First-Order Logic for Decision Problems with Preference Aggregation

2016 ◽  
Author(s):  
Arnaldo Araujo ◽  
Ana Teresa Martins ◽  
Davi Romero
Keyword(s):  
Order Logic ◽  
Decision Problems ◽  
First Order Logic ◽  
Preference Aggregation ◽  
First Order

Two variable first-order logic over ordered domains

2001 ◽  
Vol 66 (2) ◽  
pp. 685-702 ◽  
Author(s):  
Martin Otto
Keyword(s):  
Order Logic ◽  
Decision Problems ◽  
First Order Logic ◽  
Satisfiability Problem ◽  
Exponential Time ◽  
Infinite Domains ◽  
First Order ◽  
Computation Tree ◽  
Common Extension ◽  
Undecidability Result

AbstractThe satisfiability problem for the two-variable fragment of first-order logic is investigated over finite and infinite linearly ordered, respectively wellordered domains, as well as over finite and infinite domains in which one or several designated binary predicates are interpreted as arbitrary wellfounded relations.It is shown that FO2 over ordered, respectively wellordered. domains or in the presence of one well-founded relation, is decidable for satisfiability as well as for finite satisfiability. Actually the complexity of these decision problems is essentially the same as for plain unconstrained FO2. namely non-deterministic exponential time.In contrast FO2 becomes undecidable for satisfiability and for finite satisfiability, if a sufficiently large number of predicates are required to be interpreted as orderings. wellorderings. or as arbitrary wellfounded relations. This undecidability result also entails the undecidability of the natural common extension of FO2 and computation tree logic CTL.

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.

scholarly journals Formula size games for modal logic and μ-calculus

2019 ◽  
Vol 29 (8) ◽  
pp. 1311-1344 ◽  
Author(s):  
Lauri T Hella ◽  
Miikka S Vilander
Keyword(s):  
Modal Logic ◽  
Optimal Strategy ◽  
Order Logic ◽  
First Order Logic ◽  
Kripke Models ◽  
First Order ◽  
Modal Operators ◽  
Crucial Difference ◽  
Definition Of ◽  
Person Game

Abstract We propose a new version of formula size game for modal logic. The game characterizes the equivalence of pointed Kripke models up to formulas of given numbers of modal operators and binary connectives. Our game is similar to the well-known Adler–Immerman game. However, due to a crucial difference in the definition of positions of the game, its winning condition is simpler, and the second player does not have a trivial optimal strategy. Thus, unlike the Adler–Immerman game, our game is a genuine two-person game. We illustrate the use of the game by proving a non-elementary succinctness gap between bisimulation invariant first-order logic $\textrm{FO}$ and (basic) modal logic $\textrm{ML}$. We also present a version of the game for the modal $\mu $-calculus $\textrm{L}_\mu $ and show that $\textrm{FO}$ is also non-elementarily more succinct than $\textrm{L}_\mu $.

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