41 research outputs found

    Completeness and properness of refinement operators in inductive logic programming

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    AbstractWithin Inductive Logic Programming, refinement operators compute a set of specializations or generalizations of a clause. They are applied in model inference algorithms to search in a quasi-ordered set for clauses of a logical theory that consistently describes an unknown concept. Ideally, a refinement operator is locally finite, complete, and proper. In this article we show that if an element in a quasi-ordered set 〈S, ≥〉 has an infinite or incomplete cover set, then an ideal refinement operator for 〈S, ≥〉 does not exist. We translate the nonexistence conditions to a specific kind of infinite ascending and descending chains and show that these chains exist in unrestricted sets of clauses that are ordered by θ-subsumption. Next we discuss how the restriction to a finite ordered subset can enable the construction of ideal refinement operators. Finally, we define an ideal refinement operator for restricted θ-subsumption ordered sets of clauses

    Automating Diagrammatic Proofs of Arithmetic Arguments

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    Centre for Intelligent Systems and their ApplicationsThis thesis is on the automation of diagrammatic proofs, a novel approach to mechanised mathematical reasoning. Theorems in automated theorem proving are usually proved by formal logical proofs. However, there are some conjectures which humans can prove by the use of geometric operations on diagrams that somehow represent these conjectures, so called diagrammatic proofs. Insight is often more clearly perceived in these diagrammatic proofs than in the algebraic proofs. We are investigating and automating such diagrammatic reasoning about mathematical theorems.Concrete rather than general diagrams are used to prove ground instances of a universally quantified theorem. The diagrammatic proof in constructed by applying geometric operations to the diagram. These operations are in the inference steps of the proof. A general schematic proof is extracted from the ground instances of a proof. it is represented as a recursive program that consists of a general number of applications of geometric operations. When gien a particular diagram, a schematic proof generates a proof for that diagram. To verify that the schematic proof produces a correct proof of the conjecture for each ground instance we check its correctness in a theory of diagrams. We use the constructive omega-rule and schematic proofs to make a translation from concrete instances to a general argument about the diagrammatic proof.The realisation of our ideas is a diagrammatic reasoning system DIAMOND. DIAMOND allows a user to interactively construct instances of a diagrammatic proof. It then automatically abstracts these into a general schematic proof and checks the correctness of this proof using an inductive theorem prover

    String unification is essentially infinitary

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    A unifier of two terms s and t is a substitution sigma such that ssigma=tsigma and for first-order terms there exists a most general unifier sigma in the sense that any other unifier delta can be composed from sigma with some substitution lambda, i.e. delta=sigmacirclambda. This notion can be generalised to E-unificationwhere E is an equational theory, =_{E} is equality under E andsigmaa is an E-unifier if ssigma =_{E}tsigma. Depending on the equational theory E, the set of most general unifiers is always a singleton (as above), or it may have more than one, either finitely or infinitely many unifiers and for some theories it may not even exist, in which case we call the theory of type nullary. String unification (or Löb\u27s problem, Markov\u27s problem, unification of word equations or Makanin\u27s problem as it is often called in the literature) is the E-unification problem, where E = {f(x,f(y,z))=f(f(x,y),z)}, i.e. unification under associativity or string unification once we drop the fs and the brackets. It is well known that this problem is infinitary and decidable. Essential unifiers, as introduced by Hoche and Szabo, generalise the notion of a most general unifier and have a dramatically pleasant effect on the set of most general unifiers: the set of essential unifiers is often much smaller than the set of most general unifiers. Essential unification may even reduce an infinitary theory to an essentially finitary theory. The most dramatic reduction known so far is obtained for idempotent semigroups or bands as they are called in computer science: bands are of type nullary, i.e. there exist two unifiable terms s and t, but the set of most general unifiers is not enumerable. This is in stark contrast to essential unification: the set of essential unifiers for bands always exists and is finite. We show in this paper that the early hope for a similar reduction of unification under associativity is not justified: string unification is essentially infinitary. But we give an enumeration algorithm for essential unifiers. And beyond, this algorithm terminates when the considered problem is finitary

    Master index volumes 31–40

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    Equational methods in first order predicate calculus

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    We show that the application of the resolution principle to a set of clauses can be regarded as the construction of a term rewriting system confluent on valid formulas. This result allows the extension of usual properties and methods of equational theories (such as Birkhoff's theorem and the Knuth and Bendix completion algorithm) to quantifier-free first order theories. These results are extended to first order predicate calculus in an equational theory, as studied by Plotkin (1972), Slagle (1974) and Lankford (1975). This paper is a continuation of the work of Hsiang & Dershowitz (1983), who have already shown that rewrite methods can be used in first order predicate calculus. The main difference is the following: Hsiang uses rewrite methods only as a refutational proof technique, the initial set of formulas being unsatisfiable iff the equation TRUE = FALSE is generated by the completion algorithm. We generalise these methods to satisfiable theories; in particular, we show that the concept of confluent rewriting system, which is the main tool for studying equational theories, can be extended to any quantifier-free first order theory. Furthermore, we show that rewrite methods can be used even if formulas are kept in clausal form

    Logical presentations of domains

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    Bibliography: pages 168-174.This thesis combines a fairly general overview of domain theory with a detailed examination of recent work which establishes a connection between domain theory and logic. To start with, the theory of domains is developed with such issues as the semantics of recursion and iteration; the solution of recursive domain equations; and non-determinism in mind. In this way, a reasonably comprehensive account of domains, as ordered sets, is given. The topological dimension of domain theory is then revealed, and the logical insights gained by regarding domains as topological spaces are emphasised. These logical insights are further reinforced by an examination of pointless topology and Stone duality. A few of the more prominent categories of domains are surveyed, and Stone-type dualities for the objects of some of these categories are presented. The above dualities are then applied to the task of presenting domains as logical theories. Two types of logical theory are considered, namely axiomatic systems, and Gentzen-style deductive systems. The way in which these theories describe domains is by capturing the relationships between the open subsets of domains

    Fully Abstract Translations Between Functional Languages

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    We examine the problem of finding fully abstract translations between programming languages, i.e., translations that preserve code equivalence and nonequivalence. We present three examples of fully abstract translations: one from call-by-value to lazy PCF, one from call-by name to call-by-value PCF, and one from lazy to call-by-value PCF. The translations yield upper and lower bounds on decision procedures for proving equivalences of code. We finally define a notion of functional translation that captures the essence of the proofs of full abstraction, and show that some languages cannot be translated into others
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