EXPSPACE-Completeness of the Logics K4 × S5 and S4 × S5 and the Logic of Subset Spaces

2021 ◽  
Vol 22 (4) ◽  
pp. 1-71
Author(s):  
Peter Hertling ◽  
Gisela Krommes
Keyword(s):  
Upper Bounds ◽  
Satisfiability Problem ◽  
Satisfiability Problems

It is known that the satisfiability problems of the product logics K4 × S5 and S4 × S5 are NEXPTIME-hard and that the satisfiability problem of the logic SSL of subset spaces is PSPACE-hard. Furthermore, it is known that the satisfiability problems of these logics are in N2EXPTIME. We improve the lower and the upper bounds for the complexity of these problems by showing that all three problems are in ESPACE and are EXPSPACE-complete under logspace reduction.

scholarly journals An approximative inference method for solving ∃∀SO satisfiability problems

2012 ◽  
Vol 45 ◽  
pp. 79-124 ◽  
Author(s):  
H. Vlaeminck ◽  
J. Vennekens ◽  
M. Denecker ◽  
M. Bruynooghe
Keyword(s):  
Constraint Propagation ◽  
Second Order ◽  
Order Logic ◽  
Satisfiability Problem ◽  
Finite Domain ◽  
Satisfiability Problems ◽  
Conformant Planning ◽  
Inductive Definitions ◽  
Powerful Knowledge ◽  
Second Order Logic

This paper considers the fragment ∃∀SO of second-order logic. Many interesting problems, such as conformant planning, can be naturally expressed as finite domain satisfiability problems of this logic. Such satisfiability problems are computationally hard (ΣP2) and many of these problems are often solved approximately. In this paper, we develop a general approximative method, i.e., a sound but incomplete method, for solving ∃∀SO satisfiability problems. We use a syntactic representation of a constraint propagation method for first-order logic to transform such an ∃∀SO satisfiability problem to an ∃SO(ID) satisfiability problem (second-order logic, extended with inductive definitions). The finite domain satisfiability problem for the latter language is in NP and can be handled by several existing solvers. Inductive definitions are a powerful knowledge representation tool, and this moti- vates us to also approximate ∃∀SO(ID) problems. In order to do this, we first show how to perform propagation on such inductive definitions. Next, we use this to approximate ∃∀SO(ID) satisfiability problems. All this provides a general theoretical framework for a number of approximative methods in the literature. Moreover, we also show how we can use this framework for solving practical useful problems, such as conformant planning, in an effective way.

PHOTOCOMPUTING: EXPLORATIONS WITH TRANSPARENCY AND OPACITY

2007 ◽  
Vol 17 (04) ◽  
pp. 339-347 ◽  
Author(s):  
TOM HEAD
Keyword(s):  
Parallel Computing ◽  
Linear Time ◽  
Boolean Satisfiability ◽  
Satisfiability Problem ◽  
Problem Instance ◽  
Boolean Satisfiability Problem ◽  
Satisfiability Problems

We continue to search for methods of parallel computing using light. An algorithm for solving instances of the Boolean satisfiability problem is given and illustrated using a photocopying machine with plastic transparencies as medium. The algorithm solves satisfiability problems in linear time but requires the assumption that information can be stored with a density that is exponential in the number of variables in the problem instance. Consideration is given to situations in which this density limitation is not quite absolute.

On Random Betweenness Constraints

2010 ◽  
Vol 19 (5-6) ◽  
pp. 775-790 ◽  
Author(s):  
ANDREAS GOERDT
Keyword(s):  
Normal Form ◽  
High Probability ◽  
Conjunctive Normal Form ◽  
Satisfiability Problem ◽  
Linear Ordering ◽  
Satisfiability Problems ◽  
Ordering Constraints ◽  
Random Instances ◽  
Np Complete

Ordering constraints are formally analogous to instances of the satisfiability problem in conjunctive normal form, but instead of a boolean assignment we consider a linear ordering of the variables in question. A clause becomes true given a linear ordering if and only if the relative ordering of its variables obeys the constraint considered.The naturally arising satisfiability problems are NP-complete for many types of constraints. We look at random ordering constraints. Previous work of the author shows that there is a sharp unsatisfiability threshold for certain types of constraints. The value of the threshold, however, is essentially undetermined. We pursue the problem of approximating the precise value of the threshold. We show that random instances of the betweenness constraint are satisfiable with high probability if the number of randomly picked clauses is ≤0.92n, where n is the number of variables considered. This improves the previous bound, which is <0.82n random clauses. The proof is based on a binary relaxation of the betweenness constraint and involves some ideas not used before in the area of random ordering constraints.

Extension of the Spatial PDI Model to Three Dimensions

1997 ◽  
Vol 84 (1) ◽  
pp. 176-178
Author(s):  
Frank O'Brien
Keyword(s):  
Population Density ◽  
Dimensional Space ◽  
Finite Lattice ◽  
Three Dimensional ◽  
Upper Bounds ◽  
Three Dimensions ◽  
Lower And Upper Bounds ◽  
Density Index ◽  
Three Dimensional Space

The author's population density index ( PDI) model is extended to three-dimensional distributions. A derived formula is presented that allows for the calculation of the lower and upper bounds of density in three-dimensional space for any finite lattice.

Energy of spherical fuzzy graphs

2020 ◽  
pp. 321-332
Author(s):  
S. Yahya Mohamed ◽  
A. Mohamed Ali
Keyword(s):  
Adjacency Matrix ◽  
Upper Bounds ◽  
Fuzzy Graph ◽  
Lower And Upper Bounds ◽  
Fuzzy Graphs

In this paper, the notion of energy extended to spherical fuzzy graph. The adjacency matrix of a spherical fuzzy graph is defined and we compute the energy of a spherical fuzzy graph as the sum of absolute values of eigenvalues of the adjacency matrix of the spherical fuzzy graph. Also, the lower and upper bounds for the energy of spherical fuzzy graphs are obtained.

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