An arithmetical characterization of NP

Adleman and Manders (Proc. 17th IEEE Symposium on Foundations of Computer Science, 1976) obtained a near characterization of NP using Diophantine predicates with quantifier bounds. We characterize NP as the class of sets definable by a polynomial predicate preceded by a string of quantifiers where each universal variable has a bound of the form p(|n|) and each existential variable has a bound of the form 2^q(|n|) for input n and polynomials p and q.     

Integrating Standard Dependency Schemes in QCSP Solvers

Quantified constraint satisfaction problems (QCSPs) are an extension to constraint satisfaction problems (CSPs) with both universal quantifiers and existential quantifiers.In this paper we apply variab...     

Variant quantifiers in L3-valued first-order logic

Traditional first-order logic has four definitions for quantifiers, which are defined by universal and existential quantifiers. In L₃-valued (three-valued) first-order logic, there are eight kinds of definitions for quantifiers; and corresponding Gentzen deduction systems will be given and their soundness and completeness theorems will be proved.     

Dot-depth, monadic quantifier alternation, and first-order closure over grids and pictures

This paper presents results from two different areas. The first area is monadic second-order logic (MSO) over finite structures, in particular over the so-called grids. These are structures whose elements can be arranged as a matrix and which have two binary relations corresponding to vertical and horizontal successors. For this logic, we study the expressive power of the alternation of existential and universal monadic second-order quantifiers, i.e., set quantifiers. In Matz et al. (Information and Computation, LICS’ 97, 1999, to appear) it had been shown that these alternations cannot be limited to a fixed number without loss of expressiveness. In this paper, we strengthen this result in several ways. Firstly, we show that there are MSO formulas that have a very restricted form of k+1 set quantifiers but are not equivalent to a formula with k quantifiers. Secondly, we show that if we fix the number of such alternations, the expressive power of formulas that start with a block of universal quantifiers differs from the power of those that start with an existential one—this was previously known only for coloured grids. Thirdly, we investigate how an additional prefix of first-order (i.e., element) quantifiers influences the expressive power of MSO formulas. The second area that this paper is concerned with is two-dimensional formal language theory. We study how the alternation of (first- and monadic second-order) quantifications, on the one hand, relates to the dot-depth measure of two-dimensional (i.e., picture) languages, on the other hand. That measure is the two-dimensional version of the classical notion of dot-depth for (one-dimensional) starfree word languages. We show that the hierarchy induced by this dot-depth cuts through the hierarchy given by monadic second-order quantifications. In particular, beyond each level of the monadic second-order alternation hierarchy, there is a starfree picture language.     

Decision-theoretic planning with generalized first-order decision diagrams

Many tasks in AI require representation and manipulation of complex functions. First-Order Decision Diagrams (FODD) are a compact knowledge representation expressing functions over relational structures. They represent numerical functions that, when constrained to the Boolean range, use only existential quantification. Previous work has developed a set of operations for composition and for removing redundancies in FODDs, thus keeping them compact, and showed how to successfully employ FODDs for solving large-scale stochastic planning problems through the formalism of relational Markov decision processes (RMDP). In this paper, we introduce several new ideas enhancing the applicability of FODDs. More specifically, we first introduce Generalized FODDs (GFODD) and composition operations for them, generalizing FODDs to arbitrary quantification. Second, we develop a novel approach for reducing (G)FODDs using model checking. This yields – for the first time – a reduction that maximally reduces the diagram for the FODD case and provides a sound reduction procedure for GFODDs. Finally we show how GFODDs can be used in principle to solve RMDPs with arbitrary quantification, and develop a complete solution for the case where the reward function is specified using an arbitrary number of existential quantifiers followed by an arbitrary number of universal quantifiers.     

