Universal Witnesses for State Complexity of Basic Operations Combined with Reversal

We study the state complexity of boolean operations, concatenation and star with one or two of the argument languages reversed. We derive tight upper bounds for the symmetric differences and differences of such languages. We prove that the previously discovered bounds for union, intersection, concatenation and star of such languages can all be met by the recently introduced universal witnesses and their variants.Comment: 18 pages, 8 figures. LNCS forma

Linear superposition as a core theorem of quantum empiricism

Clarifying the nature of the quantum state $|\Psi\rangle$ is at the root of the problems with insight into (counterintuitive) quantum postulates. We provide a direct-and math-axiom free-empirical derivation of this object as an element of a vector space. Establishing the linearity of this structure-quantum superposition-is based on a set-theoretic creation of ensemble formations and invokes the following three principia: $(\textsf{I})$ quantum statics, $(\textsf{II})$ doctrine of a number in the physical theory, and $(\textsf{III})$ mathematization of matching the two observations with each other; quantum invariance. All of the constructs rest upon a formalization of the minimal experimental entity: observed micro-event, detector click. This is sufficient for producing the $\mathbb C$-numbers, axioms of linear vector space (superposition principle), statistical mixtures of states, eigenstates and their spectra, and non-commutativity of observables. No use is required of the concept of time. As a result, the foundations of theory are liberated to a significant extent from the issues associated with physical interpretations, philosophical exegeses, and mathematical reconstruction of the entire quantum edifice.Comment: No figures. 64 pages; 68 pages(+4), overall substantial improvements; 70 pages(+2), further improvement

Unrestricted State Complexity of Binary Operations on Regular and Ideal Languages

We study the state complexity of binary operations on regular languages over different alphabets. It is known that if $L'_m$ and $L_n$ are languages of state complexities $m$ and $n$, respectively, and restricted to the same alphabet, the state complexity of any binary boolean operation on $L'_m$ and $L_n$ is $mn$, and that of product (concatenation) is $m 2^n - 2^{n-1}$. In contrast to this, we show that if $L'_m$ and $L_n$ are over different alphabets, the state complexity of union and symmetric difference is $(m+1)(n+1)$, that of difference is $mn+m$, that of intersection is $mn$, and that of product is $m2^n+2^{n-1}$. We also study unrestricted complexity of binary operations in the classes of regular right, left, and two-sided ideals, and derive tight upper bounds. The bounds for product of the unrestricted cases (with the bounds for the restricted cases in parentheses) are as follows: right ideals $m+2^{n-2}+2^{n-1}$ ($m+2^{n-2}$); left ideals $mn+m+n$ ($m+n-1$); two-sided ideals $m+2n$ ($m+n-1$). The state complexities of boolean operations on all three types of ideals are the same as those of arbitrary regular languages, whereas that is not the case if the alphabets of the arguments are the same. Finally, we update the known results about most complex regular, right-ideal, left-ideal, and two-sided-ideal languages to include the unrestricted cases.Comment: 30 pages, 15 figures. This paper is a revised and expanded version of the DCFS 2016 conference paper, also posted previously as arXiv:1602.01387v3. The expanded version has appeared in J. Autom. Lang. Comb. 22 (1-3), 29-59, 2017, the issue of selected papers from DCFS 2016. This version corrects the proof of distinguishability of states in the difference operation on p. 12 in arXiv:1609.04439v

Complexity of Left-Ideal, Suffix-Closed and Suffix-Free Regular Languages

A language $L$ over an alphabet $\Sigma$ is suffix-convex if, for any words $x,y,z\in\Sigma^*$, whenever $z$ and $xyz$ are in $L$, then so is $yz$. Suffix-convex languages include three special cases: left-ideal, suffix-closed, and suffix-free languages. We examine complexity properties of these three special classes of suffix-convex regular languages. In particular, we study the quotient/state complexity of boolean operations, product (concatenation), star, and reversal on these languages, as well as the size of their syntactic semigroups, and the quotient complexity of their atoms.Comment: 20 pages, 11 figures, 1 table. arXiv admin note: text overlap with arXiv:1605.0669