Extension Of First-order Logic Research Articles

The expressive power of fixed-point logic with counting

AbstractWe study the expressive power in the finite of the logic Fixed-Point+Counting, the extension of first-order logic which is obtained through adding both the fixed-point constructor and the ability to count.To this end an isomorphism preserving (‘generic’) model of computation is introduced whose PTime restriction exactly corresponds to this level of expressive power, while its PSpace restriction corresponds to While+Counting. From this model we obtain a normal form which shows a rather clear separation of the relational vs. the arithmetical side of the algorithms involved.In parallel, we study the relations of Fixed-Point+Counting with the infinitary logics and the corresponding pebble games.The main result, however, involves the concept of an arithmetical invariant. By this we mean a functor taking every finite relational structure to an expansion of (an initial segment of) the standard arithmetical structure. In particular its values are linearly ordered structures. We establish the existence of a family of arithmetical invariants with the following properties:• The invariants themselves can be evaluated in polynomial time.• A class of finite relational structures is definable in Fixed-Point+Counting if and only if membership can be decided in polynomial time on the basis of the values of one of the invariants.• The invariant r classifies all finite relational structures exactly up to equivalence with respect to the logic We also give a characterization of Fixed-Point+Counting in terms of sequences of formulae in the : It corresponds exactly to the polynomial time computable families (φn)n ∈ ω in these logics.Towards a positive assessment of the expressive power of Fixed-Point+Counting, it is shown that the natural extension of fixed-point logic by Lindström quantifiers, which capture all the PTime computable properties of cardinalities of definable predicates, is strictly weaker than what we get here. This implies in particular that every extension of fixed-point logic by means of monadic Lindström quantifiers, which stays within PTime, must be strictly contained in Fixed-Point+Counting.

The Expressive Power of Finitely Many Generalized Quantifiers

We consider extensions of first order logic (FO) and fixed point logic (FP) by means of generalized quantifiers in the sense of Lindström. We show that adding a finite set of such quantifiers to FP fails to capture PTIME, even over a fixed signature, strengthening earlier results. We also prove a stronger version of this result for PSPACE, which enables us to establish that PSPACE ≠ FO on any infinite class of ordered structures, a weak version of the ordered conjecture formulated by Ph. G. Kolaitis and M. Y. Vardi (Fixpoint logic vs. infinitary logic in finite-model theory, in "Proceedings, 7th IEEE Symposium on Logic in Computer Science, 1992," pp. 46-57). These results are obtained by defining a notion of element type for bounded variable logics with finitely many generalized quantifiers. Using these, we characterize the classes of finite structures over which the infinitary logic Lω∞ω extended by a finite aw set of generalized quantifiers Q is no more expressive than first order logic extended by the quantifiers in Q.

Infinitary Logic and Inductive Definability over Finite Structures

The extensions of first-order logic with a least fixed point operator (FO + LFP) and with a partial fixed point operator (FO + PFP) are known to capture the complexity classes P and PSPACE respectively in the presence of an ordering relation over finite structures. Recently, Abiteboul and Vianu ( in "Proceedings of the 23rd ACM Symposium on the Theory of Computing," 1991) investigated the relationship of these two logics in the absence of an ordering, using a machine model of generic computation. In particular, they showed that the two languages have equivalent expressive power if and only if P = PSPACE. These languages can also be seen as fragments of an infinitary logic where each formula has a bounded number of variables, L ω ∞ω , (see, for instance, Kolaitis and Vardi, in "Proceedings of the 5th IEEE Symposium on Logic in Computer Science," pp. 156-167, 1990). We investigate this logic on finite structures and provide a normal form for it. We also present a treatment of Abiteboul and Vianu′s results from this point of view. In particular, we show that we can write a formula of FO + LFP that defines an ordering of the L k ∞ω , types uniformly over all finite structures. One consequence of this is a generalization of the equivalence of FO + LFP and P from ordered structures to classes of structures where every element is definable. We also settle a conjecture mentioned by Abiteboul and Vianu by showing that FO + LFP is properly contained in the polynomial time computable fragment of L ω ∞ω , raising the question of whether the latter fragment is a recursively enumerable class.

