Logic

This chapter considers whether mereology should rightly be thought of as a first-order theory with parthood as a binary predicate. It considers extensions of classical mereology aimed at overcoming the expressive limits of standard first-order languages, focusing on second-order and plural formulations. Relatedly, Lewis’s megethology and applications to the philosophy of mathematics are discussed. Then, several ways of modifying the framework to make room for mereological considerations involving time and modality are presented, such as the possibility that an object may have different parts at different times, or that it could have had different parts from the ones it actually has. Finally, a number of theories are expounded that can be developed in order to deal with the phenomenon of mereological indeterminacy, i.e., the fact that in some cases the very question of whether something is part of something else does not appear to have a definite answer.

Design patterns for modeling first-order expressive Bayesian networks

First-order expressive capabilities allow Bayesian networks (BNs) to model problem domains where the number of entities, their attributes, and their relationships can vary significantly between model instantiations. First-order BNs are well-suited for capturing knowledge representation dependencies, but literature on design patterns specific to first-order BNs is few and scattered. To identify useful patterns, we investigated the range of dependency models between combinations of random variables (RVs) that represent unary attributes, functional relationships, and binary predicate relationships. We found eight major patterns, grouped into three categories, that cover a significant number of first-order BN situations. Selection behavior occurs in six patterns, where a relationship/attribute identifies which entities in a second relationship/attribute are applicable. In other cases, certain kinds of embedded dependencies based on semantic meaning are exploited. A significant contribution of our patterns is that they describe various behaviors used to establish the RV’s local probability distribution. Taken together, the patterns form a modeling framework that provides significant insight into first-order expressive BNs and can reduce efforts in developing such models. To the best of our knowledge, there are no comprehensive published accounts of such patterns.

Using Prior Knowledge in Data Mining

One of the most important and challenging problems in current Data Mining research is the definition of the prior knowledge that can be originated from the process or the domain. This contextual information may help select the appropriate information, features or techniques, decrease the space of hypotheses, represent the output in a most comprehensible way and improve the process. Ontological foundation is a precondition for efficient automated usage of such information (Chandrasekaran et al., 1999). An ontology is a formal explicit specification of a shared conceptualization for a domain of interest (Gruber, 1993). Among other things, this definition emphasizes the fact that an ontology has to be specified in a language that comes with a formal semantics. Due to this formalization ontologies provide the machine interpretable meaning of concepts and relations that is expected when using a semantic-based approach (Staab & Studer, 2004). In its most prevalent use in Artificial Intelligence (AI), an ontology refers to an engineering artifact (more precisely, produced according to the principles of Ontological Engineering (Gómez-Pérez et al., 2004)), constituted by a specific vocabulary used to describe a certain reality, plus a set of explicit assumptions regarding the intended meaning of the vocabulary words. This set of assumptions has usually the form of a First-Order Logic (FOL) theory, where vocabulary words appear as unary or binary predicate names, respectively called concepts and relations. In the simplest case, an ontology describes a hierarchy of concepts related by subsumption relationships; in more sophisticated cases, suitable axioms are added in order to express other relationships between concepts and to constrain their intended interpretation. Ontologies can play several roles in Data Mining (Nigro et al., 2007). In this chapter we investigate the use of ontologies as prior knowledge in Data Mining. As an illustrative case throughout the chapter, we choose the task of Frequent Pattern Discovery, it being the most representative product of the cross-fertilization among Databases, Machine Learning and Statistics that has given rise to Data Mining. Indeed it is central to an entire class of descriptive tasks in Data Mining among which Association Rule Mining (Agrawal et al., 1993; Agrawal & Srikant, 1994) is the most popular. A pattern is considered as an intensional description (expressed in a given language L) of a subset of a data set r. The support of a pattern is the relative frequency of the pattern within r and is computed with the evaluation function supp. The task of Frequent Pattern Discovery aims at the extraction of all frequent patterns, i.e. all patterns whose support exceeds a user-defined threshold of minimum support. The blueprint of most algorithms for Frequent Pattern Discovery is the levelwise search (Mannila & Toivonen, 1997). It is based on the following assumption: If a generality order = for the language L of patterns can be found such that = is monotonic w.r.t. supp, then the resulting space (L, =) can be searched breadth-first by starting from the most general pattern in L and alternating candidate generation and candidate evaluation phases.

Provability with Finitely Many Variables

For every finite n ≥ 4 there is a logically valid sentence φn with the following properties: φn contains only 3 variables (each of which occurs many times); φn contains exactly one nonlogical binary relation symbol (no function symbols, no constants, and no equality symbol); φn has a proof in first-order logic with equality that contains exactly n variables, but no proof containing only n − 1 variables. This result was first proved using the machinery of algebraic logic developed in several research monographs and papers. Here we replicate the result and its proof entirely within the realm of (elementary) first-order binary predicate logic with equality. We need the usual syntax, axioms, and rules of inference to show that φn has a proof with only n variables. To show that φn has no proof with only n − 1 variables we use alternative semantics in place of the usual, standard, set-theoretical semantics of first-order logic.