Some quantifiers require two-predicate scopes

Predicate calculus has long served as the basis of mathematical logic and has more recently achieved widespread use in artificial intelligence. This system of logic expresses propositions in terms of quantifications, restricting itself to the universal and existential quantifiers “all” and “some,” which appear to be adequate for formalizing mathematics. Systems that aspire to deal with natural language or everyday reasoning, however, must attempt to deal with the full range of quantifiers that occur in such language and reasoning, including, in particular, plurality quantifiers, such as “most,” “many,” and “few.” The logic of such quantifiers forces an extension of the predicate-calculus framework to a system of representation that involves more than one predicate in each quantification. In this paper, we prove this result for the specific case of “most.” Unlike some other arguments that attempt to establish the inadequacy of standard predicate calculus on the basis of intuitive plausibility judgements as to the likely character of human reasoning [11, 19], our result is a theorem of logic itself.     

基于视图机制和分组统计实现 SQL 中全称量词的转换

SQL语言不支持全称量词，当查询涉及“全部”语义时，需要将全称量词等价转换为存在量词，转换及量词的使用都是难点，需要有相关的逻辑推理及思维能力，从而使程序员难以理解。基于视图机制和分组统计来实现对全称量词的对应语义的转换，思路清晰，容易掌握。     

The complexity of the satisfiability problem for Krom formulas

We study the computational complexity of the satisfiability problem for Krom (CNF-2) formulas. Upper and lower bounds are obtained for several decidable classes of formulas determined by quantifier prefix and degree of predicate letters. For example, we show that determining satisfiability of Krom formulas with quantifier prefix of the form ……… is complete for deterministic exponential time, but that if the number of universal quantifiers is bounded then polynomial time suffices.     

Arity and alternation: a proper hierarchy in higher order logics

We study the effect of simultaneously bounding the maximal-arity of the higher-order variables and the alternation of quantifiers in higher-order logics, as to their expressive power on finite structures (or relational databases). Let $\mathit{AA}^i(r,m)$ be the class of (i?+?1)-th order logic formulae where all quantifiers are grouped together at the beginning of the formulae, forming m alternating blocks of consecutive existential and universal quantifiers, and such that the maximal-arity (a generalization of the concept of arity, not just the maximal of the arities of the quantified variables) of the higher-order variables is bounded by r. Note that, the order of the quantifiers in the prefix may be mixed. We show that, for every i?≥?1, the resulting $\mathit{AA}^i(r,m)$ hierarchy of formulae of (i?+?1)-th order logic is proper. This extends a result by Makowsky and Pnueli who proved that the same hierarchy in second-order logic is proper. In both cases the strategy used to prove the results consists in considering formulae which, represented as finite structures, satisfy themselves. As the well known diagonalization argument applies here, this gives rise, for each order i and each level of the $\mathit{AA}^i(r,m)$ hierarchy of arity and alternation, to a class of formulae which is not definable in that level, but which is definable in a higher level of the same hierarchy. We then use a similar argument to prove that the classes of $\Sigma^i_m \cup \Pi^i_m$ formulae in which the higher-order variables of all orders up to i?+?1 have maximal-arity at most r, also induce a proper hierarchy in each higher-order logic of order i?≥?3. It is not known whether the correspondent hierarchy in second-order logic is proper. Using the concept of finite model truth definitions introduced by M. Mostowski, we give a sufficient condition for that to be the case.     

Relaxing the universal quantifier of the division in fuzzy relational databases

In a previous paper, we presented an approach to calculate relational division in fuzzy databases, starting with the GEFRED model. This work centered on dealing with fuzzy attributes and fuzzy values and only the universal quantifier was taken into account since it is the inherent quantifier in classical relational division. In this paper, we present an extension of that division to relax the universal quantifier. With this new system we can use both absolute quantifiers and relative quantifiers irrespective of how the function of the fuzzy quantifier is defined. We also include a comparison with other fuzzy division approaches to relax the universal quantifier that have been published. Furthermore, in this paper we have extended the fuzzy SQL language to express any kind of fuzzy division. © 2001 John Wiley & Sons, Inc.     

