related) methods or sets of methods which are usually called syntactical and semantical, respectively. In this paper, I shall outline some basic ideas of a semantical theory of modal logic, including quantified modal logic. Since a fuller treatment is easy to carry out on the basis of this outline, I shall omit most of the proofs.1

Abstract: Prepositional logic formulas containing implications can suffer from antecedent failure, in which the formula is true trivially because the pre-condition of the implication is not satisfiable. In other words, the post-condition of the implication does not affect the truth value of the formula. We call this a vacuous pass, and extend the definition of vacuity to cover other kinds of trivial passes in temporal logic. We define w-ACTL, a subset of CTL and show by construction that for every w-ACTL formula ϕ there is a formula w(ϕ), such that: both ϕ and w(ϕ) are true in some model M iff ϕ passes vacuously. A useful side-effect of w(ϕ) is that if false, any counter-example is also a non-trivial witness of the original formula ϕ.