First-order spectra with one variable

First-order spectra with one binary predicate

A poor signature consists of one (unary) function symbol, a symbol for equality and a number of constant symbols. Due to the lack of internal structure, interpretations of poor signatures differ from interpretations of richer signatures in some respects. One of them is the completeness of Hoare-like systems relative to the first-order theory of the interpretation.

The paper gives a logical characterisation of the class NTIME(n) of problems that can be solved on a nondeterministic Turing machine in linear time. It is shown that a set L of strings is in this class if and only if there is a formula of the form ∃f 1··∃f k ∃R 1··∃R m ∀xϕ that is true exactly for all strings in L. In this formula the f i are unary function symbols, the R i are unary relation symbols and ϕ is a quantifier-free formula. Furthermore, the quantification of functions is restricted to non-crossing, decreasing functions and in ϕ no equations in which different functions occur are allowed. There are a number of variations of this statement, e.g., it holds also for k = 3. From these results we derive an Ehrenfeucht game characterisation of NTIME(n).KeywordsLinear TimeTuring MachineDerivation TreeRelational AtomLogical CharacterisationThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

We describe a simplifier for use in program manipulation and verification. The simplifier finds a normal form for any expression over the language consisting of individual variables, the usual boolean connectives, the conditional function cond (denoting if-then-else), the integers (numerals), the arithmetic functions and predicates +, - and ≤, the LISP constants, functions and predicates nil, car, cdr, cons and atom, the functions store and select for storing into and selecting from arrays, and uninterpreted function symbols. Individual variables range over the union of the rationals, the set of arrays, the LISP s-expressions and the booleans true and false. The constant, function and predicate symbols take their natural interpretations.The simplifier is complete; that is, it simplifies every valid formula to true. Thus it is also a decision procedure for the quantifier-free theory of rationals, arrays and s-expressions under the above functions and predicates.The organization of the simplifier is based on a method for combining decision algorithms for several theories into a single decision algorithm for a larger theory containing the original theories. More precisely, given a set S of functions and predicates over a fixed domain, a satisfiability program for S is a program which determines the satisfiability of conjunctions of literals (signed atomic formulas) whose predicates and function signs are in S. We give a general procedure for combining satisfiability programs for sets S and T into a single satisfiability program for S ∪ T, given certain conditions on S and T. We show how a satisfiability program for a set S can be used to write a complete simplifier for expressions containing functions and predicates of S as well as uninterpreted function symbols.The simplifier described in this paper is currently used in the Stanford Pascal Verifier.

A lambda calculus schema is an expression of the lambda calculus augmented by uninterpreted constant and function symbols and thus is an abstraction of programming languages such as LISP which permit functions to be passed to or returned from other functions. We then consider two natural implementation strategies: the retention strategy in which all variable bindings are retained until no longer needed (implying the use of some sort of garbage collected store) and the deletion strategy, modelled after the usual stack implementation of ALGOL-60, in which variable bindings are destroyed when control leaves the procedure (or block) in which they were created. Berry shows that the deletion strategy implementation is not “correct” for a wide class of languages in the sense that it is not equivalent to natural extensions of the copy rule of ALGOL to such languages, whereas the retention strategy is correct in that sense. We show, however, that no real power is lost in restricting oneself to a deletion strategy implementation, for any program can be translated into an equivalent one which will work correctly under such an implementation. The proof makes no use of the particular primitive functions and data of the language and hence is true of the corresponding schemata under all interpretations.

It is undecidable in general whether or not a term-rewriting system is confluent on a given congruence class. This result is shown to hold even when the term-rewriting systems under consideration contain unary function symbols only, and all their rules are length-reducing. On the other hand, for certain subclasses of these systems confluence on a given congruence class is decidable.KeywordsWord ProblemCritical PairCongruence ClassReduction SequencePushdown AutomatonThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

For a class of extensions of lattice theory by unary function symbols and equational axioms (including modularity) algorithms for the computation of the corresponding initial algebra (if this is finite) are presented. In the case of finitely presented lattices the method amounts to a decision procedure for term equality and can be used for the computation of finite sublattices. Several examples and performance comparisons are provided.

A function f: Fn2 → {0, 1} is odd-cycle-free if there are no x1,..., xk e Fn2 with k an odd integer such that f(x1) =... = f(xk) = 1 and x1 +... + xk = 0. We show that one can distinguish odd-cycle-free functions from those e-far from being odd-cycle-free by making poly(1/e) queries to an evaluation oracle. We give two proofs of this result, each shedding light on a different connection between testability of properties of Boolean functions and of dense graphs.The first problem we study is directly reducing testing linear-invariant properties of Boolean functions to testing associated graph properties. We show a black-box reduction from testing odd-cycle-freeness to testing bipartiteness of graphs. Such reductions have been shown previously (Kral-Serra-Vena, Israel J. Math 2011; Shapira, STOC 2009) for monotone linear-invariant properties defined by forbidding solutions to a finite number of equations. But for odd-cycle-freeness whose description involves an infinite number of forbidden equations, a reduction to graph property testing was not previously known. If one could show such a reduction more generally for any linear-invariant property closed under restrictions to subspaces, then it would likely lead to a characterization of the one-sided testable linear-invariant properties, an open problem raised by Sudan.The second issue we study is whether there is an efficient canonical tester for linear-invariant properties of Boolean functions. A canonical tester for linear-invariant properties operates by picking a random linear subspace and then checking if the restriction of the input function to the subspace satisfies a fixed property or not. The question is whether for every linear-invariant property, there is a canonical tester for which there is only a polynomial blowup from the optimal query complexity. We answer the question affirmatively for odd-cycle-freeness. The general question still remains open.

