Syntactic Complexity of Prefix-, Suffix-, Bifix-, and Factor-Free Regular Languages

The syntactic complexity of a regular language is the cardinality of its syntactic semigroup. The syntactic complexity of a subclass of the class of regular languages is the maximal syntactic complexity of languages in that class, taken as a function of the state complexity $n$ of these languages. We study the syntactic complexity of prefix-, suffix-, bifix-, and factor-free regular languages. We prove that $n^{n-2}$ is a tight upper bound for prefix-free regular languages. We present properties of the syntactic semigroups of suffix-, bifix-, and factor-free regular languages, conjecture tight upper bounds on their size to be $(n-1)^{n-2}+(n-2)$, $(n-1)^{n-3} + (n-2)^{n-3} + (n-3)2^{n-3}$, and $(n-1)^{n-3} + (n-3)2^{n-3} + 1$, respectively, and exhibit languages with these syntactic complexities.Comment: 28 pages, 6 figures, 3 tables. An earlier version of this paper was presented in: M. Holzer, M. Kutrib, G. Pighizzini, eds., 13th Int. Workshop on Descriptional Complexity of Formal Systems, DCFS 2011, Vol. 6808 of LNCS, Springer, 2011, pp. 93-106. The current version contains improved bounds for suffix-free languages, new results about factor-free languages, and new results about reversa

Weakly Submodular Functions

Submodular functions are well-studied in combinatorial optimization, game theory and economics. The natural diminishing returns property makes them suitable for many applications. We study an extension of monotone submodular functions, which we call {\em weakly submodular functions}. Our extension includes some (mildly) supermodular functions. We show that several natural functions belong to this class and relate our class to some other recent submodular function extensions. We consider the optimization problem of maximizing a weakly submodular function subject to uniform and general matroid constraints. For a uniform matroid constraint, the "standard greedy algorithm" achieves a constant approximation ratio where the constant (experimentally) converges to 5.95 as the cardinality constraint increases. For a general matroid constraint, a simple local search algorithm achieves a constant approximation ratio where the constant (analytically) converges to 10.22 as the rank of the matroid increases

A Formal Theory for the Complexity Class Associated with the Stable Marriage Problem

Subramanian defined the complexity class CC as the set of problems log-space reducible to the comparator circuit value problem. He proved that several other problems are complete for CC, including the stable marriage problem, and finding the lexicographical first maximal matching in a bipartite graph. We suggest alternative definitions of CC based on different reducibilities and introduce a two-sorted theory VCC* based on one of them. We sharpen and simplify Subramanian\u27s completeness proofs for the above two problems and formalize them in VCC*

On The Complexity Of The Evaluation Of Transient Extensions Of Boolean Functions

Electronic version of an article published as International Journal of Foundations of Computer Science, 23(01), 2012, 21–35. http://dx.doi.org/10.1142/S0129054112400023 © World Scientific Publishing Company http://www.worldscientific.com/Transient algebra is a multi-valued algebra for hazard detection in gate circuits. Sequences of alternating 0's and 1's, called transients, represent signal values, and gates are modeled by extensions of boolean functions to transients. Formulas for computing the output transient of a gate from the input transients are known for NOT, AND, OR and XOR gates and their complements, but, in general, even the problem of deciding whether the length of the output transient exceeds a given bound is NP-complete. We propose a method of evaluating extensions of general boolean functions. We study a class of functions for which, instead of evaluating the extensions on a given set of transients, it is possible to get the same values by using transients derived from the given ones, but having length at most 3. We prove that all functions of three variables, as well as certain other functions, have this property, and can be efficiently evaluated.Natural Sciences and Engineering Research Council of Canada [OGP0000871]Department of Computer Science, University of Toront

Allocating capital-associated CO₂ emissions along the full lifespan of capital investments helps diffuse emission responsibility

Capital assets such as machinery and infrastructure contribute substantially to CO2 emissions over their lifetime. Unique features of capital assets such as their long durability complicate the assignment of capital-associated CO2 emissions to final beneficiaries. Whereas conventional approaches allocate emissions required to produce capital assets to the year of formation, we propose an alternative perspective through allocating required emissions from the production of assets over their entire lifespans. We show that allocating CO2 emissions embodied in capital assets over time relieves emission responsibility for the year of formation, with 25‒46% reductions from conventional emission accounts. This temporal allocation, although virtual, is important for assessing the equity of CO2 emissions across generations due to the inertia of capital assets. To re-allocate emission responsibilities to the future, we design three capital investment scenarios with different investment purposes until 2030. Overall, the existing capital in 2017 will still carry approximately 10% responsibilities of China’s CO2 emissions in 2030, and could reach more than 40% for capital-intensive service sectors.</p