Monotonicity and Collective Quantification

This article studies the monotonicity behavior of plural determinersthat quantify over collections. Following previous work, we describe thecollective interpretation of determiners such as all, some andmost using generalized quantifiers of a higher type that areobtained systematically by applying a type shifting operator to thestandard meanings of determiners in Generalized Quantifier Theory. Twoprocesses of counting and existential quantification thatappear with plural quantifiers are unified into a single determinerfitting operator, which, unlike previous proposals, both capturesexistential quantification with plural determiners and respects theirmonotonicity properties. However, some previously unnoticed factsindicate that monotonicity of plural determiners is not always preservedwhen they apply to collective predicates. We show that the proposedoperator describes this behavior correctly, and characterize themonotonicity of the collective determiners it derives. It is proved thatdeterminer fitting always preserves monotonicity properties ofdeterminers in their second argument, but monotonicity in the firstargument of a determiner is preserved if and only if it is monotonic inthe same direction in the second argument. We argue that this asymmetryfollows from the conservativity of generalized quantifiers innatural language.     

Exact certification in global polynomial optimization via sums-of-squares of rational functions with rational coefficients

We present a hybrid symbolic-numeric algorithm for certifying a polynomial or rational function with rational coefficients to be non-negative for all real values of the variables by computing a representation for it as a fraction of two polynomial sum-of-squares (SOS) with rational coefficients. Our new approach turns the earlier methods by Peyrl and Parrilo at SNC’07 and ours at ISSAC’08 both based on polynomial SOS, which do not always exist, into a universal algorithm for all inputs via Artin’s theorem.Furthermore, we scrutinize the all-important process of converting the numerical SOS numerators and denominators produced by block semidefinite programming into an exact rational identity. We improve on our own Newton iteration-based high precision refinement algorithm by compressing the initial Gram matrices and by deploying rational vector recovery aside from orthogonal projection. We successfully demonstrate our algorithm on (1) various exceptional SOS problems with necessary polynomial denominators from the literature and on (2) very large (thousands of variables) SOS lower bound certificates for Rump’s model problem (up to n=18, factor degree=17).     

The Hamiltonian Syllogistic

This paper undertakes a re-examination of Sir William Hamilton’s doctrine of the quantification of the predicate. Hamilton’s doctrine comprises two theses. First, the predicates of traditional syllogistic sentence-forms contain implicit existential quantifiers, so that, for example, All p is q is to be understood as All p is some q. Second, these implicit quantifiers can be meaningfully dualized to yield novel sentence-forms, such as, for example, All p is all q. Hamilton attempted to provide a deductive system for his language, along the lines of the classical syllogisms. We show, using techniques unavailable to Hamilton, that such a system does exist, though with qualifications that distinguish it from its classical counterpart.     

有序实数加法理论新的判定过程与多项式谱

推广VolkerWeispfenning关于正的有序实数加法理论的量词消去方法,得到有序实数加法理论的一个量词消去的判定过程.在此基础上构造出一个新的、更为精细的判定方法.并且利用这一结果证明了固定量词长度的子类属于相应计算复杂性的多项式谱.与E.D.Sontag的类似结论比较,从这种简洁的方式可以得到一个较优的结果.这个结果实际上将N.Megiddo的关于正实数理论的结论推广到了一般实数理论.     

A combinatorial characterization of resolution width

We provide a characterization of the resolution width introduced in the context of propositional proof complexity in terms of the existential pebble game introduced in the context of finite model theory. The characterization is tight and purely combinatorial. Our first application of this result is a surprising proof that the minimum space of refuting a 3-CNF formula is always bounded from below by the minimum width of refuting it (minus 3). This solves a well-known open problem. The second application is the unification of several width lower bound arguments, and a new width lower bound for the dense linear order principle. Since we also show that this principle has resolution refutations of polynomial size, this provides yet another example showing that the relationship between size and width cannot be made subpolynomial.     

