Abstract We present a uniform description of sets of m linear forms in n variables over the field of rational numbers whose computation requires m(n – 1) additions.

We present two deductively equivalent calculi for non-deterministic many-valued logics. One is defined by axioms and the other - by rules of inference. The two calculi are obtained from the truth tables of the logic under consideration in a straightforward manner. We prove soundness and strong completeness theorems for both calculi and also prove the cut elimination theorem for the calculi defined by rules of inference.

We present a uniform description of sets of m linear forms in n variables over the field of rational numbers whose computation requires m(n - 1) additions. Our result is based on bounds on the height of the annihilating polynomials in the Perron theorem and an effective form of the Lindemann-Weierstrass theorem which is due to Sert (1999).

Let μq2(n,k) denote the minimum number of multiplications required to compute the coefficients of the product of two degree n k - 1 polynomials modulo the kth power of an irreducible polynomial of degree n over the q2element field \BBF q2. It is shown that for all odd q and all n = 1,2,..., liminfk → ∞[( μq2(n,k))/ k n] ≤ 2 (1 + [ 1/( q - 2)] ). For the proof of this upper bound, we show that for an odd prime power q, all algebraic function fields in the Garcia-Stichtenoth tower over \BBF q2have places of all degrees and apply a Chudnovsky like algorithm for multiplication of polynomials modulo a power of an irreducible polynomial.

It is shown that the emptiness problemfor two-pebble automata languages is undecidable and that two-pebble automata are weaker than three-pebble automata.

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It is shown that computing the coefficients of the product of two degree-n polynomials over a q-element field by means of a quadratic algorithm requires at least $(3 + \frac{\scriptstyle (q - 1)^2}{\scriptstyle q^5 + (q - 1)^3})n - o(n)$ multiplications.

In this paper we present an alternative ‘equivalent’ semantics of the Temporal Logic of Actions. This semantics is defined with respect to the order of non-negative real numbers and is simple and intuitive.

It is shown that a stutter‐invariant property is expressible in Temporal Logic of Actions if and only if it is expressible in Second‐order Temporal Logic. In particular, validity questions can be translated from one logic to the other. The proof is based on equivalence transformations between the formulas of Temporal Logic of Actions and Second‐order Temporal Logic. The translation from Second‐order Temporal Logic into Temporal Logic of Actions is linear and the translation from Temporal Logic of Actions into Second‐order Temporal Logic is quadratic.

In this article, we introduce first-order autoepistemic logic. Our definition is semantical and is based on the intuition similar to that lying behind the definition of first-order default logic. Thus, our definition of first-order autoepistemic logic well complies with that of first-order default logic and circumscription, providing a substantial evidence for its acceptance.

In this paper we introduce first-order non-monotonic modal logic and study its properties. Our definition is semantical and is based on the intuition similar to that lying behind the definition of first-order default logic. Thus, our definition of first-order non-monotonic modal logic well complies with that of first-order default logic and circumscription, providing a substantial evidence for its acceptance. In addition, our definition implies all properties of propositional non-monotonic modal logic, which supplies an additional support for its adequacy.

In this paper we analyse the semantical definition of extensions for open default theories. We argue that this definition reflects the domain closure assumption and show how the domain closure assumption for countable and finite domains can be expressed in first-order default logic extended with the Carnap rule of inference. Also we give examples of the domain dependence of extensions for open default theories. In particular, we show that such extensions do not possess the minimality property.

It it shown that the validity questions in prepositional temporal logic of actions can be translated into validity questions of a temporal logic and vice versa. In particular, actions of prepositional temporal logic of actions can be equivalently replaced by prepositional symbols.

A model of computation dealing with infinite alphabets is proposed. This model is based on replacing the equality test by substitution. It appears to be a natural generalization of the classical Rabin-Scott finite-state automata and possesses many of their closure and decision properties. Also, when restricted to finite alphabets the model is equivalent to finite-state automata.

