We present a novel definition of an algorithm and its corresponding algorithm language called CoLweb. The merit of CoLweb [1] is that it makes algorithm design so versatile. That is, it forces us to a high-level, proof-carrying, distributed-style approach to algorithm design for both non-distributed computing and distributed one. We argue that this approach simplifies algorithm design. In addition, it unifies other approaches including recursive logical/functional algorithms, imperative algorithms, object-oriented imperative algorithms, neural-nets, interaction nets, proof-carrying code, etc. As an application, we refine Horn clause definitions into two kinds: blind-univerally-quantified (BUQ) ones and parallel-universally-quantified (PUQ) ones. BUQ definitions corresponds to the traditional ones such as those in Prolog where knowledgebase is $not$ expanding and its proof procedure is based on the backward chaining. On the other hand, in PUQ definitions, knowledgebase is $expanding$ and its proof procedure leads to forward chaining and {\it automatic memoization}.

We propose a purely extensional semantics for higher-order logic programming. In this semantics program predicates denote sets of ordered tuples, and two predicates are equal iff they are equal as sets. Moreover, every program has a unique minimum Herbrand model which is the greatest lower bound of all Herbrand models of the program and the least fixed-point of an immediate consequence operator. We also propose an SLD-resolution proof procedure which is proven sound and complete with respect to the minimum model semantics. In other words, we provide a purely extensional theoretical framework for higher-order logic programming which generalizes the familiar theory of classical (first-order) logic programming.

We present an unified methodology for representation and development of dialectical proof procedures in both abstract and assumption-based argumentation based on the notions of legal environments and dispute derivation. A legal environment specifies the legal moves of the dispute parties while a dispute derivation describes the procedure structure. A key insight of this paper is that the opponent moves determine the soundness of a dispute while its completeness depends on the proponent moves.

We present the CIFF proof procedure for abductive logic programming with constraints, and we prove its correctness. CIFF is an extension of the IFF proof procedure for abductive logic programming, relaxing the original restrictions over variable quantification (allowedness conditions) and incorporating a constraint solver to deal with numerical constraints as in constraint logic programming. Finally, we describe the CIFF System, comparing it with state of the art abductive systems and answer set solvers and showing how to use it to program some applications.

To not only proving new mathematical obtain broader input, especially from countries results by computer but also formally verifying outside North America, a call for commentaries the correctness of (certain properties of) computer was issued to the automated deduction community.

Inference systems Τ and search strategies E for T are distinguished from proof procedures β = (T,E) The completeness of procedures is studied by studying separately the completeness of inference systems and of search strategies. Completeness proofs for resolution systems are obtained by the construction of semantic trees. These systems include minimal α-restricted binary resolution, minimal α-restricted M-clash resolution and maximal pseudo-clash resolution. Certain refinements of hyper-resolution systems with equality axioms are shown to be complete and equivalent to refinements of the pararmodulation method for dealing with equality. The completeness and efficiency of search strategies for theorem-proving problems is studied in sufficient generality to include the case of search strategies for path-search problems in graphs. The notion of theorem-proving problem is defined abstractly so as to be dual to that of and" or tree. Special attention is given to resolution problems and to search strategies which generate simpler before more complex proofs. For efficiency, a proof procedure (T,E) requires an efficient search strategy E as well as an inference system T which admits both simple proofs and relatively few redundant and irrelevant derivations. The theory of efficient proof procedures outlined here is applied to proving the increased efficiency of the usual method for deleting tautologies and subsumed clauses. Counter-examples are exhibited for both the completeness and efficiency of alternative methods for deleting subsumed clauses. The efficiency of resolution procedures is improved by replacing the single operation of resolving a clash by the two operations of generating factors of clauses and of resolving a clash of factors. Several factoring methods are investigated for completeness. Of these the m-factoring method is shown to be always more efficient than the Wos-Robinson method. The University of Edinburgh

A robot, in order to act intelligently, must be able to reason from facts which its sensors detect to conclusions which govern its actions. This reasoning process is so central to human intelligence that it seems immediately relevant to the problems of robot design to consider its properties, how it might be analysed and imitated.

"It is well known to be impossible to tile with dominoes a checkerboard with two opposite corners deleted. This fact is readily stated in the first order predicate calculus, but the usual proof which involves a parity and counting argument does not readily translate into predicate calculus. We conjecture that this problem will be very difficult for programmed proof procedures."Stanford Artificial Intelligence Project Memo No. 16

This work is partly included in a project sponsored by Statens tekniska forskningsråd (Sweden). Berg, and Mr Voghera for reading the manuscript and making valuable suggestions (see also n. 11). Use the link below to share a full-text version of this article with your friends and colleagues.

The hope that mathematical methods employed in the investigation of formal logic would lead to purely computational methods for obtaining mathematical theorems goes back to Leibniz and has been revived by Peano around the turn of the century and by Hilbert's school in the 1920's. Hilbert, noting that all of classical mathematics could be formalized within quantification theory, declared that the problem of finding an algorithm for determining whether or not a given formula of quantification theory is valid was the central problem of mathematical logic. And indeed, at one time it seemed as if investigations of this "decision" problem were on the verge of success. However, it was shown by Church and by Turing that such an algorithm can not exist. This result led to considerable pessimism regarding the possibility of using modern digital computers in deciding significant mathematical questions.