Equational formulae are first-order formulae over an alphabet F of function symbols whose only predicate symbol is syntactic equality. Unification problems are an important special case of equational formulae, where no universal quantifiers and no negation occur. By the negation elimination problem we mean the problem of deciding whether a given equational formula is semantically equivalent to a unification problem. This decision problem has many interesting applications in machine learning, logic programming, functional programming, constrained rewriting, etc. In this work we present a new algorithm for the negation elimination problem of equational formulae with purely existential quantifier prefix. Moreover, we prove the coNP-completeness for equational formulae in DNF and the Π2p-hardness in case of CNF.

Adding datatypes to ontology-mediated queries (OMQs) often makes query answering hard, even for lightweight languages. As a consequence, the use of datatypes in ontologies, e.g. in OWL 2 QL, has been severely restricted. We propose a new, non-uniform, way of analyzing the data-complexity of OMQ answering with datatypes. Instead of restricting the ontology language we aim at a classification of the patterns of datatype atoms in OMQs into those that can occur in non-tractable OMQs and those that only occur in tractable OMQs. To this end we establish a close link between OMQ answering with datatypes and constraint satisfaction problems (CSPs) over the datatypes. Given that query answering in this setting is undecidable in general already for very simple datatypes, we introduce, borrowing from the database literature, a property of OMQs called the Bounded Match Depth Property (BMDP). We apply the link to CSPs– using results and techniques in universal algebra and model theory–to prove PTIME/co-NP dichotomies for OMQs with the BDMP over Horn-ALCHI extended with (1) all finite datatypes, (2) rational numbers with linear order and (3) certain families of datatypes over the integers with the successor relation.

In this article we investigate the family of independence-friendly (IF) logics in the equality-free setting, concentrating on questions related to expressive power. Various natural equality-free fragments of logics in this family translate into existential second-order logic with prenex quantification of function symbols only and with the first-order parts of formulae equality-free. We study this fragment of existential second-order logic. Our principal technical result is that over finite models with a vocabulary consisting of unary relation symbols only, this fragment of second-order logic is weaker in expressive power than first-order logic (with equality). Results about the fragment could turn out useful for example in the study of independence-friendly modal logics. In addition to proving results of a technical nature, we address issues related to a perspective from which IF logic is regarded as a specification framework for games, and also discuss the general significance of understanding fragments of second-order logic in investigations related to non-classical logics.

In general, planning supply use the information from combination varying capacity and inventory to plan the aggregate rates of production by time period for groups of product. This research objective is PT. Tiga Pilar Sejahtera’s (TPS) that produces wafersticks. In the company sometimes occured the wafersticks product can be not fulfilled. This is happened because waferstick inventory unstable. This causes problems that PT. Tiga Pilar Sejahtera (TPS) was less capable to fulfill the market’s demands. To solve this PT TPS should be managing the working hours, workforces, production capacity and finished goods inventory, they are called aggregate planning. This research are determine safety stocks, planning supply and material inventory planning.Managing supply of waferstick product is started by forecast the demand. The forecast result is used to estimate safety stock of waferstick. Then the forecasting result and safety stock become of input Integer linier programming model to determine aggregate planning. The aggregate planning is used to find optomal lot size of waferstick raw material. Then lot sizing become input Planned Order Receipt in the Material Planning Requirements (MRP). Forecast method used to forecast the demand is decomposition, this method is he best method find estimation demand because it have more less MAD than the other. The determination of the safety stock is actually more accurate with the 90% by Romy & Yuli waferstick safety inventory is 1106 cartons and 2364 cartons for Pio. From aggregate planning there is back order in 13 th is 11 cartons. The planning cost for Romy & Yuli waferstick Rp 625.670.902,33 and Cost for Pio waferstick Rp 1.462.296.180,00. Waferstick material planning by optimal lot size between 695 kilograms up to 8487 kilograms by waferstick materials cost is Rp 6.315.520,00.

We show that the first-order theory of any Knuth-Bendix order in the case of the signatures consisting of unary function symbols and constants is decidable. Our decision procedure uses interpretation of unary terms as trees and uses decidability of the weak monadic second-order theory of binary trees. One area of applications of our result is automated deduction, since using the first-order theory of the Knuth-Bendix orders we can decide an important class of ordering constraints. umaiy uccs.

Let o-:S--->to be a similarity type, i.e., S is a set of funct ion symbols, to ={0, 1 ,2 . . . . } and for every FeS , o-(F) is the rank of F. Let us assume for a while that o'(F) = 0 for some F e S (such F are called constant symbols*). Let Q~ denote the set of all constant terms, i.e., the smallest set such that { F e S: o(F)= 0}_ Q~ and F-r~ 9 9 9 ~'~<~)e Q~, for every F e S and r 1 . . . . . Ttr(F)~ Q~. We put

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.