The quantifier complexity of polynomial-size iterated definitions in first-order logic

Levesque proposed a generalization of a database called a proper knowledge base (KB), which is equivalent to a possibly infinite consistent set of ground literals. In contrast to databases, proper KBs do not make the closed-world assumption and hence the entailment problem becomes undecidable. Levesque then proposed a limited but efficient inference method V for proper KBs, which is sound and, when the query is in a certain normal form, also logically complete. He conjectured that for every first-order query there is an equivalent one in normal form. In this note, we show that this conjecture is false. In fact, we show that any class of formulas for which V is complete must be strictly less expressive than full first-order logic. Moreover, in the propositional case it is very unlikely that a formula always has a polynomial-size normal form.

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A Logical Characterization of Small 2NFAs

International Journal of Foundations of Computer Science ◽

10.1142/s0129054117400019 ◽

2017 ◽

Vol 28 (05) ◽

pp. 445-464

Author(s):

Christos A. Kapoutsis ◽

Lamana Mulaffer

Keyword(s):

Transitive Closure ◽

Closure Operator ◽

Finite Automata ◽

Disjunctive Normal Form ◽

Order Logic ◽

First Order Logic ◽

First Order ◽

Polynomial Size ◽

Fixed Input

Let 2N be the class of families of problems solvable by families of two-way nondeterministic finite automata of polynomial size. We characterize 2N in terms of families of formulas of transitive-closure logic. These formulas apply the transitive-closure operator on a quantifier-free disjunctive normal form of first-order logic with successor and constants, where (i) apart from two special variables, all others are equated to constants in every clause, and (ii) no clause simultaneously relates these two special variables and refers to fixed input cells. We prove that automata with polynomially many states are as powerful as formulas with polynomially many clauses and polynomially large constants. This can be seen as a refinement of Immerman’s theorem that nondeterministic logarithmic space matches positive transitive-closure logic ([Formula: see text]).

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First-Order Logic Reasoning Support for the Semantic Web

Journal of Software ◽

10.3724/sp.j.1001.2008.03091 ◽

2009 ◽

Vol 19 (12) ◽

pp. 3091-3099 ◽

Cited By ~ 1

Author(s):

Gui-Hong XU ◽

Jian ZHANG

Keyword(s):

Semantic Web ◽

Order Logic ◽

First Order Logic ◽

First Order ◽

Logic Reasoning

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Boolean-valued structures

10.1093/oso/9780198790396.003.0013 ◽

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.

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GAMES AND CARDINALITIES IN INQUISITIVE FIRST-ORDER LOGIC

The Review of Symbolic Logic ◽

10.1017/s1755020321000198 ◽

2021 ◽

pp. 1-28

Author(s):

IVANO CIARDELLI ◽

GIANLUCA GRILLETTI

Keyword(s):

Order Logic ◽

First Order Logic ◽

First Order

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Extensions in graph normal form

Logic Journal of IGPL ◽

10.1093/jigpal/jzaa054 ◽

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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Relational Chase Procedures Interpreted as Resolution with Paramodulation1

Fundamenta Informaticae ◽

10.3233/fi-1991-15203 ◽

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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Formula size games for modal logic and μ-calculus

Journal of Logic and Computation ◽

10.1093/logcom/exz025 ◽

2019 ◽

Vol 29 (8) ◽

pp. 1311-1344 ◽

Cited By ~ 2

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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