Constant Symbols Research Articles

Similarity-based SLD resolution and its role for web knowledge discovery

This work presents the implementation of an extension of SLD resolution towards approximate reasoning and its implementation in an extended Prolog system. The proposed refutation procedure overcomes failures in the unification process by exploiting similarity relations defined between predicate and constant symbols. This enables to compute approximate solutions, with an associated approximation degree, when failures of the exact inference process occur. In this paper we outline the main ideas of this approach and we present an extended PROLOG interpreter, named SiLog, which implements this inference procedure. Then we point out on a web-based platform, usable for knowledge discovery, that exploits as inner feature the similarity-based SLD resolution.

Optimal design and placement of pilot symbols for channel estimation

The problem of designing and placing pilot symbols for the estimation of frequency-selective random channels is considered. The channel is assumed to be a block-fading model with finite impulse response (FIR). For both single-input single-output (SISO) and multiple-input multiple-output (MIMO) channels, under the assumption of independent and identical distributed channel taps, the Cramer-Rao bound (CRB) on the mean square error (MSE) of semi-blind channel estimators is derived and minimized with respect to pilot symbols and their placement. It is shown that the optimal strategy is to place pilot symbols satisfying certain orthogonality condition in such a way that data and pilot symbols with higher power are in the middle of the packet. The results also indicate that the optimal pilot placements are independent of channel probability distribution. For constant modulus symbols, we show that the quasi-periodic placement and its generalization in the multiuser case turn out to be optimal. We further consider estimating channels with correlated taps and show that the previous placement strategy is also optimal among orthogonal pilot sequences.

A reduced complexity channel estimation for OFDM systems with transmit diversity in mobile wireless channels

A reduced complexity channel estimation for OFDM systems with transmit diversity is proposed by exploiting the correlation of the adjacent subchannel responses. The sizes of the matrix inverse and the FFTs required in the channel estimation at every OFDM data symbol are reduced by half of the existing method for OFDM systems with nonconstant modulus subcarrier symbols or constant modulus subcarrier symbols with some guard tones. The complexity reduction of half FFTs size and some matrix multiplications is still achieved for constant modulus subcarrier symbols with no guard tones. The price for the complexity reduction is a slight BER degradation and for the channels with small relative delay spreads, the BER performance of the reduced complexity method becomes quite comparable to the existing method. An alternative approach for the number of significant taps required in the channel estimation is described which achieves a comparable performance to the case with the known suitable number of significant taps. A simple modification which reduces the lost leakage of the nonsample-spaced channel paths is also proposed. This modification achieves a substantial performance improvement over the existing method without any added complexity.

Single Elementary Associative-Commutative Matching

A single elementary associative-commutative (AC) matching problem has a pattern term that consists of a single (variadic) AC function symbol with only variable symbols as arguments and a subject term that consists of a single (variadic) AC function symbol with only constant symbols as arguments. We show that even this very restricted formulation of AC matching has an NP-complete decision problem. We consider a number of methods to contain the growth in the search space, including a lookup table for the solubility of subproblems, a digraph reformulation of the problem, and a search tree pruning method that uses failure information together with a partial ordering on branches. We give empirical results for the method that seems to work best in practice, and we list some ‘hard’ problem instances.

Complexity and Approximability of Quantified and Stochastic Constraint Satisfaction Problems