Alternation on cellular automata

In this paper we consider several notions of alternation in cellular automata: non-uniform, uniform and weak alternation. We study relations among these notions and with alternating Turing machines. It is proved that the languages accepted in polynomial time by alternating Turing machines are those accepted by alternating cellular automata in polynomial time for all the proposed alternating cellular automata. In particular, this is true for the weak model where the difference between existential and universal states is omitted for all the cells except the first one. It is proved that real time alternation in cellular automata is strictly more powerful than real time alternation in Turing machines, with only one read-write tape. Moreover, it is shown that in linear time uniform and weak models agree.     

Nondeterministic flowchart programs with recursive procedures: Semantics and correctness II

In the second part of this work, we formulate a new inductive assertion method applying to the class of nondeterministic flowchart programs with recursive procedures studied in part 1. Using results on unfolding proved in part 1, we prove that this method is sound and complete with a finite number of assertions. We study four notions of correctness: two notions of partial correctness (existential and universal) and the corresponding notions of total correctness. We also formalize two notions of extension and equivalence (existential and universal) in the second-order predicate calculus.     

Fuzzy relation equations (I): the general and specialized solving algorithms

 In this article, we develop a new method and an algorithm to solve a system of fuzzy relation equations. We first introduce a solution-base-matrix and then give a tractable mathematical logic representation of all minimal solutions. Next, we design a new universal algorithm to get them. Two simplification rules are found to simplify the solution-base-matrix. We show that a polynomial time algorithm to find all minimal solutions for a general system of fuzzy relation equations simply does not exist with expectation of P=N P. Hence, the problem of solving fuzzy relation equations is an N P-hard problem in terms of computational complexity. Our universal algorithm is still useful when one does not solve a large number of equations. In many real applications the problem of solving fuzzy relation equations can be simplified in polynomial time problems. In this article, we will discuss several cases of practical applications which have such polynomial algorithms.     

A proof procedure for the logic of hereditary Harrop formulas

A proof procedure is presented for a class of formulas in intuitionistic logic. These formulas are the so-calledgoal formulas in the theory of hereditary Harrop formulas. Proof search in intuitionistic logic is complicated by the nonexistence of a Herbrand-like theorem for this logic: formulas cannot in general be preprocessed into a form such as the clausal form, and the construction of a proof is often sensitive to the order in which the connectives and quantifiers are analyzed. An interesting aspect of the formulas we consider here is that this analysis can be carried out in a relatively controlled manner in their context. In particular, the task of finding a proof can be reduced to one of demonstrating that a formula follows from a set of assumptions, with the next step in this process being determined by the structure of the conclusion formula. An acceptable implementation of this observation must utilize unification. However, since our formulas may contain universal and existential quantifiers in mixed order, care must be exercised to ensure the correctness of unification. One way of realizing this requirement involves labelling constants and variables and then using these labels to constrain unification. This form of unification is presented and used in a proof procedure for goal formulas in a first-order version of hereditary Harrop formulas. Modifications to this procedure for the relevant formulas in a higher-order logic are also described. The proof procedure that we present has a practical value in that it provides the basis for an implementation of the logic programming language Prolog.     

Two-dimensional specification of universal quantification in agraphical database query language

A technique is proposed for specifying universal quantification and existential quantification (combined with negation) in a two-dimensional (graphical) database query language. Unlike other approaches that provide set operators to simulate universal quantification, this technique allows a direct representation of universal quantification. Syntactic constructs for specifying universal and existential quantifications, two-dimensional translation of universal quantification to existential quantification (with negation), and translation of existentially quantified two-dimensional queries to relational queries are presented. The resulting relational queries can be processed directly by many existing database systems. The authors claim that this technique renders universal quantifications easy to understand. To substantiate this claim, they provide a simple, easy-to-follow guideline for constructing universally quantified queries