Tomasz Pietrzykowski

• Dalhousie University

this report we give formal presenta4 tions of the parametrization and instantiation methods. To do so requires formalizing some common assumptions about the form of system descriptions. General principles for organizing system descriptions have been discussed by [Genesereth 1984]. We make explicit the relationships between the background theory (lo...

One method of proving theorems in Horn clause theories is surface deduction (also known as the modification method). Surface deduction yields interpretations of unification failures in terms of residual hypotheses needed for unification to succeed. This suggests that it can be used for abductive reasoning with equality. In surface deduction the inp...

Equality can be added to logic programming by using surface deduction. Surface deduction yields interpretations of unification failures in terms of residual hypotheses needed for unification to succeed. It can therefore be used for abductive reasoning with equality. In surface deduction the input clauses are first transformed to a flat form (involv...

A critique of textual programming languages and software development environments is presented, linking them to the development of hardware and discussing their connection with natural languages and mathematical formalisms. The authors then outline criteria for modern integrated programming languages and environments based on the use of graphics. T...

A definition is presented of the concept of 'cause' for an observation and of the properties basicness, minimality, nontriviality, and acceptability which characterize useful causes. A procedure is presented for computing, from an observation, causes with these properties. The application of this method to the diagnosis problem is then illustrated...

We describe a uniform deduction mechanism called surface deduction, for Horn clauses with equality in a special form called flat. In this flat form arguments of non-equality predicates must be variables, and each equality has at most one non-variable subterm: this subterm can occur only on the right hand side of the equality predicate, and itself h...

A formal definition is presented for the cause for an event with a given knowledge base. An event is understood to be any closed formula; a knowledge base is a set of closed formulae; and a cause is a prenex formula, the matrix of which is a conjunction of literals. The properties of minimality, basicness, consistency and nontriviality are defined...

PROGRAPH is a functional, data flow oriented language, expressed graphically in the form of pictographs. Pictographs are created by computer graphics and are directly executable. PROGRAPH contains a data base subsystem which is also functional in nature (including update operations). This report provides a presentation of the main features of the l...

This paper develops a method of mechanical deduction based on a graphical representation of the structure of proofs. Attempts to find a refutation(s) are recorded in the form of plans, corresponding to portions of an AND/OR graph search space and representing a purely deductive structure of derivation. This method can be applied to any initial base...

The graphical functional language PROGRAPH is briefly described, followed by a more extensive discussion of prograph development facilities. In particular, the PROGRAPH editor/interpreter is described, which allows prographs to be simultaneously constructed and executed, using sophisticated graphics in interactions with the user. The micro-PROLOG i...

An algorithm is presented which, for an arbitrary literal containing Skolem functions, outputs a set of closed quantified literals with the following properties. If a and b are formulae we define a ⊃ b iff {sk(a),dsk(b)} is unifiable where sk denotes Skolemization and dsk denotes the dual operation, where the roles of ∀ and ∃ are reversed. If d is...

The paper presents a method of mechanical deduction along the lines indicated in [3]. Attempts to find refutation (s) are recorded in the form of triples: plan, constraints, conflicts. A plan corresponds to a portion of AND/OR graph search space and represents purely deductive structure of derivation. Constraints form a graph recording the attempts...

The paper presents the data structure enabling an implementation of an efficient backtracking method for plan-based deduction. The structure consists of two parts. Static structure combines information about the initial set of clauses and unifiers. Dynamic structure records information about the history of the deduction process (plan graph), about...

A proof procedure is described that relies on the construction of certain directed graphs called ``deduction plans.'' Plans represent the structure of proofs in such a way that problem reduction may be used without imposing any ordering on the solution of subproblems, as required by other systems. The structure also allows access to all clauses ded...

Consider the following nonlinear optimization problem: find a local minimum $x_0 $ of the function f on the set $F = \{ {x \in R^n |\phi _i (x) \geqq 0,i = 1, \cdots ,M} \}$. A penalty function p is defined as follows: \[ p(x) = \mu f(x) - \sum _{i = 1}^M {\min \left( {0,\phi _i (x)} \right)} ,\qquad x \in R^n .\] Under some natural assumptions p h...

This paper deals with the problems of applying a unification procedure in a mechanization of the full type theory. The early sections describe the unification problem and an algorithm which produces a complete set of unifiers. The special case of the 2nd order problem is differentiated. Then there is presented an extensive proof of completeness of...

A generalization of the resolution method for higher order logic is presented. The languages acceptable for the method are phrased in a theory of types of order w (all finite types)—including the &lgr;-operator, propositional functors, and quantifiers. The resolution method is, of course, a machine-oriented theorem search procedure based on refutat...

We present a complete unification algorithm for logic with the language of all finite types. In addition, we show how this procedure can be used with a generalized resolution principle which does not require expressions to be pre-Skolemized. Both of these steps are essential for an effective and complete mechanization of higher order logic.

The paper discusses conditions of the convergence of a potential method. The method consists of approximating a constrained maximum by unconstrained maxima of a potential function. A proof of the convergence is given when dealing with the local nonsingular maximum of a locally concave functional on a locally convex set in a Banach reflexive space....

In the paper there is presented a method of reducing the constrained maximization problem to an unconditional one. The method consists of introducing a function, called potential, which at an unconditional maximum converges to the constrained maximum when a positive parameter of the potential tends to zero. A proof of such convergence for the local...

The main result of the paper consists of the theorem that under certain, natural assumptions the local conditional maximum $x_0 $ of the function f on the set \[ A = \left\{ {x \notin R^n |\phi _i (x) \geqq 0,\psi _j (x) = 0,i = 1, \cdots ,k,j = 1, \cdots ,l} \right\} \] is identical with the unconditional maximum of the potential function \[ p(x,\...