Models of Relevant Arithmetic

It is well known that the relevant arithmetic R# admits finite models whose domains are the integers modulo n rather than the expected natural numbers. Less well appreciated is the fact that the logic of these models is much more subtle than that of the three-valued structure in which they are usually presented. In this paper we consider the DeMorgan monoids in which R# can be modelled, deriving a fairly complete account of those modelling the stronger arithmetic RM# modulo n and a partial account for the case of R# modulo a prime number. The more general case in which the modulus is arbitrary is shown to lead to infinite propositional structures even with the additional constraint that '0=1' implies everything

Models of Relevant Arithmetic

It is well known that the relevant arithmetic R# admits finite models whose domains are the integers modulo n rather than the expected natural numbers. Less well appreciated is the fact that the logic of these models is much more subtle than that of the three-valued structure in which they are usually presented. In this paper we consider the DeMorgan monoids in which R# can be modelled, deriving a fairly complete account of those modelling the stronger arithmetic RM# modulo n and a partial account for the case of R# modulo a prime number. The more general case in which the modulus is arbitrary is shown to lead to infinite propositional structures even with the additional constraint that '0=1' implies everything

Set-theoretic duality: A fundamental feature of combinatorial optimisation

The duality between conflicts and diagnoses in the field of diagnosis, or between plans and landmarks in the field of planning, or between unsatisfiable cores and minimal co-satisfiable sets in SAT or CSP solving, has been known for many years. Recent wo

Classical versions of BCI, BCK and BCIW logics

The question is, is there a formula X, independent of B,C,K1, I and W that creates distinct subclassical logics BCIX,BCKX and BCIWX, while BCKWX is the full classical implicational logic TV

A Logic for Vagueness

This paper presents F, substructural logic designed to treat vagueness. Weaker than Lukasiewicz’s infinitely valued logic, it is presented first in a natural deduction system, then given a Kripke semantics in the manner of Routley and Meyer's ternary relational semantics for R and related systems, but in this case, the points are motivated as degrees to which the truth could be stretched. Soundness and completeness are proved, not only for the propositional system, but also for its extension with first-order quantifiers. The first-order models allow not only objects with vague properties, but also objects whose very existence is a matter of degree

Incremental Lower Bounds for Additive Cost Planning Problems

We present a novel method for computing increasing lower bounds on the cost of solving planning problems, based on repeatedly solving and strengthening the delete relaxation of the problem. Strengthening is done by compiling select conjunctions into new atoms, similar to the P*m construction. Because it does not rely on search in the state space, this method does not suffer some of the weaknesses of admissible search algorithms and therefore is able to prove higher lower bounds for many problems that are too hard for optimal planners to solve, thus narrowing the gap between lower bound and cost of the best known plan, providing better assurances of plan quality

Computers and relevant logic : a project in computing matrix model structures for propositional logics

I present and discuss four classes of algorithm designed as solutions to the problem of generating matrix representations of model structures for some non-classical propositional logics. I then go on to survey the output from implementations of these algorithms and finally exhibit some logical investigations suggested by that output. All four algorithms traverse a search tree depthfirst. In the case of the first and fourth methods the tree is fixed by imposing a lexicographic order on possible matrices, while the second and third create their search tree dynamically as the job progresses. The first algorithm is a simple "backtrack" with some pruning of the tree in response to refutations of possible matrices. The fourth, the most efficient we have for time, maximises the amount of pruning while keeping the same basic form. The second, which uses a large number of special properties of the logics in question, and so requires some logical and algebraic knowledge on the part of the programmer, finds the matrices at the tips of branches only, while the third, due to P.A. Pritchard, is far easier to program and tests a matrix at every node of the search tree. The logics with which I am concerned are in the "relevant" group first seriously investigated by A.R. Anderson and N.D. Belnap (see their Entailment: the logic of relevance and necessity, 1975). The most surprising observation in my preliminary survey of the numbers of matrices validating such systems is that the typical models are not much like the models normally taken as canonical for the logics. In particular the proportion of inconsistent models (validating some cases of the scheme 'A & ~A') is much higher than might have been expected. Among the logical investigations already suggested by the quasi-empirical data now available in the form of matrices are some work on the system R-W, including my theorem, proved in chapter 2.3, that with the law of excluded middle it suffices to trivialise naive set theory, and the little-noticed subject of Ackermann constants (sentential constants) in these logics. The formula which collapses naive set theory in R-W plus A v ~A is the most damaging set-theoretic antinomy known. The theorem that there are at least 3088 Ackermann constants in the logic R (chapter 2.4) could not reasonably have been proved without the aid of a computer. My major conclusion is that this work on applications of computers in logical research has reached a point where we are able not only to relieve logicians of some drudgery, but to suggest theorems and insights of new and possibly important kinds