Interpolation in propositional Horn logic

The axiom of choice asserts that if 41 is a set of non-null, disjoint sets mi, then there exists a set S, called a set of choice, which contains one and only one element from each of the mni. The existence of such a collection as S seems assured on the basis of intuitive reasoning, However, it must be recalled that the word set is undefined, and only attains meaning through its occurrences in the axioms of set theory. An equivalent assertion is that of the well-ordering theorem which states that every class can be so ordered that each of its non-empty subclasses has a first element. This article gives a. brief description of the significance of the axiom of choice and the well-orderingtheorem in the development of abstract set theory. The birth of Cantor's theory of sets. The use of the procedure of cuts by Dedekind [11i to define irrational numbers made it clear that the real numbers could only be attained (conceptually) by conceiving of an actual infinity of numbers previously introduced (rationals). Thus, it became necessary to admit reasoning which dared to use an actual infinity of propositions as premises, each corresponding in a one-to-one fashion to the rationals 1221. Manyof thepredecessors of Cantor had tried to reason on the infinite, but so many contradictions were met that the actual infinite was relegated to a place outside of mathematics. Cantor taught mathematicians to consider certain distinctions such as that between cardinal and ordinal, which only the grammarians had previously observed. Cantor also pointed out that, although there is no real distinction in the case of finite collections, it becomes necessary to distinguish between these conceptions in the case of infinite aggregates [41]. It can be shown that it is by means of the wellordering theorem (or, equivalently, the axiom of choice) that the concepts of cardinal and ordinal are united for the infinite case too. Many difficulties arose when analysis was made of objects whose

This paper presents a recent formalization of a Henkin-style completeness proof for the propositional modal logic S5 using the Lean theorem prover. The proof formalized is close to that of Hughes and Cresswell, but the system, based on a different choice of axioms, is better described as a Mendelson system augmented with axiom schemes for K, T, S4, and B, and the necessitation rule as a rule of inference. The language has the false and implication as the only primitive logical connectives and necessity as the only primitive modal operator. The full source code is available online at https://github.com/bbentzen/mpl/ and has been typechecked with Lean 3.4.2.

In this paper, we focus on regularity and set-valued functions. The regularity was first introduced by S.C. Kleene (1952) into the propositional connectives of a ternary logic. Then, M. Mukaidono (1986) expanded the regularity of Kleene into n-variable ternary functions, and a ternary function which is regular is called a regular ternary logic function. Some studies expanded regular ternary logic functions into /spl tau/-valued functions, and studied properties of them. In this paper, we propose another extension of the concepts of the regularity in the sense of Kleene and Mukaidono. That is, we introduce regularity into r-valued set-valued functions. Further, we give properties of the set-valued functions with the regularity.

We define the concept of continuum-wise expansiveness for set-valued functions. We prove that for a compact metric space X X with positive topological dimension, if a set-valued function is c w cw -expansive, then its topological entropy is positive. We also introduce the notion of point-wise specification property for set-valued functions and prove that set-valued functions with this property have positive entropy.

On the Computational Feasibility of Abductive Diagnosis for Practical Applications

On extremal solutions of inclusion problems with applications to game theory

After introducing the method of indeterminate coefficients, by which we can derive all finite models satisfying a given set of axioms, we derive all models of 8 elements of Kleene algebra by using the method, and find out many examples of a set of independent and complete axioms of Kleene algebra by checking whether each axiom in Kleene algebra is independent from others or not based on the method.

We show that Kleene algebra with tests (KAT) subsumes propositional Hoare logic (PHL). Thus the specialized syntax and deductive apparatus of Hoare logic are inessential and can be replaced by simple equational reasoning. In addition, we show that all relationally valid inference rules are derivable in KAT and that deciding the relational validity of such rules is PSPACE -complete.