Let D be an arbitrary (not necessarily finite) nonempty set, let C be a finite set of constant symbols denoting arbitrary elements of D, and let S be an arbitrary finite set of finite-arity relations on D. We denote the problem of determining the satisfiability of finite conjunctions of relations in S applied to variables (to variables and symbols in C) by SAT(S) (by SAT{sub c}(S)). Here, we study simultaneously the complexity of and the existence of efficient approximation algorithms for a number of variants of the problems SAT(S) and SAT{sub c}(S), and for many different D, C, and S. These problem variants include decision and optimization problems, for formulas, quantified formulas stochastically-quantified formulas. We denote these problems by Q-SAT(S), MAX-Q-SAT(S), S-SAT(S), MAX-S-SAT(S) MAX-NSF-Q-SAT(S) and MAX-NSF-S-SAT(S). The main contribution is the development of a unified predictive theory for characterizing the the complexity of these problems. Our unified approach is based on the following basic two basic concepts: (i) strongly-local replacements/reductions and (ii) relational/algebraic representability. Let k {ge} 2. Let S be a finite set of finite-arity relations on {Sigma}{sub k} with the following condition on S: All finite arity relations on {Sigma}{sub k} can be represented as finite existentially-quantified conjunctionsmore » of relations in S applied to variables (to variables and constant symbols in C), Then we prove the following new results: (1) The problems SAT(S) and SAT{sub c}(S) are both NQL-complete and {le}{sub logn}{sup bw}-complete for NP. (2) The problems Q-SAT(S), Q-SAT{sub c}(S), are PSPACE-complete. Letting k = 2, the problem S-SAT(S) and S-SAT{sub c}(S) are PSPACE-complete. (3) {exists} {epsilon} > 0 for which approximating the problems MAX-Q-SAT(S) within {epsilon} times optimum is PSPACE-hard. Letting k =: 2, {exists} {epsilon} > 0 for which approximating the problems MAX-S-SAT(S) within {epsilon} times optimum is PSPACE-hard. (4) {forall} {epsilon} > 0 the problems MAX-NSF-Q-SAT(S) and MAX-NSF-S-SAT(S), are PSPACE-hard to approximate within a factor of n{sup {epsilon}} times optimum. These results significantly extend the earlier results by (i) Papadimitriou [Pa851] on complexity of stochastic satisfiability, (ii) Condon, Feigenbaum, Lund and Shor [CF+93, CF+94] by identifying natural classes of PSPACE-hard optimization problems with provably PSPACE-hard {epsilon}-approximation problems. Moreover, most of our results hold not just for Boolean relations: most previous results were done only in the context of Boolean domains. The results also constitute as a significant step towards obtaining a dichotomy theorems for the problems MAX-S-SAT(S) and MAX-Q-SAT(S): a research area of recent interest [CF+93, CF+94, Cr95, KSW97, LMP99].« less

Learning one-variable pattern languages very efficiently on average, in parallel, and by asking queries

A pattern is a finite string of constant and variable symbols. The language generated by a pattern is the set of all strings of constant symbols which can be obtained from the pattern by substituting non-empty strings for variables. We study the learnability of one-variable pattern languages in the limit with respect to the update time needed for computing a new single hypothesis and the expected total learning time taken until convergence to a correct hypothesis. Our results are as follows. First, we design a consistent and set-driven learner that, using the concept of descriptive patterns, achieves update time O(n 2 logn) , where n is the size of the input sample. The best previously known algorithm for computing descriptive one-variable patterns requires time O(n 4 logn) (cf. Angluin, J. Comput. Systems Sci. 21(1) (1980) 46–62). Second, we give a parallel version of this algorithm that requires time O( logn) and O(n 3/ log n) processors on an EREW-PRAM. Third, using a modified version of the sequential algorithm as a subroutine, we devise a learning algorithm for one-variable patterns whose expected total learning time is O(ℓ 2 logℓ) provided the sample strings are drawn from the target language according to a probability distribution with expected string length ℓ. The probability distribution must be such that strings of equal length have equal probability, but can be arbitrary otherwise. Thus, we establish the first algorithm for learning one-variable pattern languages having an expected total learning time that provably differs from the update time by a constant factor only. Finally, we show how the algorithm for descriptive one-variable patterns can be used for learning one-variable patterns with a polynomial number of superset queries with respect to the one-variable patterns as query language.

Inductive inference of unbounded unions of pattern languages from positive data

A pattern is a string consisting of constant symbols and variables. The language of a pattern is the set of constant strings obtained by substituting nonempty constant strings for variables in the pattern. In this paper we consider the inductive inference of unbounded unions of pattern languages from positive data. For any fixed k, the class of unions of at most k pattern languages is already shown to be inferable from positive data. The class of unbounded unions is not inferable, because any pattern with at least one variable defines an infinite language and any constant string defines a singleton set consisting of itself, and therefore, the class of unions becomes super-finite, that is, it contains all the finite languages and at least one infinite language. In this paper, we consider several restrictions on patterns to investigate the inferability of unbounded unions of their languages from positive data. A proper pattern contains at least one variable. A regular pattern contains at most one occurrence of every variable. Although the class of unbounded unions of proper pattern languages is not super-finite, it is shown not to be inferable from positive data, even if patterns are restricted to be regular or one-variable. When regular patterns are restricted to be of the form “ xwy”, where x and y are variables and w is a constant string, the class of unbounded unions is shown to be inferable from positive data. When regular patterns do not contain more than l consecutive occurrences of constant symbols, for some fixed l, the class of unbounded unions is shown to be inferable from positive data. Extended pattern languages, where substitutions of the empty string for some variables are allowed, are also considered from the viewpoint of relationship to the inferability of unbounded unions of non-extended pattern languages.

First Order Common Knowledge Logics

In this paper we investigate first order common knowledge logics; i.e., modal epistemic logics based on first order logic with common knowledge operators. It is shown that even rather weak fragments of first order common knowledge logics are not recursively axiomatizable. This applies, for example, to fragments which allow to reason about names only; that is to say, fragments the first order part of which is based on constant symbols and the equality symbol only. Then formal properties of "quantifying into" epistemic contexts are investigated. The results are illustrated by means of epistemic representations of Nash Equilibria for finite games with mixed strategies.