Conclusions by failure to prove the opposite are frequently used in reasoning about an incompletely specdied world.This naturally leads to logics for default reasoning that, in general, are nonmonotonic; that is, introducing new facts can invalidate previously made conclusions.Accordingly, a nonmonotonic theory is called (nonmonotonically) degenerate, if adding new axioms does not invalidate already-proved theorems.Nonmonotonic logics are studied on the basis of various sets of defaults and a necessary and sufficient condition is presented for a nonmonotonic modal theory to be degenerate.In particular, this condition provides several alternative descriptions of degenerate theories.Also some closure properties of sets of defaults defining a nonmonotonic modal logic are established.

A model of computation dealing with infinite alphabets is proposed. The model is based on replacing the equality test by unification. It appears to be a natural generalization of the classical Rabin-Scott finite-state automata and possesses many of their properties.>

We define various kinds of branching time logic consisting of path quantifiers and second-order (ω-regular) linear time logic which are extensions of CTL* (computation tree logic). These logics are decidable for the computation tree semantics. However, if we adopt the non-tree semantics reflecting the possible computations of a looping program eventually returning to the same state of computation, the picture is quite different, and one of the logics under consideration becomes highly undecidable. Nevertheless, this semantics allows a simpler model definition, including finite models corresponding to a finite state machine (program), and circular path finite models reflecting programs, which are really deterministic finite state machines, but the model's view does not include all the details of program state. Since circular path finite model may have only countably many possible paths (computations), this semantics is much simpler than the usual one based on sets of all infinite paths in computation tree. We show that for the finite models the circular path semantics is equivalent to the usual one, i.e. that for a finite structure a formula is true in the model of all the possible paths if and only if it is true in the model of all the possible paths if and only if it is true in the model of all the circular paths.

The authors prove the $2.5 n - o(n)$ lower bound on the number of multiplications/divisions required to compute the coefficients of the product of two polynomials of degree n over a finite field by means of straight-line algorithms.

Probabilistic algorithms are presented for testing the result of the product of two n -bit integers in O ( n ) bit operations and for testing the result of the product of two polynomials of degree n over any integral domain in 4 n + o ( n ) algebraic operations with the error probability o (l/ n 1-ε ) for any ε > 0. The last algorithm does not depend on the constants of the underlying domain.

Let M q ( n ) denote the number of multiplications required to compute the coefficients of the product of two polynomials of degree n over a q -element field by means of bilinear algorithms. It is shown that M q ( n ) ≱ 3 n - o ( n ). In particular, if q /2 < n ⪇ q + 1, we establish the tight bound M q ( n ) = 3 n + 1 [ q /2].The technique we use can be applied to analysis of algorithms for multiplication of polynomials modulo a polynomial as well.

Let Mq(n) denote the number of multiplications required to compute the coefficients of the product of two polynomials of degree n over a q-element field by means of bilinear algorithms. It is shown that Mq(n) ≥ 3n - o(n). In particular, if q/2 ≪ n ≤ q + 1, we establish the tight bound Mq(n) = 3n + 1 - ⌊q/2⌋. The technique we use can be applied to analysis of algorithms for multiplication of polynomials modulo a polynomial as well.

An algorithm is presented to compute the residue of a polynomial over a finite field of degree n modulo a polynomial of degree O (log n ) in O ( n ) algebraic operations. This algorithm can be implemented on a Turing machine. The implementation is based on Turing machine procedure that divides a polynomial of degree n by a sparse polynomial with k nonzero coefficients in O ( kn ) steps. This algorithm can be adapted to compute the residue of a number of length n modulo a number of length O (log n ) in O ( n ) bit operations.

Let R be a ring, and let f(/spl alpha/), g(/spl alph/) /spl epsi/ R[/spl alpha/] be univariate polynomials over R of degree n. We Present an algorithm for computing the coefficients of the product f(/spl alpha/)G (/spl alpha/) by O (nlgn) multiplications. This algorithm is based on an algorithm for multiplying polynomials over the ring of integers, and does not depend on R. Also we prove that multiplying the third degree polynomials over the ripg of integers requires at least nine multiplications. This bound is tight.