Relational Chase Procedures Interpreted as Resolution with Paramodulation1

1991 ◽  
Vol 15 (2) ◽  
pp. 123-138
Author(s):  
Joachim Biskup ◽  
Bernhard Convent
Keyword(s):  
Alternative Interpretation ◽  
Order Logic ◽  
First Order Logic ◽  
Inference Problem ◽  
First Order ◽  
Inference Problems ◽  
The One ◽  
The Relationship ◽  
Theoretical Foundations ◽  
Insight Into

In this paper the relationship between dependency theory and first-order logic is explored in order to show how relational chase procedures (i.e., algorithms to decide inference problems for dependencies) can be interpreted as clever implementations of well known refutation procedures of first-order logic with resolution and paramodulation. On the one hand this alternative interpretation provides a deeper insight into the theoretical foundations of chase procedures, whereas on the other hand it makes available an already well established theory with a great amount of known results and techniques to be used for further investigations of the inference problem for dependencies. Our presentation is a detailed and careful elaboration of an idea formerly outlined by Grant and Jacobs which up to now seems to be disregarded by the database community although it definitely deserves more attention.

Notions of locality and their logical characterizations over finite models

1999 ◽  
Vol 64 (4) ◽  
pp. 1751-1773 ◽  
Author(s):  
Lauri Hella ◽  
Leonid Libkin ◽  
Juha Nurmonen
Keyword(s):  
Small Degree ◽  
Order Logic ◽  
First Order Logic ◽  
Bounded Degree ◽  
First Order ◽  
Finite Models ◽  
Order Case ◽  
The Relationship

AbstractMany known tools for proving expressibility bounds for first-ordér logic are based on one of several locality properties. In this paper we characterize the relationship between those notions of locality. We note that Gaifman's locality theorem gives rise to two notions: one deals with sentences and one with open formulae. We prove that the former implies Hanf's notion of locality, which in turn implies Gaifman's locality for open formulae. Each of these implies the bounded degree property, which is one of the easiest tools for proving expressibility bounds. These results apply beyond the first-order case. We use them to derive expressibility bounds for first-order logic with unary quantifiers and counting. We also characterize the notions of locality on structures of small degree.

Cognitive Modelling Applied to Aspects of Schizophrenia and Autonomic Computing

2011 ◽  
pp. 220-234
Author(s):  
Lee Flax
Keyword(s):  
Autonomic Computing ◽  
Signs And Symptoms ◽  
Order Logic ◽  
First Order Logic ◽  
Object Language ◽  
Cognitive Modelling ◽  
Belief Contraction ◽  
Algebraic Version ◽  
First Order ◽  
The One

We give an approach to cognitive modelling, which allows for richer expression than the one based simply on the firing of sets of neurons. The object language of the approach is first-order logic augmented by operations of an algebra, PSEN. Some operations useful for this kind of modelling are postulated: combination, comparison, and inhibition of sets of sentences. Inhibition is realised using an algebraic version of AGM belief contraction (Gärdenfors, 1988). It is shown how these operations can be realised using PSEN. Algebraic modelling using PSEN is used to give an account of an explanation of some signs and symptoms of schizophrenia due to Frith (1992) as well as a proposal for the cognitive basis of autonomic computing. A brief discussion of the computability of the operations of PSEN is also given.

On the first-order logic of terms

1973 ◽  
Vol 38 (2) ◽  
pp. 177-188
Author(s):  
Lars Svenonius
Keyword(s):  
Order Logic ◽  
First Order Logic ◽  
Identity Function ◽  
First Order ◽  
Elementary Condition ◽  
Combinatorial Function ◽  
The One ◽  
Definition Of ◽  
Image Position

By an elementary condition in the variablesx1, …, xn, we mean a conjunction of the form x1 ≤ i < j ≤ naij where each aij is one of the formulas xi = xj or xi ≠ xj. (We should add that the formula x1 = x1 should be regarded as an elementary condition in the one variable x1.)Clearly, according to this definition, some elementary conditions are inconsistent, some are consistent. For instance (in the variables x1, x2, x3) the conjunction x1 = x2 & x1 = x3 & x2 ≠ x3 is inconsistent.By an elementary combinatorial function (ex. function) we mean any function which can be given a definition of the formwhere E1(x1, …, xn), …, Ek(x1, …, xn) is an enumeration of all consistent elementary conditions in x1, …, xn, and all the numbers d1, …, dk are among 1, …, n.Examples. (1) The identity function is the only 1-ary e.c. function.(2) A useful 3-ary e.c. function will be called J. The definition is

Compactly expandable models and stability

1995 ◽  
Vol 60 (2) ◽  
pp. 673-683 ◽  
Author(s):  
Enrique Casanovas
Keyword(s):  
Order Logic ◽  
First Order Logic ◽  
Unary Predicate ◽  
Similarity Type ◽  
First Order ◽  
Related Notion ◽  
Special Part ◽  
Consistent Extension ◽  
The Relationship ◽  
Close Parallelism