Quantification and ontology

Quineans have taken the basic expression of ontological commitment to be an assertion of the form '∃ x φ', assimilated to theEnglish 'there is something that is a φ'. Here I take the existential quantifier to be introduced, not as an abbreviation for an expression of English, but via Tarskian semantics. I argue, contrary to the standard view, that Tarskian semantics in fact suggests a quite different picture: one in which quantification is of a substitutional type apparently first proposed by Geach. The ontological burden is borne by constant symbols, and truth is defined separately from reference.

Spline qualocation methods for variable-coefficient elliptic equations on curves

Here the stability and convergence results of oqualocation methods providing additional orders of convergence are extended from the special class of pseudodifferential equations with constant coefficient symbols to general classical pseudodifferential equations of strongly and of oddly elliptic type. The analysis exploits localization in the form of frozen coefficients, pseudohomogeneous asymptotic symbol representation of classical pseudodifferential operators, a decisive commutator property of the qualocation projection and requires qualocation rules which provide sufficiently many additional degrees of precision for the numerical integration of smooth remainders. Numerical examples show the predicted high orders of convergence.

Computational Complexity of Simultaneous Elementary Matching Problems

The simultaneous elementary E-matching problem for an equational theory E is to decide whether there is an E-matcher for a given system of equations in which the only nonconstant function symbols occurring in the terms to be matched are the ones constrained by the equational axioms of E. We study the computational complexity of simultaneous elementary matching problems for the equational theories A of semigroups, AC of commutative semigroups, and ACU of commutative monoids. In each case, we delineate the boundary between NP-completeness and solvability in polynomial time by considering two parameters, the number of equations in the systems and the number of constant symbols in the signature. Moreover, we analyze further the intractable cases of simultaneous elementary AC-matching and ACU-matching by also taking into account the maximum number of occurrences of each variable. Using combinatorial optimization techniques, we show that if each variable is restricted to having at most two occurrences, then several cases of simultaneous elementary AC-matching and ACU-matching can be solved in polynomial time.

Circumscribing datalog: Expressive power and complexity

In this paper we study a generalization of datalog, the language of function-free definite clauses. It is known that standard datalog semantics (i.e., least Herbrand model semantics) can be obtained by regarding programs as theories to be circumscribed with all predicates to be minimized. The extension proposed here, called datalog circ , consists in considering the general form of circumscription, where some predicates are minimized, some predicates are fixed, and some vary. We study the complexity and the expressive power of the language thus obtained. We show that this language (and, actually, its non-recursive fragment) is capable of expressing all the queries in DB-co-NP and, as such, is much more powerful than standard datalog, whose expressive power is limited to a strict subset of PTIME queries. Both data and combined complexities of answering datalog circ queries are studied. Data complexity is proved to be co-NP-complete. Combined complexity is shown to be in general hard for co-NE and complete for co-NE in the case of Herbrand bases containing k distinct constant symbols, where k is bounded.

On multidimensional coded modulations having uniform error property for generalized decoding and flat-fading channels

We consider the problem of uniform error property (UEP) for a coded modulation with constant energy multidimensional symbols, transmitted over the additive white Gaussian noise (AWGN) or fading channels and received by a broad class of decoders. This class includes coherent, partially coherent, double differential, and noncoherent decoders, decoders designed for fading channels, decoders using one or multiple-symbol observations, and many more. These decoders are described as special cases of a general decoder model. This decoder operates by maximizing an arbitrary likelihood function that its arguments are front-end correlator (matched-filter) outputs, A group code structure that guarantees UEP is developed by using the theory of geometrically uniform codes and applying it to the general decoder. These codes are defined over groups (commonly nonbinary) with isometric mapping to channel symbols. We show the code construction for the specific case of Lth-dimensional M-ary phase-shift keying (MPSK). An additional interesting property of these general uniform error codes is related to the case of noncoherent decoding. We show that when using codes of this family, if a code is noncoherently catastrophic, then it is also rotationally invariant. Then, the use of preceding of the input such that the code becomes rotationally transparent will also make it noncatastrophic.

Model theory of Boolean products of subdirectly irreducible heyting algebras

It is a longstanding open problem in algebraic model theory to determine the model companions of the varieties of relative Stone algebras. Following Weispfenning's general model theory of Boolean products of structures we obtain various theories of Heyting algebras which are model and substructure complete. This works by adding only finitely many constant symbols to the language of Heyting algebras, one of which denoting a global dual atom. Thereby we especially obtain quantifier elimination for theories of atomless Post algebras of order n.