A note on valuation definable expansions of fields

In this note, we consider models of the theories of valued algebraically closed fields and convexly valued real closed fields, their reducts to the pure field or ordered field language respectively, and expansions of these by predicates which are definable in the valued field. We show that, in terms of definability, there is no structure properly between the pure (ordered) field and the valued field. Our results are analogous to several other definability results for reducts of algebraically closed and real closed fields; see [9], [10], [11] and [12]. Throughout this paper, definable will mean definable with parameters.Theorem A. Let ℱ = (F, +, ×, V) be a valued, algebraically closed field, where V denotes the valuation ring. Let A be a subset ofFndefinable in ℱv. Then either A is definable in ℱ = (F, +, ×) or V is definable in.Theorem B. Let ℛv = (R, <, +, ×, V) be a convexly valued real closed field, where V denotes the valuation ring. Let Abe a subset ofRndefinable in ℛv. Then either A is definable in ℛ = (R, <, +, ×) or V is definable in.The proofs of Theorems A and B are quite similar. Both ℱv and ℛv admit quantifier elimination if we adjoin a definable binary predicate Div (interpreted by Div(x, y) if and only if v(x) ≤ v(y)). This is proved in [14] (extending [13]) in the algebraically closed case, and in [4] in the real closed case. We show by direct combinatorial arguments that if the valuation is not definable then the expanded structure is strongly minimal or o-minimal respectively. Then we call on known results about strongly minimal and o-minimal fields to show that the expansion is not proper.

Undecidability of modal and intermediate first-order logics with two individual variables

The interest in fragments of predicate logics is motivated by the well-known fact that full classical predicate calculus is undecidable (cf. Church [1936]). So it is desirable to find decidable fragments which are in some sense “maximal”, i.e., which become undecidable if they are “slightly” extended. Or, alternatively, we can look for “minimal” undecidable fragments and try to identify the vague boundary between decidability and undecidability. A great deal of work in this area concerning mainly classical logic has been done since the thirties. We will not give a complete review of decidability and undecidability results in classical logic, referring the reader to existing monographs (cf. Suranyi [1959], Lewis [1979], and Dreben, Goldfarb [1979]). A short summary can also be found in the well-known book Church [1956]. Let us recall only several facts. Herein we will consider only logics without functional symbols, constants, and equality.(C1) The fragment of the classical logic with only monadic predicate letters is decidable (cf. Behmann [1922]).(C2) The fragment of the classical logic with a single binary predicate letter is undecidable. (This is a consequence of Gödel [1933].)(C3) The fragment of the classical logic with a single individual variable is decidable; in fact it is equivalent to Lewis S5 (cf. Wajsberg [1933]).(C4) The fragment of the classical logic with two individual variables is decidable (Segerberg [1973] contains a proof using modal logic; Scott [1962] and Mortimer [1975] give traditional proofs.)(C5) The fragment of the classical logic with three individual variables and binary predicate letters is undecidable (cf. Surańyi [1943]). In fact this paper considers formulas of the following typeφ,ψ being quantifier-free and the set of binary predicate letters which can appear in φ or ψ being fixed and finite.

Preservation theorem and relativization theorem for cofinal extensions

One of the typical preservation theorems in a first order classical predicate logic with equality L is the following theorem due to J. Łoś [4] and A. Tarski [9] (also cf. [1, p. 139]).Theorem A (Łoś-Tarski). For any sentences A and B in L, the following two conditions (i) and (ii) are equivalent.(i) Every extension of any model of A is a model of B.(ii) The two sentences A ⊃ C and C ⊃ B are provable in L for some existential sentence C in L.In [2], S. Feferman obtained a similar preservation theorem for outer extensions. In the following, we assume that L has a fixed binary predicate symbol <. Then Σ-formulas are formulas in L which are constructed from atomic formulas and their negations by applying ∧ (conjunctions), ∨ (disjunctions), ∀x < y (bounded universal quantifications), and ∃ (existential quantifications). An extension of an L-structure is said to be an outer extension of if ⊨ a < b and b ϵ ∣∣ imply a ϵ ∣∣ for any elements a, b in ∣∣.Theorem B (Feferman). For any sentences A and B in L, the following two conditions (i) and (ii) are equivalent.(i) Every outer extension of any model of A is a model of B.(ii) The two sentences A ⊃ C and C ⊃ B are provable in L for some Σ-sentence C in L.

A normal form theorem for first order formulas and its application to Gaifman's splitting theorem

Let L be a first order predicate calculus with equality which has a fixed binary predicate symbol <. In this paper, we shall deal with quantifiers Cx, ∀x ≦ y, ∃x ≦ y defined as follows: CxA(x) is ∀y∃x(y ≦ x ∧ A(x)), ∀x ≦ yA{x) is ∀x(x ≦ y ⊃ A(x)), and ∃x ≦ yA(x) is ∃x(x ≦ y ∧ 4(x)). The expressions x̄, ȳ, … will be used to denote sequences of variables. In particular, x̄ stands for 〈x1, …, xn〉 and ȳ stands for 〈y1,…, ym〉 for some n, m. Also ∃x̄, ∀x̄ ≦ ȳ, … will be used to denote ∃x1 ∃x2 … ∃xn, ∀x1 ≦ y1 ∀x2 ≦ y2 … ∀xn ≦ yn, …, respectively. Let X be a set of formulas in L such that X contains every atomic formula and is closed under substitution of free variables and applications of propositional connectives ¬(not), ∧(and), ∨(or). Then, ∑(X) is the set of formulas of the form ∃x̄B(x̄), where B ∈ X, and Φ(X) is the set of formulas of the form.Since X is closed under ∧, ∨, the two sets Σ(X) and Φ(X) are closed under ∧, ∨ in the following sense: for any formulas A and B in Σ(X) [Φ(X)], there are formulas in Σ(X)[ Φ(X)] which are obtained from A ∧ B and A ∨ B by bringing some quantifiers forward in the usual manner.