It is shown that an orthomodular lattice can be axiomatized as an ortholattice with a unique operation of identity (bi-implication) instead of the operation of implication, and a corresponding algebraic unified quantum logic is formulated. A statistical YES-NO physical interpretation of the quantum logical propositions is then provided to establish a support for a novel YES-NO representation of quantum logic which prompts a conjecture about a possible completion of quantum logic by means of probabilistic forcing. In this paper we shall anwser two basic questions of quantum' logics, present a novel probabilistic representation of the logic and of quantum measurements, and propose a probabilistic forcing as a possible tool for completing quantum logic. The questions are, first, whether there is a unique object language operation which can take over the role of the unique classical operation of implication (conditional, set-theoretic inclusion), and, second, whether there is a relation which is more appropriate for set-theoretic representation of quantum-theoretic measurements than the usual irreflexive and symmetric orthogonality relation. We answer the first question in the positive by substituting the biimplication for the implication and the equality for the ordering relation, thus at the same time making the ordering within quantum sets irrelevant. This renders the usual techniques of quantum logic as a deductive inferential theory inappropriate and ascribes quantum deductive logic a particular equational meaning. The result is obtained in Section 2.

Deriving conclusions in expert systems when knowledge is incomplete

On the hydra number of disconnected graphs

Traditionally, formal languages are defined as sets of words. More recently, the alternative coalgebraic or coinductive representation as infinite tries, i.e., prefix trees branching over the alphabet, has been used to obtain compact and elegant proofs of classic results in language theory. In this article, we study this representation in the Isabelle proof assistant. We define regular operations on infinite tries and prove the axioms of Kleene algebra for those operations. Thereby, we exercise corecursion and coinduction and confirm the coinductive view being profitable in formalizations, as it improves over the set-of-words view with respect to proof automation.

Based on a given set of uncertain scenarios, robust optimization aims to determine the best possible choices. The best solution in robust optimization is usually found by identifying the best out of the worst-case analysis or by using the min–max counterpart. The min–max approach is a pessimistic approach and commonly discards the effect of some uncertain scenarios in the process of finding an optimum solution. Thus, in this article, a new robust counterpart is introduced with the help of the generalized ordered weighted aggregation (GOWA) operator to solve uncertain multi-objective optimization problems with an uncertainty set of finite cardinality. After introducing GOWA robustness, the elementary properties of the GOWA robust objective function, such as continuity and local Lipschitz continuity, are analysed. Then, the relationship is discussed between the concept of GOWA robustness and (the pessimistic) min–max and (the optimistic) min–min robustness. It is shown that min–min and min–max robustness are particular cases of GOWA robustness. The GOWA robust counterpart is found to be an optimization problem with a set-valued objective mapping. Two approaches are provided to find robust solutions for GOWA: weighted-sum and epsilon-constraint. Finally, an objective-wise GOWA robust counterpart is provided to solve uncertain multi-objective optimization problems and relate it to min–max robustness. Several geometrical and numerical illustrations support the entire article.

Interval temporal logics formalize reasoning about interval structures over (usually) linearly ordered domains, where time intervals are the primitive ontological entities and truth of formulae is defined relative to time intervals, rather than time points. In this paper, we introduce and study Metric Propositional Neighborhood Logic (MPNL) over natural numbers. MPNL features two modalities referring, respectively, to an interval that is “met by” the current one and to an interval that “meets” the current one, plus an infinite set of length constraints, regarded as atomic propositions, to constrain the lengths of intervals. We argue that MPNL can be successfully used in different areas of artificial intelligence to combine qualitative and quantitative interval temporal reasoning, thus providing a viable alternative to well-established logical frameworks such as Duration Calculus. We show that MPNL is decidable in double exponential time and expressively complete with respect to a well-defined subfragment of the two-variable fragment FO2[N, =, <, s] of first-order logic for linear orders with successor function, interpreted over natural numbers. Moreover, we show that MPNL can be extended in a natural way to cover full FO2[N, =, <, s], but, unexpectedly, the latter (and hence the former) turns out to be undecidable.

We present a thorough study of propositional dynamic logic over a variation of labeled transition systems, called accelerated labelled transition systems, which are transition systems labeled with regular expressions over action labels. We study the model checking and satisfiability decision problems. Through a notion of regular expression rewriting, we reduce these two problems to the corresponding ones of PDL in the traditional semantics (w.r.t. LTS). As for the complexity, both of problems are proved to be ExPSPACE-complete. Moreover, the program complexity of model checking problem turns out to be NLOGSPACE-complete. Furthermore, we provide an axiomatization for PDL which involves Kleene Algebra as an Oracle. The soundness and completeness are shown.