Results 1 
4 of
4
2004).On the notion of assumption in logical systems
 In R
"... When a logical system is specified and the notion of a derivation or formal proof is explained, we are told (i) which formulas can be used to start a derivation and (ii) which formulas can be derived given that certain other formulas have already been derived. Formulas of the sort (i) are either ass ..."
Abstract

Cited by 3 (3 self)
 Add to MetaCart
When a logical system is specified and the notion of a derivation or formal proof is explained, we are told (i) which formulas can be used to start a derivation and (ii) which formulas can be derived given that certain other formulas have already been derived. Formulas of the sort (i) are either assumptions or axioms, formulas of the sort (ii) are conclusions of (proper) inference rules. Axioms may be viewed as conclusions of (improper) inference rules, viz. inference rules without premisses. In what follows I refer to conclusions of proper or improper inference rules as assertions. 1 In natural deduction systems, inference rules deal both with assumptions and assertions, as the assumptions on which the conclusion of an inference rule depends, are not necessarily given by the collection of all assumptions on which the premisses depend, in case the rule permits the discharging of assumptions. For example, the rule of implication introduction
A Definitional Approach to Functional Logic Programming
 Extensions of Logic Programming 5th International Workshop, ELP'96, number 1050 in Lecture Notes in Artificial Intelligence
, 1996
"... . We describe a definitional approach to the combination of functional and logic programming based on the theory of Partial Inductive Definitions. The described method produces programs directly executable in the definitional programming language GCLA. We show both a basic calculus for functional lo ..."
Abstract

Cited by 2 (2 self)
 Add to MetaCart
. We describe a definitional approach to the combination of functional and logic programming based on the theory of Partial Inductive Definitions. The described method produces programs directly executable in the definitional programming language GCLA. We show both a basic calculus for functional logic program definitions and discuss a refined version where the rules definitional resolution, definitional reflection, and definitional axiom are altered to be better suited for functional evaluation and equation solving. 1 Introduction Through the years there have been numerous attempts to combine the two main declarative programming paradigms functional and logic programming into one framework providing the benefits of both. The proposed methods varies from different kinds of translations, embedding one of the methods into the other, [21, 26], to more integrated approaches such as narrowing languages [9, 13, 20, 24] based on Horn clause logic with equality [23], some kind of higher order...
Functional logic programming in GCLA
 Proceedings of the 6th Nordic Workshop on Programming Theory. Aarhus
, 1994
"... We describe a definitional approach to functional logic programming, based on the theory of Partial Inductive Definitions and the programming language GCLA. It is shown how functional and logic programming are easily integrated in GCLA using the features of the language, that is combining functions ..."
Abstract

Cited by 2 (2 self)
 Add to MetaCart
We describe a definitional approach to functional logic programming, based on the theory of Partial Inductive Definitions and the programming language GCLA. It is shown how functional and logic programming are easily integrated in GCLA using the features of the language, that is combining functions and predicates in programs becomes a matter of programming methodology. We also give a brief description of a way to automatically generate efficient procedural parts to the described definitions. 1
Section A.2 Philosophical Logic DEFINITIONAL REFLECTION AND CIRCULAR REASONING
"... The theory of definitional reflection provides a novel framework for studying logical features of circular, and especially paradoxical reasoning. Definitional reflection originated from reading clauses for atoms as definitions, thereby extending ideas concerning elimination rules in natural deductio ..."
Abstract
 Add to MetaCart
The theory of definitional reflection provides a novel framework for studying logical features of circular, and especially paradoxical reasoning. Definitional reflection originated from reading clauses for atoms as definitions, thereby extending ideas concerning elimination rules in natural deduction ⎧[2, 3]. In the simplest (propositional) case, given a definition for an atom a of the ⎪ ⎨ a ⇐ ∆1 form D:, the rule of definitional reflection (D ⊢) a ⇐ ∆n {Γ, ∆i ⊢ C}i is associated Γ, a ⊢ C with a as a left introduction rule. If individual variables are present, and for computational purposes, the rule becomes more complicated [3, 5]. (D ⊢) is considered as introducing an atomic assumption a according to its definitional meaning given by D. This is the specific way of introducing a as an assumption, which is distinguished from the unspecific way by means