51,508 research outputs found
Automated Reasoning over Deontic Action Logics with Finite Vocabularies
In this paper we investigate further the tableaux system for a deontic action
logic we presented in previous work. This tableaux system uses atoms (of a
given boolean algebra of action terms) as labels of formulae, this allows us to
embrace parallel execution of actions and action complement, two action
operators that may present difficulties in their treatment. One of the
restrictions of this logic is that it uses vocabularies with a finite number of
actions. In this article we prove that this restriction does not affect the
coherence of the deduction system; in other words, we prove that the system is
complete with respect to language extension. We also study the computational
complexity of this extended deductive framework and we prove that the
complexity of this system is in PSPACE, which is an improvement with respect to
related systems.Comment: In Proceedings LAFM 2013, arXiv:1401.056
New developments in the theory of Groebner bases and applications to formal verification
We present foundational work on standard bases over rings and on Boolean
Groebner bases in the framework of Boolean functions. The research was
motivated by our collaboration with electrical engineers and computer
scientists on problems arising from formal verification of digital circuits. In
fact, algebraic modelling of formal verification problems is developed on the
word-level as well as on the bit-level. The word-level model leads to Groebner
basis in the polynomial ring over Z/2n while the bit-level model leads to
Boolean Groebner bases. In addition to the theoretical foundations of both
approaches, the algorithms have been implemented. Using these implementations
we show that special data structures and the exploitation of symmetries make
Groebner bases competitive to state-of-the-art tools from formal verification
but having the advantage of being systematic and more flexible.Comment: 44 pages, 8 figures, submitted to the Special Issue of the Journal of
Pure and Applied Algebr
Min-Rank Conjecture for Log-Depth Circuits
A completion of an m-by-n matrix A with entries in {0,1,*} is obtained by
setting all *-entries to constants 0 or 1. A system of semi-linear equations
over GF(2) has the form Mx=f(x), where M is a completion of A and f:{0,1}^n -->
{0,1}^m is an operator, the i-th coordinate of which can only depend on
variables corresponding to *-entries in the i-th row of A. We conjecture that
no such system can have more than 2^{n-c\cdot mr(A)} solutions, where c>0 is an
absolute constant and mr(A) is the smallest rank over GF(2) of a completion of
A. The conjecture is related to an old problem of proving super-linear lower
bounds on the size of log-depth boolean circuits computing linear operators x
--> Mx. The conjecture is also a generalization of a classical question about
how much larger can non-linear codes be than linear ones. We prove some special
cases of the conjecture and establish some structural properties of solution
sets.Comment: 22 pages, to appear in: J. Comput.Syst.Sci
Symmetry in Critical Random Boolean Network Dynamics
Using Boolean networks as prototypical examples, the role of symmetry in the
dynamics of heterogeneous complex systems is explored. We show that symmetry of
the dynamics, especially in critical states, is a controlling feature that can
be used both to greatly simplify analysis and to characterize different types
of dynamics. Symmetry in Boolean networks is found by determining the frequency
at which the various Boolean output functions occur. There are classes of
functions that consist of Boolean functions that behave similarly. These
classes are orbits of the controlling symmetry group. We find that the symmetry
that controls the critical random Boolean networks is expressed through the
frequency by which output functions are utilized by nodes that remain active on
dynamical attractors. This symmetry preserves canalization, a form of network
robustness. We compare it to a different symmetry known to control the dynamics
of an evolutionary process that allows Boolean networks to organize into a
critical state. Our results demonstrate the usefulness and power of using the
symmetry of the behavior of the nodes to characterize complex network dynamics,
and introduce a novel approach to the analysis of heterogeneous complex
systems
Construction and analysis of causally dynamic hybrid bond graphs
Engineering systems are frequently abstracted to models with discontinuous behaviour (such as a switch or contact),
and a hybrid model is one which contains continuous and discontinuous behaviours. Bond graphs are an established
physical modelling method, but there are several methods for constructing switched or ‘hybrid’ bond graphs, developed
for either qualitative ‘structural’ analysis or efficient numerical simulation of engineering systems. This article proposes a
general hybrid bond graph suitable for both. The controlled junction is adopted as an intuitive way of modelling a discontinuity in the model structure. This element gives rise to ‘dynamic causality’ that is facilitated by a new bond graph notation. From this model, the junction structure and state equations are derived and compared to those obtained by
existing methods. The proposed model includes all possible modes of operation and can be represented by a single set
of equations. The controlled junctions manifest as Boolean variables in the matrices of coefficients. The method is more
compact and intuitive than existing methods and dispenses with the need to derive various modes of operation from a
given reference representation. Hence, a method has been developed, which can reach common usage and form a platform for further study
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