An alternative proof method for possibilistic logic and its application to terminological logics

Possibilistic logic, an extension of first-order logic, deals with uncertainty that can be estimated in terms of possibility and necessity measures. Syntactically, this means that a first-order formula is equipped with a possibility degree or a necessity degree that expresses to what extent the formula is possibly or necessarily true. Possibilistic resolution yields a calculus for possibilistic logic which respects the semantics developed for possibilistic logic. A drawback, which possibilistic resolution inherits from classical resolution, is that it may not terminate if applied to formulas belonging to decidable fragments of first-order logic. Therefore we propose an alternative proof method for possibilistic logic. The main feature of this method is that it completely abstracts from a concrete calculus but uses as basic operation a test for classical entailment. We then instantiate possibilistic logic with a terminological logic, which is a decidable subclass of first-order logic but nevertheless much more expressive than propositional logic. This yields an extension of terminological logics towards the representation of uncertain knowledge which is satisfactory from a semantic as well as algorithmic point of view.

Generalized Quantifiers and Logical Reducibilities

We consider extensions of first order logic (FO) and least fixed point logic (LFP) with generalized quantifiers in the sense of Lindstrom [Lin66]. We show that adding a finite set of such quantifiers to LFP fails to capture all polynomial time properties of structures, even over a fixed signature. We show that this strengthens results in [Hel92] and [KV92a]. We also consider certain regular infinite sets of Lindstrom quantifiers, which correspond to a natural notion of logical reducibility. We show that if there is any recursively enumerable set of quantifiers that can be added to FO (or LFP) to capture P, then there is one with strong uniformity conditions. This is established through a general result, linking the existence of complete problems for complexity classes with respect to the first order translations of [Imm87] or the elementary reductions of [LG77] with the existence of recursive index sets for these classes. Comments University of Pennsylvania Department of Computer and Information Science Technical Report No. MSCIS-92-85. This technical report is available at ScholarlyCommons: http://repository.upenn.edu/cis_reports/348 Generalized Quantifiers and Logical Reducibilities MS-CIS-92-85 LINC LAB 240

An Extension Principle for Fuzzy Logics

AbstractLet S be a set, P(S) the class of all subsets of S and F(S) the class of all fuzzy subsets of S. In this paper an “extension principle” for closure operators and, in particular, for deduction systems is proposed and examined. Namely we propose a way to extend any closure operator J defined in P(S) into a fuzzy closure operator J* defined in F(S). This enables us to give the notion of canonical extension of a deduction system and to give interesting examples of fuzzy logics. In particular, the canonical extension of the classical propositional calculus is defined and it is showed its connection with possibility and necessity measures. Also, the canonical extension of first order logic enables us to extend some basic notions of programming logic, namely to define the fuzzy Herbrand models of a fuzzy program. Finally, we show that the extension principle enables us to obtain fuzzy logics related to fuzzy subalgebra theory and graded consequence relation theory.Mathematics Subject Classification: 03B52.

Monotonicity and the Expressibility of NP Operators

We investigate why similar extensions of first-order logic using operators (that is, generalized quantifiers) corresponding to NP-complete decision problems apparently differ in expressibility: the logics capture either NP or LNP. It had been conjectured that the complexity class captured is NP if and only if the operator is monotone. We show that this conjecture is false. However, we provide evidence supporting a revised conjecture involving finite variations of monotone problems. Mathematics Subject Classification: 68Q15, 03D15, 03C13.

A sense-based, process model of belief

A process-oriented model of belief is presented which permits the representation of nested propositional attitudes within first-order logic. The model (NIM, for ‘nested intensional model’) is axiomatized, sense-based (via intensions), and sanctions inferences involving nested epistemic attitudes, with different agents and different times. Because NIM is grounded upon senses, it provides a framework in which agents may reason about the beliefs of another agent while remaining neutral with respect to the syntactic forms used to express the latter agent's beliefs. Moreover, NIM provides agents with a conceptual map, interrelating the concepts of ‘knowledge’, ‘belief’, ‘truth’, and a number of cognate concepts, such as ‘infers’, ‘retracts’, and ‘questions’. The broad scope of NIM arises in part from the fact that its axioms are represented in a novel extension of first-order logic, ℐ-FOL (presented herein). ℐ-FOL simultaneously permits the representation of truth ascriptions, implicit self-reference, and arbitrarily embedded sentences within a first-order setting. Through the combined use of principles derived from Frege, Montague, and Kripke, together with context-sensitive semantic conventions, ℐ-FOL captures the logic of truth inferences, while avoiding the inconsistencies exhibited by Tarski. Applications of ℐ-FOL and NIM to interagent reasoning are described and the soundness and completeness of ℐ-FOL are established herein.