The Decidability of the Existential Theory of the Poset of Recursively Enumerable Degrees with Jump Relations

We show that the existential theory of the recursively enumerable degrees in the language L containing predicates for order andn-jump comparability for alln, and constant symbols for least and greatest elements, is decidable. The decidability follows from our main theorem, where we show that any finite L-structure which is consistent with the order relation, the order-preserving property of the jump operator, and the property of the jump operator that the jump of an element is strictly greater than the element, can be embedded into the r.e. degrees.

Asymptotic Conditional Probabilities: The Unary Case

Motivated by problems that arise in computing degrees of belief, we consider the problem of computing asymptotic conditional probabilities for first-order sentences. Given first-order sentences $\phi$ and $\theta$, we consider the structures with domain $\{1,\ldots, N\}$ that satisfy $\theta$, and compute the fraction of them in which $\phi$ is true. We then consider what happens to this fraction as $N$ gets large. This extends the work on 0-1 laws that considers the limiting probability of first-order sentences, by considering asymptotic conditional probabilities. As shown by Liogon'kii [Math. Notes Acad. USSR, 6 (1969), pp. 856--861] and by Grove, Halpern, and Koller [Res. Rep. RJ 9564, IBM Almaden Research Center, San Jose, CA, 1993], in the general case, asymptotic conditional probabilities do not always exist, and most questions relating to this issue are highly undecidable. These results, however, all depend on the assumption that $\theta$ can use a nonunary predicate symbol. Liogonkii [Math. Notes Acad. USSR, 6 (1969), pp. 856--861] shows that if we condition on formulas $\theta$ involving unary predicate symbols only (but no equality or constant symbols), then the asymptotic conditional probability does exist and can be effectively computed. This is the case even if we place no corresponding restrictions on $\phi$. We extend this result here to the case where $\theta$ involves equality and constants. We show that the complexity of computing the limit depends on various factors, such as the depth of quantifier nesting, or whether the vocabulary is finite or infinite. We completely characterize the complexity of the problem in the different cases, and show related results for the associated approximation problem.

Complexity and undecidability results for logic programming

This paper surveys complexity, degree of uncomputability, and expressive power results for logic programming. Some major decision problem complexity results and other results for logic programming are also covered. It also proves several new results filling in previous gaps in the literature. The paper considers seven logic programming semantics: the van Emden-Kowalski semantics for definite (Horn) logic programs; the perfect model semantics for stratified and for locally stratified logic programs; and the two- and three-valued program completion semantics, the well-founded semantics, and the stable semantics, all for normal logic programs, under skeptical inference. The main results concern expressibility and query complexity/uncomputability in five contexts: for propositional logic programs, for first order logic programs with infinite Herbrand universes on their Herbrand universes (a closed domain assumption), for first order logic programs with infinite Herbrand universes on those universes extended with infinitely many new elements (an open domain assumption), and for logic programs without function or constant symbols evaluated over varying extensional databases (DATALOG-type results, data complexity results only) under both closed and open domain assumptions. Several of the open domain assumption results are new to this paper. Other results surveyed are (1) results about the family of all stable models of a program and (2) decision questions about when a logic program has nice properties with respect to a semantics (e.g., a unique stable model). One decision result, for well-founded semantics, is new to this paper.

Constants in Kripke Models for Intuitionistic Logic

AbstractWe present a technique to extend a Kripke structure (for intuitionistic logic) into an elementary extension satisfying some property (cardinality, saturation, etc.) which can be “axiomatized” by a family of sets of sentences, where, most often, many constant symbols occur. To that end, we prove extended theorems of completeness and compactness. Also, a section of the paper is devoted to the back‐and‐forth construction of isomorphisms between Kripke structures.

Combining Symbolic Constraint Solvers on Algebraic Domains

In the context of constraint logic programming and theorem proving, the development of constraint solvers on algebraic domains and their combination is of prime interest. As an example, a constraint solver in finite algebras is presented for a constraint language including for instance equations, disequations and inequations. By extending techniques used for the combination of unification in disjoint equational theories, we show how to combine constraint solvers on different algebraic domains that may share some constant symbols. We illustrate this technique by combining the constraint solver in finite algebras with other unification algorithms, and with another constraint solver on a different finite algebra.

Ternary Operations as Primitive Notions for Constructive Plane Geometry IV

AbstractIn this paper we provide a quantifier‐free constructive axiomatization for Euclidean planes in a first‐order language with only ternary operation symbols and three constant symbols (to be interpreted as ‘points’). We also determine the algorithmic theories of some ‘naturally occurring’ plane geometries.Mathematics Subject Classification: 03F65, 51M05, 51M15, 03B30.