Grant Pogosyan

Post classes characterized by functional terms

Abstract The classes of Boolean functions closed under classical compositions form an algebraic lattice which was completely described in 1941 in a pioneering work of Post (Ann. Math. Stud. (5) (1941)). These classes and the lattice are often referred to as the Post Classes and the Post Lattice, respectively. There are several approaches to present a Post class. Being a set of functions it can be characterized by a traditional set-theoretic description of its members. Since the Post classes are closed under certain operations, they are also often presented by sets of generators. A remarkable approach, which has been widely used in Universal Algebra is a characterization of classes by means of preservation of polyrelations (Kibernetika (3)(Pt1) (1969) 1, (5)(PtII) (1969) 1). Recently, there appeared several new methods for the characterization of classes of logic functions. These methods are based on special formal expression, which in general define a much larger variety of classes particularly including all Post classes. One such result is by Ekin et al. (Discrete Math. 211 (2000) 27), which presents the characterization of classes by a set (possibly infinite) of certain equational identities. The approach developed in [Ekin et al. (2000)] was soon extended to several directions. Pippenger (Discrete Math. 254 (2002) 405) presented the classes through pairs of relations (constrains) in the setting of a Galois Theory. Pogosyan (Multiple Valued Logic, Gordon and Breach, London, 2001, pp. 417–448, Vol. 7) has defined each such class by one functional term (possibly of infinite length), and has introduced the notion of rank for a term as well as for a class. In (Algebra Universalis 44 (2000) 309) established a connection with the Birkhoff-Tarski HSP Theorem. This paper presents a complete characterization of the Post Classes by means of functional terms (as in [Pogosyan, 2001]). We also give a constructive criterion which establishes the minimal ranks for all Post classes.

The number of orthogonal permutations

A problem on maximal clones in universal algebra leads to the natural concept of orthogonal orders and their characterization. Two (partial) orders on the same set P are orthogonal if they share only trivial endomorphisms, i.e. if the identity self-map of P is the sole non-constant self-map preserving (i.e. compatible with) both orders. We start with a neat and easy characterization of orthogonal pairs of chains (i.e. linear or total orders) and then proceed to the study of the number q(k) of chains on {0, 1, …, k − 1} orthogonal to the natural chain 0 < 1 < ⋯ < k − 1. We obtain a recurrence formula for q(k) and prove that the ratio q(k)k! (of such chains among all chains) goes to e−2 = 0.1353 ⋯ as k → ∞. Results are formulated in terms of permutations.

Semirigid sets of central relations over a finite domain

A set of central h-ary relations on a set A is called semirigid if the clones of k-valued logic functions determined by the relations share only the clone K/sub h-1/ consisting of all projections and all functions assuming at most h-1 values (12; K/sub 1/ is the set of trivial functions, i.e., the clone consisting of all constants and all projections). The problem of determining semirigid sets of central relations is studied. For the set of h-ary central relations with the centers of the largest size, it is shown that the set consisting of all such relations is the only semirigid set. It is also shown that the minimum size of a semirigid set of central h-ary relations is h+1. For k=4, semirigid sets of binary central relations are investigated in detail.<<ETX>>

On the Size Complexity of Deterministic Frequency Automata

Austinat, Diekert, Hertrampf, and Petersen [2] proved that every language L that is (m,n)-recognizable by a deterministic frequency automaton such that m > n/2 can be recognized by a deterministic finite automaton as well. First, the size of deterministic frequency automata and of deterministic finite automata recognizing the same language is compared. Then approximations of a language are considered, where a language L′ is called an approximation of a language L if L′ differs from L in only a finite number of strings. We prove that if a deterministic frequency automaton has k states and (m,n)-recognizes a language L, where m > n/2, then there is a language L′ approximating L such that L′ can be recognized by a deterministic finite automaton with no more than k states.

Semirigid sets of diamond orders

Abstract An order relation ⩽ab on a set A is a diamond provided x ⩽aby holds exactly if x = a or y = b. A set R of diamonds on A is semirigid if the identity map on A and all constant self-maps of A are the only self-maps of A that are (jointly) isotone for all diamonds from R. The study of such sets is motivated by the classification of bases in multiple-valued logics. We give a simple semirigidity criterion. For A finite we describe all semirigid sets of diamonds of the least possible cardinality and give their number. We also give nonsemirigid sets of diamonds of the maximum possible cardinality. We find the total number of semirigid sets of diamonds and their ratio among all sets of diamonds. This ratio converges fast to 1; e.g. for a 26-element set A the probability that a randomly chosen set of diamonds is semirigid is 0.999 9992 …

Generation of the Post Lattice by irreducible clones

Clones of logic functions form an algebraic lattice, which in the case of Boolean functions was completely described by E.L.Post in 1941. This lattice, often referred to as the Post Lattice, has been well studied from various angles, particularly, the generation of the Post Lattice by its subsets. This paper discusses the results about clones that are irreducible by means of meet- and/or join-operations of the lattice. We show that the join-irreducible clones generate the Post Lattice, and dually, the meet-irreducible clones generate the Post Lattice. We present a complete description of such generation in both cases. We observe that any clone can be presented as a join or meet of at most two irreducible clones. In the former case each clone is the join of at most four join-irreducible clones and in the latter case each clone is the meet of at most three meet-irreducible ones.

Efficiently irreducible bases in multiple-valued logic

Basis is a functionally complete set of multiple-valued logic functions that is irreducible, i.e. contains no complete proper subsets. Functional completeness of a set means that for any function in MVL there exists a formula over this set that implements it. However, this classical definition of basis does not consider the efficiency of implementation, particularly, it does not guarantee the existence of an efficient implementation regarding the complexity of formal expressions. In this note the notion of efficiently irreducible basis is introduced, and is termed /spl epsiv/-basis. A criterion for the basic set of operations to be efficiently irreducible is given. In the cases of Boolean and ternary logic functions complete enumeration and description of /spl epsiv/-bases are presented.

Hereditary clones of multiple valued logic algebra

We discuss relationships between properties and operations over the set /spl Omega/ of MVL functions. Closed properties are those invariant under the classical closure operation. A new type of properties, called hereditary, is defined, as well as hereditary closure. We calculate the ratio of hereditary properties, describe the families of maximal hereditary clones, and give a formula for their enumeration. We show that there are exactly eleven such clones in ternary logic. For Boolean algebra the lattice of all hereditary classes is finite, and we describe it completely. Meanwhile, starting from the three valued case there are still a continuum number of clones.<<ETX>>

A map from the lower-half of the n -cube onto the ( n −1)-cube which preserves intersecting antichains

Abstract We prove that there is a 1–1 correspondence between the set of intersecting antichains in the lower-half of the n-cube and the set of intersecting antichains in the (n−1)-cube. This reduces the enumeration of intersecting antichains contained in the former set to that in the latter.

Semirigid sets of quasilinear clones

Let k be a prime and G a Galois field on k:=(0,1,. . .,k-1). The set of all quasilinear (or affine with respect to G) k-valued logic functions is a maximal clone called quasilinear. A family of quasilinear clones on k is semirigid if the clones of the family share exactly the constant functions and the projections. Semirigid sets of quasilinear clones are needed for the classification of bases of k-valued logic, which is unknown for k>3. The authors characterize all semirigid sets of quasilinear clones. In particular, for k=5 they describe all semirigid triples of quasilinear clones and show that no such pair exists. For every prime k>5 they exhibit a semirigid pair of quasi-linear clones. The techniques used are based on elementary number theory and on polynomials over G.<<ETX>>