In analogy to ω-logic, one defines M-logic for an arbitrary structure M (see [5],[6]). In M-logic only those structures are considered in which a special part, determined by a fixed unary predicate, is isomorphic to M. Let L be the similarity type of M and T its complete theory. We say that M-logic is κ-compact if it satisfies the compactness theorem for sets of < κ sentences. In this paper we introduce the related notion of compactness for expandability: a model M is κ-compactly expandable if for every extension T′ ⊇ T of cardinality < κ, if every finite subset of T′ can be satisfied in an expansion of M, then T′ can also be satisfied in an expansion of M. Moreover, M is compactly expandable if it is ∥M∥+-compactly expandable. It turns out that M-logic is κ-compact iff M is κ-compactly expandable.Whereas for first-order logic consistency and finite satisfiability are the same, consistency with T and finite satisfiability in M are, in general, no longer the same thing. We call the model Mκ-expandable if every consistent extension T′ ⊇ T of cardinality < κ can be satisfied in an expansion of M. We say that M is expandable if it is ∥M∥+-expandable. Here we study the relationship between saturation, expandability and compactness for expandability. There is a close parallelism between our results about compactly expandable models and some theorems of S. Shelah about expandable models, which are in fact expressed in terms of categoricity of PC-classes (see [7, Th. VI.5.3, VI.5.4 and VI.5.5]). Our results could be obtained directly from these theorems of Shelah if expandability and compactness for expandability were the same notion.

scholarly journals A Definability Theorem for First Order Logic

1997 ◽  
Vol 4 (3) ◽  
Author(s):  
Carsten Butz ◽  
Ieke Moerdijk
Keyword(s):  
Boolean Algebra ◽  
Topological Space ◽  
Model Theory ◽  
Order Logic ◽  
First Order Logic ◽  
Infinitary Logic ◽  
First Order ◽  
Definable Subset ◽  
Language L ◽  
The One

In this paper, we will present a definability theorem for first order logic.<br />This theorem is very easy to state, and its proof only uses elementary tools. To explain the theorem, let us first observe that if M is a model of a theory T in a language L, then, clearly, any definable subset S M (i.e., a subset S = {a | M |= phi(a)} defined by some formula phi) is invariant under all<br />automorphisms of M. The same is of course true for subsets of M" defined<br />by formulas with n free variables.<br /> Our theorem states that, if one allows Boolean valued models, the converse holds. More precisely, for any theory T we will construct a Boolean valued model M, in which precisely the T-provable formulas hold, and in which every (Boolean valued) subset which is invariant under all automorphisms of M is definable by a formula of L.<br />Our presentation is entirely selfcontained, and only requires familiarity<br />with the most elementary properties of model theory. In particular, we have added a first section in which we review the basic definitions concerning<br />Boolean valued models.<br />The Boolean algebra used in the construction of the model will be presented concretely as the algebra of closed and open subsets of a topological space X naturally associated with the theory T. The construction of this space is closely related to the one in [1]. In fact, one of the results in that paper could be interpreted as a definability theorem for infinitary logic, using topological rather than Boolean valued models.

Measuring Understandability of Organizational Policies: A Metric Based Approach

2018 ◽  
Vol 27 (02) ◽  
pp. 1850003
Author(s):  
Deepika Prakash ◽  
N. Parimala
Keyword(s):  
Structural Complexity ◽  
Theoretical Framework ◽  
Order Logic ◽  
First Order Logic ◽  
Policy Compliance ◽  
Organizational Policies ◽  
First Order ◽  
Health Domain ◽  
Order Representation ◽  
The Relationship

In order to ensure policy compliance, it is important for all stakeholders to understand the policy. One of the ways in which policies are represented in an organization is first-order logic. We propose a metric-based approach to measure understandability by measuring the structural complexity of the first-order representation of a policy. In this regard, we define a two-step approach that first calculates the complexity of an individual policy and then computes the complexity of a set of policies or a policy set. A running example of a policy set of six policies taken from the health domain is used. Finally, we evaluate the metrics using theoretical framework of Zuse. The relationship between structural complexity of a policy set and understandability is established by performing empirical validations. This was done by formulating policy sets from 23 domains.

First-Order Logic Reasoning Support for the Semantic Web

2009 ◽  
Vol 19 (12) ◽  
pp. 3091-3099 ◽  
Author(s):  
Gui-Hong XU ◽  
Jian ZHANG
Keyword(s):  
Semantic Web ◽  
Order Logic ◽  
First Order Logic ◽  
First Order ◽  
Logic Reasoning

Boolean-valued structures

2018 ◽  
Author(s):  
Tim Button ◽  
Sean Walsh
Keyword(s):  
Model Theory ◽  
Order Logic ◽  
First Order Logic ◽  
Inference Rules ◽  
Mathematical Concepts ◽  
Semantic Framework ◽  
First Order ◽  
Standard Models ◽  
Boolean Valued Model ◽  
Semantic Values

Chapters 6-12 are driven by questions about the ability to pin down mathematical entities and to articulate mathematical concepts. This chapter is driven by similar questions about the ability to pin down the semantic frameworks of language. It transpires that there are not just non-standard models, but non-standard ways of doing model theory itself. In more detail: whilst we normally outline a two-valued semantics which makes sentences True or False in a model, the inference rules for first-order logic are compatible with a four-valued semantics; or a semantics with countably many values; or what-have-you. The appropriate level of generality here is that of a Boolean-valued model, which we introduce. And the plurality of possible semantic values gives rise to perhaps the ‘deepest’ level of indeterminacy questions: How can humans pin down the semantic framework for their languages? We consider three different ways for inferentialists to respond to this question.

scholarly journals Extensions in graph normal form

2020 ◽  
Author(s):  
Michał Walicki
Keyword(s):  
Normal Form ◽  
Fixed Points ◽  
Propositional Logic ◽  
Order Logic ◽  
First Order Logic ◽  
First Order ◽  
Special Cases

Abstract Graph normal form, introduced earlier for propositional logic, is shown to be a normal form also for first-order logic. It allows to view syntax of theories as digraphs, while their semantics as kernels of these digraphs. Graphs are particularly well suited for studying circularity, and we provide some general means for verifying that circular or apparently circular extensions are conservative. Traditional syntactic means of ensuring conservativity, like definitional extensions or positive occurrences guaranteeing exsitence of fixed points, emerge as special cases.

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