The KE inference system is a tableau method developed by Marco Mondadori which was presented as an improvement, in the computational efficiency sense, over Analytic Tableaux. In the literature, there is no description of a theorem prover based on the KE method for the C"1 paraconsistent logic. Paraconsistent logics have several applications, such as in robot control and medicine. These applications could benefit from the existence of such a prover. We present a sound and complete KE system for C"1, an informal specification of a strategy for the C"1 prover as well as problem families that can be used to evaluate provers for C"1. The C"1KE system and the strategy described in this paper will be used to implement a KE based prover for C"1, which will be useful for those who study and apply paraconsistent logics.
In @cite_7 , tableau systems for several logics of the @math hierarchy were presented. The tableau system presented there is also based on AT. While in the previous systems @math was applied whenever necessary to generate the branches of the tableau, this system has specific rules to directly deal with all operators, including @math ''. However, as it is based on the analytic tableau method, it also has too many (six) branching rules. We also do not know of any implementation of this method.
@cite_7
2057199660 1819951499 2135625884 2167352300
In this paper we present a new hierarchy of analytical tableaux systems TNDC"n,1@?n =1, as primitive operators, differently to what has been done in the literature, where these operators are usually defined operators. We prove a version of Cut Rule for the TNDC"n,1@?n<@w, and also prove that these systems are logically equivalent to the corresponding systems C"n,1@?n<@w. The systems TNDC"n constitute completely automated theorem proving systems for the systems of da Costa's hierarchy C"n,1@?n<@w. A generic tableau prover has been implemented and integrated with Isabelle (Paulson, 1994). Compared with classical first-order logic provers, it has numerous extensions that allow it to reason with any supplied set of tableau rules. It has a higherorder syntax in order to support user-defined binding operators, such as those of set theory. The unification algorithm is first-order instead of higher-order, but it includes modifications to handle bound variables. The proof, when found, is returned to Isabelle as a list of tactics. Because Isabelle verifies the proof, the prover can cut corners for efficiency’s sake without compromising soundness. For example, the prover can use type information to guide the search without storing type information in full. Categories: F.4, I.1 1. Summary In Part I, four ostensibly different theoretical models of induction are presented, in which the problem dealt with is the extrapolation of a very long sequence of symbols—presumably containing all of the information to be used in the induction. Almost all, if not all problems in induction can be put in this form. Some strong heuristic arguments have been obtained for the equivalence of the last three models. One of these models is equivalent to a Bayes formulation, in which a priori probabilities are assigned to sequences of symbols on the basis of the lengths of inputs to a universal Turing machine that are required to produce the sequence of interest as output. Though it seems likely, it is not certain whether the first of the four models is equivalent to the other three. Few rigorous results are presented. Informal investigations are made of the properties of these models. There are discussions of their consistency and meaningfulness, of their degree of independence of the exact nature of the Turing machine used, and of the accuracy of their predictions in comparison to those of other induction methods. In Part II these models are applied to the solution of three problems—prediction of the Bernoulli sequence, extrapolation of a certain kind of Markov chain, and the use of phrase structure grammars for induction. Though some approximations are used, the first of these problems is treated most rigorously. The result is Laplace's rule of succession. The solution to the second problem uses less certain approximations, but the properties of the solution that are discussed, are fairly independent of these approximations. The third application, using phrase structure grammars, is least exact of the three. First a formal solution is presented. Though it appears to have certain deficiencies, it is hoped that presentation of this admittedly inadequate model will suggest acceptable improvements in it. This formal solution is then applied in an approximate way to the determination of the “optimum” phrase structure grammar for a given set of strings. The results that are obtained are plausible, but subject to the uncertainties of the approximation used. Three temporal logics are introduced which induce on labeled transition systems the same identifications as branching bisimulation. The first is an extension of Hennessy-Milner logic with a kind of unit operator. The second is another extension of Hennessy-Milner logic which exploits the power of backward modalities. The third is CTL* with the next-time operator interpreted over all paths, not just over maximal ones. A relevant side effect of the last characterization is that it sets a bridge between the state- and event-based approaches to the semantics of concurrent systems. >