8,562 research outputs found
Non-Deterministic Matching Algorithm for Net Transformations
Modeling and simulating dynamic systems require to represent their processes and the system changes within one model. To that effect, reconfigurable Petri nets consist of a place/transition net and a set of rules that can modify the Petri net. The application of a rule is based on finding a suitable match of the rule in the given net. This match is an isomorphic subnet that has to be located meeting requirements of the rule application as well as the simulation. In this paper a non-deterministic algorithm is presented for the matching in reconfigurable Petri nets. It is an extension of the VF2 algorithm for graph (sub-)isomorphisms. We show that this extension is correct and complete. Non-determinism ensures that during simulation different matches can be found for each transformation step and is hence crucial for the simulation. But non-determinism has not been present in the VF2 algorithm. For the matching algorithm non-determinism is proven
Physical portrayal of computational complexity
Computational complexity is examined using the principle of increasing
entropy. To consider computation as a physical process from an initial instance
to the final acceptance is motivated because many natural processes have been
recognized to complete in non-polynomial time (NP). The irreversible process
with three or more degrees of freedom is found intractable because, in terms of
physics, flows of energy are inseparable from their driving forces. In
computational terms, when solving problems in the class NP, decisions will
affect subsequently available sets of decisions. The state space of a
non-deterministic finite automaton is evolving due to the computation itself
hence it cannot be efficiently contracted using a deterministic finite
automaton that will arrive at a solution in super-polynomial time. The solution
of the NP problem itself is verifiable in polynomial time (P) because the
corresponding state is stationary. Likewise the class P set of states does not
depend on computational history hence it can be efficiently contracted to the
accepting state by a deterministic sequence of dissipative transformations.
Thus it is concluded that the class P set of states is inherently smaller than
the set of class NP. Since the computational time to contract a given set is
proportional to dissipation, the computational complexity class P is a subset
of NP.Comment: 16, pages, 7 figure
Deterministic Polynomial Time Algorithms for Matrix Completion Problems
We present new deterministic algorithms for several cases of the maximum rank
matrix completion problem (for short matrix completion), i.e. the problem of
assigning values to the variables in a given symbolic matrix as to maximize the
resulting matrix rank. Matrix completion belongs to the fundamental problems in
computational complexity with numerous important algorithmic applications,
among others, in computing dynamic transitive closures or multicast network
codings (Harvey et al SODA 2005, Harvey et al SODA 2006).
We design efficient deterministic algorithms for common generalizations of
the results of Lovasz and Geelen on this problem by allowing linear functions
in the entries of the input matrix such that the submatrices corresponding to
each variable have rank one. We present also a deterministic polynomial time
algorithm for finding the minimal number of generators of a given module
structure given by matrices. We establish further several hardness results
related to matrix algebras and modules. As a result we connect the classical
problem of polynomial identity testing with checking surjectivity (or
injectivity) between two given modules. One of the elements of our algorithm is
a construction of a greedy algorithm for finding a maximum rank element in the
more general setting of the problem. The proof methods used in this paper could
be also of independent interest.Comment: 14 pages, preliminar
Complex Networks and Symmetry I: A Review
In this review we establish various connections between complex networks and
symmetry. While special types of symmetries (e.g., automorphisms) are studied
in detail within discrete mathematics for particular classes of deterministic
graphs, the analysis of more general symmetries in real complex networks is far
less developed. We argue that real networks, as any entity characterized by
imperfections or errors, necessarily require a stochastic notion of invariance.
We therefore propose a definition of stochastic symmetry based on graph
ensembles and use it to review the main results of network theory from an
unusual perspective. The results discussed here and in a companion paper show
that stochastic symmetry highlights the most informative topological properties
of real networks, even in noisy situations unaccessible to exact techniques.Comment: Final accepted versio
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