Algebra and Logic最新文献

We study temporal multi-agent logics using a new approach to defining time for individual agents. It is assumed that in any time state each agent (in a sense) generates its own future time, which will only be available for analysis in the future. That is, the defined time interval depends both on the agent and on the initial state where the agent starts to act. It is also assumed that the agent may have intervals of forgotten (lost) time. We investigate problems of unification and problems of computable recognizing admissible inference rules. An algorithm is found for solving these problems based on the construction of a finite computable set of formulas which is a complete set of unifiers. We use the technique of projective formulas developed by S. Ghilardi. It is proved that any unifiable formula is in fact projective and an algorithm is constructed which creates its projective unifier. Thereby we solve the unification problem, and based at this, find the solution to the open problem of computable recognizing admissible inference rules.

ℵ₁-free groups, Abelian groups for which every countable subgroup is free, exhibit a number of interesting algebraic and set-theoretic properties. We will give a complete proof that the property of being ℵ₁-free is absolute; that is, if an Abelian group G is ℵ₁-free in some transitive model M of ZFC, then it is ℵ₁-free in any transitive model of ZFC containing G. The absoluteness of ℵ₁-freeness has the following remarkable consequence: an Abelian group G is ℵ₁-free in some transitive model of ZFC if and only if it is (countable and) free in some model extension. This set-theoretic characterization will be a starting point for further exploring the relationship between the set-theoretic and algebraic properties of ℵ₁-free groups. In particular, we will demonstrate how proofs may be dramatically simplified using model extensions for ℵ₁-free groups.

It is shown that any low linear order of the form (mathcal{L})+ω^∗, where (mathcal{L}) is some η-presentation, has a computable copy. This result contrasts with there being low η-presentations not having a computable copy.

We look at the complexity of the existence problem for a Horn sentence equivalent to a given one. It is proved that for a signature consisting of one unary function symbol and any finite number of unary predicate symbols, the problem is computable. For a signature with at least two unary function symbols, it is stated that the problem mentioned is an m-complete ({sum }_{1}^{0}mathrm{set}).

The Lambek calculus was introduced as a tool for examining linguistic constructions. Then this calculus was complemented with both new connectives and structural rules like contraction, weakening, and permutation. The structural rules are allowed only for formulas under the symbol of a specific modality called exponential. The Lambek calculus itself is a noncommutative structural logic, and for arbitrary formulas, the structural rules mentioned are not allowed. The next step is the introduction of a system of subexponentials: under the symbol of such a modality, only certain structural rules are admitted. The following question arises: is it possible to formulate a system with a certain version of the local contraction rule (for formulas under subexponential) to recover the cut elimination property? We consider two approaches to solving this problem: one can both weaken the rule of introducing ! in the right-hand side of a sequent (LSCLC) and extend the local contraction rule from individual formulas to their subsequents (LMCLC). It is also worth mentioning that in commutative calculi, such a problem is missing since formulas in a sequent are allowed to be permuted (i.e, the local contraction rule coincides with the nonlocal one). The following results are proved: cut eliminability in the LMCLC and LSCLC calculi; algorithmic decidability of fragments of these calculi in which ! is allowed to be applied only to variables; algorithmic undecidability of LMCLC (for LSCLC, decidability remains an open question); correctness and absence of strong completeness of LSCLC with respect to a class of relational models; various results on equivalence for the calculi in question and the calculi with other versions of the contraction subexponential.

Many important properties are identified and criteria are developed for the existence of subquasigroups in finite quasigroups. Based on these results, we propose an effective method that concludes the nonexistence of proper subquasigroups in a given finite quasigroup, or finds all its proper subquasigroups. This has an important application in checking the cryptographic suitability of a quasigroup. Using arithmetic of finite fields, we introduce a binary operation to construct quasigroups of order p^r. Criteria are developed under which the quasigroups mentioned have desirable cryptographic properties, such as polynomial completeness and absence of proper subquasigroups. Effective methods are given for constructing cryptographically suitable quasigroups. The efficiency of the methods is illustrated by some academic examples and implementation of all proposed algorithms in the computer algebra system Singular.

We give an algebraic description of Boolean algebras autostable relative to n-decidable presentations. Also, autostable I_λ,μ-algebras are described.