scholarly journals Results on the Propositional µ-Calculus

1982 ◽  
Vol 11 (146) ◽  
Author(s):  
Dexter Kozen
Keyword(s):  
Representation Theorem ◽  
Decision Procedure ◽  
Deductive System ◽  
Algebraic Semantics ◽  
Model Property ◽  
Exponential Time ◽  
Small Model

We define a propositional version of the µ-calculus, and give an exponential-time decision procedure, small model property, and complete deductive system. We also show that it is strictly more expressive than PDL. Finally we give an algebraic semantics and prove a representation theorem.

Small substructures and decidability issues for first-order logic with two variables

2012 ◽  
Vol 77 (3) ◽  
pp. 729-765 ◽  
Author(s):  
Emanuel Kieroński ◽  
Martin Otto
Keyword(s):  
Order Logic ◽  
First Order Logic ◽  
Equivalence Relations ◽  
Finite Model ◽  
Model Property ◽  
Finite Model Property ◽  
First Order ◽  
Surrounding Structure ◽  
Small Model ◽  
Exponential Size

AbstractWe study first-order logic with two variables FO2 and establish a small substructure property. Similar to the small model property for FO2 we obtain an exponential size bound on embedded substructures, relative to a fixed surrounding structure that may be infinite. We apply this technique to analyse the satisfiability problem for FO2 under constraints that require several binary relations to be interpreted as equivalence relations. With a single equivalence relation, FO2 has the finite model property and is complete for non-deterministic exponential time, just as for plain FO2. With two equivalence relations, FO2 does not have the finite model property, but is shown to be decidable via a construction of regular models that admit finite descriptions even though they may necessarily be infinite. For three or more equivalence relations, FO2 is undecidable.

The finite model property for knotted extensions of propositional linear logic

2005 ◽  
Vol 70 (1) ◽  
pp. 84-98 ◽  
Author(s):  
C. J. van Alten
Keyword(s):  
Linear Logic ◽  
Finite Model ◽  
Algebraic Semantics ◽  
Structural Rule ◽  
Model Property ◽  
Algebraic Models ◽  
Finite Model Property ◽  
Finite Embeddability Property ◽  
Full Algebra ◽  
Intuitionistic Linear Logic

AbstractThe logics considered here are the propositional Linear Logic and propositional Intuitionistic Linear Logic extended by a knotted structural rule: . It is proved that the class of algebraic models for such a logic has the finite embeddability property, meaning that every finite partial subalgebra of an algebra in the class can be embedded into a finite full algebra in the class. It follows that each such logic has the finite model property with respect to its algebraic semantics and hence that the logic is decidable.

Monadic NM-algebras

2019 ◽  
Vol 27 (6) ◽  
pp. 812-835
Author(s):  
Juntao Wang ◽  
Pengfei He ◽  
Yanhong She
Keyword(s):  
Representation Theorem ◽  
Predicate Logic ◽  
Algebraic Semantics ◽  
Subdirectly Irreducible ◽  
Approximation Spaces ◽  
Rough Approximation ◽  
Subdirect Representation ◽  
Algebraic Foundation ◽  
The Relationship ◽  
Monadic Filters

Abstract In this paper, we investigate universal and existential quantifiers on NM-algebras. The resulting class of algebras will be called monadic NM-algebras. First, we show that the variety of monadic NM-algebras is algebraic semantics of the monadic NM-predicate logic. Moreover, we discuss the relationship among monadic NM-algebras, modal NM-algebras and rough approximation spaces. Second, we introduce and investigate monadic filters in monadic NM-algebras. Using them, we prove the subdirect representation theorem of monadic NM-algebras, and characterize simple and subdirectly irreducible monadic NM-algebras. Finally, we present the monadic NM-logic and prove its (chain) completeness with respect to (strong) monadic NM-algebras. These results constitute a crucial first step for providing an algebraic foundation for the monadic NM-predicate logic.

scholarly journals The Small Model Property: How Small Can It Be?

2002 ◽  
Vol 178 (1) ◽  
pp. 279-293 ◽  
Author(s):  
Amir Pnueli ◽  
Yoav Rodeh ◽  
Ofer Strichman ◽  
Michael Siegel
Keyword(s):  
Model Property ◽  
Small Model

Logical questions concerning the μ-calculus: Interpolation, Lyndon and Łoś-Tarski

2000 ◽  
Vol 65 (1) ◽  
pp. 310-332 ◽  
Author(s):  
Giovanna D'agostino ◽  
Marco Hollenberg
Keyword(s):  
Decision Procedure ◽  
Sequent Calculus ◽  
Theoretical Perspective ◽  
Order Logic ◽  
Rite Of Passage ◽  
Finite Model ◽  
Finite Model Property ◽  
Exponential Time ◽  
Second Order Logic ◽  
Monadic Second Order Logic

The (modal) μ-calculus ([14]) is a very powerful extension of modal logic with least and greatest fixed point operators. It is of great interest to computer science for expressing properties of processes such as termination (every run is finite) and fairness (on every infinite run, no action is repeated infinitely often to the exclusion of all others).The power of the μ-calculus is also evident from a more theoretical perspective. The μ-calculus is a fragment of monadic second-order logic (MSO) containing only formulae that are invariant for bisimulation, in the sense that they cannot distinguish between bisimilar states. Janin and Walukiewicz prove the converse: any property which is invariant for bisimulation and MSO-expressible is already expressible in the μ-calculus ([13]). Yet the μ-calculus enjoys many desirable properties which MSO lacks, like a complete sequent-calculus ([29]), an exponential-time decision procedure, and the finite model property ([25]). Switching from MSO to its bisimulation-invariant fragment gives us these desirable properties.In this paper we take a classical logician's view of the μ-calculus. As far as we are concerned a new logic should not be allowed into the community of logics without at least considering the standard questions that any logic is bothered with. In this paper we perform this rite of passage for the μ-calculus. The questions we will be concerned with are the following.

scholarly journals Erratum to “The small model property: how small can it be?” [Inform. Comput. 178 (2002) 279–293]

2003 ◽  
Vol 184 (1) ◽  
pp. 227 ◽  
Author(s):  
Amir Pnueli ◽  
Yoav Rodeh ◽  
Ofer Strichman ◽  
Michael Siegel
Keyword(s):  
Model Property ◽  
Small Model
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