scholarly journals Simulating reachability using first-order logic with applications to verification of linked data structures

2009 ◽  
Vol 5 (2) ◽  
Author(s):  
Tal Lev-Ami ◽  
Neil Immerman ◽  
Thomas Reps ◽  
Mooly Sagiv ◽  
Siddharth Srivastava ◽  
...  
Keyword(s):  
Data Structures ◽  
Linked Data ◽  
Order Logic ◽  
First Order Logic ◽  
First Order

A NORMAL FORM FOR FIRST-ORDER LOGIC OVER DOUBLY-LINKED DATA STRUCTURES

2008 ◽  
Vol 19 (01) ◽  
pp. 205-217 ◽  
Author(s):  
STEVEN LINDELL
Keyword(s):  
Normal Form ◽  
Data Structures ◽  
Linked Data ◽  
Linear Time ◽  
Order Logic ◽  
Information Storage ◽  
First Order Logic ◽  
Total Size ◽  
Bounded Degree ◽  
First Order

We use singulary vocabularies to analyze first-order definability over doubly-linked data structures. Singulary vocabularies contain only monadic predicate and monadic function symbols. A class of mathematical structures in any vocabulary can be elementarily interpreted in a singulary vocabulary, while preserving notions of total size and degree. Doubly-linked data structures are a special case of bounded-degree finite structures in which there are reciprocal connections between elements, corresponding closely with physically feasible models of information storage. They can be associated with logical models involving unary relations and bijective functions in what we call an invertible singulary vocabulary. Over classes of these models, there is a normal form for first-order logic which eliminates all quantification of dependent variables. The paper provides a syntactically based proof using counting quantifiers. It also makes precise the notion of implicit calculability for arbitrary arity first-order formulas. Linear-time evaluation of first-order logic over doubly-linked data structures becomes a direct corollary. Included is a discussion of why these special data structures are appropriate for physically realizable models of information.

scholarly journals Question Answering over Linked Data Using First-order Logic

2014 ◽  
Author(s):  
Shizhu He ◽  
Kang Liu ◽  
Yuanzhe Zhang ◽  
Liheng Xu ◽  
Jun Zhao
Keyword(s):  
Linked Data ◽  
Question Answering ◽  
Order Logic ◽  
First Order Logic ◽  
First Order

scholarly journals A First-Order Logic with Frames

2020 ◽  
pp. 515-543 ◽  
Author(s):  
Adithya Murali ◽  
Lucas Peña ◽  
Christof Löding ◽  
P. Madhusudan
Keyword(s):  
Data Structures ◽  
Order Logic ◽  
Separation Logic ◽  
First Order Logic ◽  
Program Logic ◽  
First Order ◽  
Proof Rules ◽  
Local Specifications ◽  
Recursive Definitions ◽  
The Universe

AbstractWe propose a novel logic, called Frame Logic (FL), that extends first-order logic (with recursive definitions) using a construct $$\textit{Sp}(\cdot )$$ Sp ( · ) that captures the implicit supports of formulas— the precise subset of the universe upon which their meaning depends. Using such supports, we formulate proof rules that facilitate frame reasoning elegantly when the underlying model undergoes change. We show that the logic is expressive by capturing several data-structures and also exhibit a translation from a precise fragment of separation logic to frame logic. Finally, we design a program logic based on frame logic for reasoning with programs that dynamically update heaps that facilitates local specifications and frame reasoning. This program logic consists of both localized proof rules as well as rules that derive the weakest tightest preconditions in FL.

scholarly journals Logtk : A Logic ToolKit for Automated Reasoning and its Implementation

10.29007/4z1m ◽  
2018 ◽  
Author(s):  
Simon Cruanes
Keyword(s):  
Data Structures ◽  
Automated Reasoning ◽  
Order Logic ◽  
First Order Logic ◽  
First Order ◽  
Design And Implementation ◽  
Data Structures And Algorithms ◽  
Index Terms

We describe the design and implementation of Logtk, an OCaml library for writing automated reasoning tools that deal with (possibly typed) first-order logic. The library provides data structures and algorithms to represent terms, formulas, substitutions, perform unification, index terms, parse problems, as well as a few tools to demonstrate itsuse. It is the basis of a full-fledged superposition prover.

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.

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.

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