Frank M. Brown

Optimization of the ABCD Formula Used for Melanoma Diagnosis

The ABCD formula is used for computing a new attribute, called TDS, to help with melanoma diagnosis. In our research four discretization techniques were used, two of them never published before. We found four corresponding new ABCD formulas to compute TDS by applying more than 163 thousand experiments of variable ten-fold cross validation. Diagnosis of melanoma with each of these new ABCD formulas, when used with an appropriate discretization technique, is significantly more accurate (with the level of significance 5%) than diagnosis using the traditional ABCD formula. Finally, the rule sets, induced from data sets obtained using four new ABCD formulas and the traditional ABCD formula, were graded by an experienced melanoma diagnostician.

An experimental logic based on the fundamental deduction principle

Abstract Experimental logic can be viewed as a branch of logic dealing with the actual construction of useful deductive systems and their application to various scientific disciplines. In this paper we describe an experimental deductive system called the SYMbolic EVALuator (i.e. SYMEVAL) which is based on a rather simple, yet startling principle about deduction, namely that deduction is fundamentally a process of replacing expressions by logically equivalent expressions. This principle applies both to logical and domain-dependent axioms and rules. Unlike more well-known logical inference systems which do not satisfy this principle, herein is described a system of logical axioms and rules called the SYMMETRIC LOGIC which is based on this principle. Evidence for this principle is given by proving theorems and performing deduction in the areas of set theory, logic programming, natural language analysis, program verification, automatic complexity analysis, and inductive reasoning.

A Commonsense Theory of Nonmonotonic Reasoning

A commonsense theory of nonmonotonic reasoning is presented which models our intuitive ability to reason about defaults. The concepts of this theory do not involve mathematical fixed points, but instead are explicitly defined in a monotonic modal quantificational logic which captures the modal notion of logical truth. The axioms and inference rules of this modal logic are described herein along with some basic theorems about nonmonotonic reasoning. An application to solving the frame problem in robot plan formation is presented.

A MODAL LOGIC FOR THE REPRESENTATION OF KNOWLEDGE

ABSTRACT A modal logic which axiomatizes the modal concept of logical truth is presented. This modal logic is stronger than S5 modal logic and contains axioms allowing individual facts to be proven to be logically possible with respect to a body of knowledge. An extentional semantics is described and the soundness of the logic is proven.

ACTION, REFLECTIVE POSSIBILITY, AND THE FRAME PROBLEM

A solution to the frame problem is proposed. This solution is based on the idea that the frame problem can essentially be reduced to the concepts of reflection and logical possibility. Both a very general action-frame law and a more specific sequential action-frame law are described and exemplified.

Solving modal equivalences

Nonmonotonic logics such as reflective logic, default logic, and autoepistemic logic, are usually defined in terms of set-theoretic fixedpoint equations defined over deductively closed sets of sentences of first order logic. Such systems may also be represented as necessary equivalences in a modal logic stronger than S5 with the added advantages that such representations may be generalized to allow quantified variables crossing modal scopes whereby both the Barcan formula and its converse hold. Herein, we discuss the problem of solving such necessary equivalences. Methods for solving equations and subsidiary methods for deducing what is logically possible are discussed. Solutions obtained by this method are then compared to related results obtained in the literature by circumscription in 2/sup nd/ order logic since the disjunction of all the solutions of a necessary equivalences containing just normal defaults is equivalent to circumscription in 2/sup nd/ order logic.

A theory of nonmonotonic planning

A theory of planning that uses nonmonotonic reasoning based on the modal quantificational logic Z is developed. It does forward reasoning and backward planning to reach the goal. The theory we are proposing uses the frame axiom and the modal quantifieational logic Z to propagate the facts from the current situation to the next situation. Only the explicit results of an action are provided; no delete list is needed. The facts are automatically added and deleted from one situation to the next by nonmonotonic reasoning as the actions are performed. A couple of examples are given to illustrate the technique.

Logistica 2.0: A technology for implementing automatic Deduction systems

The Logistica 2.0 Deduction System Implementation Technology is a programming language extension to R5RS Scheme which automatically computes all possible combinations of values of multiply valued subexpressions. Multiple values are generated by multiple definitions of a symbol and by allowing Second Order patterns such as segment variables, which may match in different ways, as the parameters of lambda abstractions. This technology is briefly illustrated with an extensible deduction system involving the derivation of an axiom schema.

A comparison of autoepistemic logic and default logic both generalized so as to allow quantified variables to cross modal scopes

Default logic and autoepistemic logic are both generalized so as to allow universally quantified variables to cross modal scopes whereby the Barcan formula and its converse hold. This is done by representing the fixed point equation for default logic and the fixed point equation for the kernel of autepistemic logic both as reflective equivalences of the modal quantificational Logic Z. The two resulting systems, called quantified default logic and quantified autepistemic logic, are then compared by deriving metatheorems of Z that express their relationships. The main result is to show that every solution to the reflective equivalence for quantified default logic is a strongly grounded solution to the reflective equivalence for quantified autepistemic logic. This generalizes previous work relating default logic and autepistemic logic to the case where universally quantified variables cross modal scopes. It is further shown that quantified default logic and quantified autepistemic logic have exactly the same solutions when no default has an entailment condition. Finally it is noted that quantified default logic and quantified autepistemic logic are particularly powerful logics since the disjunction of all the solutions of a particular subcase of each is equivalent to parallel circumscription with circumscribed, variable, and fixed predicates.

Decision procedures for the propositional cases of Second Order Logic and Z Modal Logic representations of a first Order L-Predicate nonmonotonic Logic

Decision procedures for the propositional cases of two different logical representations for an L-Predicate Logic generalizing Autoepistemic Logic to handle quantified variables over modal scopes are described. The first representation is Second Order Logic. The second is Z Modal Logic which extends its S5 modal laws with laws stating what is logically possible. It is suggested that certain problems are more easily solved using one representation whereas other problems are more easily solved using the other.