Making deductive databases a practical technology: a step forward

Deductive databases provide a formal framework to study rule-based query languages that are extensions of first-order logic. However, deductive database languages and their current implementations do not seem appropriate for improving the development of real applications or even sample of them. Our goal is to make deductive database technology practical. The design and implementation of the RDL1 system, presented in this paper, constitute a step toward this goal. Our approach is based on the integration of a production rule language within a relational database system, the development of a rule-based programming environment and the support of system extensibility using Abstract Data Types. We discuss important practical experience gained during the implementation of the system. Also, comparisons with related work such as LDL, STARBURST and POSTGRES are given.

Circumscription within monotonic inferences

AbstractA conservative extension of first order logic, suitable for circumscriptive inference, is introduced.

A simple solution to Friedman's fourth problem

AbstractIt is shown that Friedman's problem, whether there exists a proper extension of first order logic satisfying the compactness and interpolation theorems, has extremely simple positive solutions if one considers extensions by generalized (finitary) propositional connectives. This does not solve, however, the problem of whether such extensions exist which are also closed under relativization of formulas.

Query processing in deductive databases with incomplete information

We study here automated deduction in databases in the presence of various types of inference rules of the form of Horn Clauses with Skolem functions. These inference rules are typical for databases with incomplete information. We demonstrate a number of results related to processing of conjunctive queries for different types of database intensions. In particular, we show that when a database intension is built from possibly cyclic inclusion dependencies and view definitions any conjunctive query can be translated to the an equivalent form which can be evaluated directly over the database extension (disregarding inference rules). We also demonstrate that the complexity of query processing significantly grows when we mix incomplete information with recursive rules. In particular, we demonstrate here that even the power of least fixpoint extension of first order logic may be not sufficient to process queries in the presence of incomplete data and recursive rules. The same is demonstrated in case disjunctive information is allowed in the database.

Filter Logics on ω

AbstractLogics LF(M) are considered, in which M (“most”) is a new first-order quantifier whose interpretation depends on a given filter F of subsets of ω. It is proved that countable compactness and axiomatizability are each equivalent to the assertion that F is not of the form {(⋂F) ∪ X: ∣ω − X∣ < ω} with ∣ω − ⋂F∣ = ω. Moreover the set of validities of LF (M) and even of depends only on a few basic properties of F. Similar characterizations are given of the class of filters F for which LF (M) has the interpolation or Robinson properties. An omitting types theorem is also proved. These results sharpen the corresponding known theorems on weak models (, where the collection q is allowed to vary. In addition they provide extensions of first-order logic which possess some nice properties, thus escaping from contradicting Lindström's Theorem [1969] only because satisfaction is not isomorphism-invariant (as it is tied to the filter F). However, Lindström's argument is applied to characterize the invariant sentences as just those of first-order logic.

Some contributions to definability theory for languages with generalized quantifiers

One of the first results in model theory [12] asserts that a first-order sentence is preserved in extensions if and only if it is equivalent to an existential sentence.In the first section of this paper, we analyze a natural program for extending this result to a class of languages extending first-order logic, notably including L(Q) and L(aa), respectively the languages with the quantifiers “there exist un-countably many” and “for almost all countable subsets”.In the second section we answer a question of Bruce [3] by showing that this program cannot resolve the question for L(Q). We also consider whether the natural class of “generalized Σ-sentences” in L(Q) characterizes the class of sentences preserved in extensions, refuting the relativized version but leaving the unrestricted question open.In the third section we show that the analogous class of L(aa)-sentences preserved in extensions does not include (up to elementary equivalence) all such sentences. This particular candidate class was nominated, rather tentatively, by Bruce [3].In the fourth section we show that under rather general conditions, if L is a countably compact extension of first-order logic and T is an ℵ1-categorical first-order theory, then L is trivial relative to T.

Decidability and Definability Results Concerning Well‐Orderings and Some Extensions of First Order Logic

Decidability and Definability Results Concerning Well‐Orderings and Some Extensions of First Order Logic