Predicate Symbols Research Articles

Decidability questions for a ring of Laurent polynomials

We prove that the additive structure of the ring of Laurent polynomials augmented by the predicate symbol P, where P(x) if and only if x is a power of t, is decidable. We also prove that the first-order theory of the previous structure together with the relation ∣t, where x∣ty if and only if ∃s∈Zy=x⋅ts, is undecidable.

Generalized stability of torsion-free Abelian groups

Stable theories were introduced in [1] for constructing classification theory and are a generalization of the concept of a totally transcendental theory as defined in [2]. In [3], it was proved that the property of being stable for a theory is equivalent to being definable for every complete type of the theory. This property plays a fundamental part in research on stable theories. In [4], the notion of E∗-stability (generalized stability) was introduced, and it was proved that types for E∗-stable theories are definable. A consequence of that result was stating, along with definability of types for stable theories (see [3]), that types over any P -sets in P -stable theories likewise are definable (which had been established in [5] for types over P -models). The notion of E∗-stability is a new stability scale, whose basic parameter is a mapping of types of a complete theory into types of another theory. An interesting example of E∗-stability is (P, a)-stability, defined by adding to a language a unary predicate symbol and adding to types the condition of being algebraically closed for that predicate. In this paper, we work to prove a (P, a)-stability theorem for theories of torsion-free Abelian groups. In so doing, use is made of quantifier elimination down to positive primitive formulas in Abelian groups with a predicate distinguishing a subgroup. By virtue of this fact, the question of being (P, a)-stable for a theory reduces to asking if a system of linear equations with integer coefficients has a solution in the algebraic closure of constants involved in the system. To ∗Supported by RFBR (project No. 09-01-00336-a).

Characterizing minimal semantics-preserving slices of predicate-linear, free, liberal program schemas

A program schema defines a class of programs, all of which have identical statement structure, but whose functions and predicates may differ. A schema thus defines an entire class of programs according to how its symbols are interpreted. A subschema of a schema is obtained from a schema by deleting some of its statements. We prove that given a schema S which is predicate-linear, free and liberal, such that the true and false parts of every if predicate satisfy a simple additional condition, and a slicing criterion defined by the final value of a given variable after execution of any program defined by S, the minimal subschema of S which respects this slicing criterion contains all the function and predicate symbols ‘needed’ by the variable according to the data dependence and control dependence relations used in program slicing, which is the symbol set given by Weiser’s static slicing algorithm. Thus this algorithm gives predicate-minimal slices for classes of programs represented by schemas satisfying our set of conditions. We also give an example to show that the corresponding result with respect to the slicing criterion defined by termination behaviour is incorrect. This complements a result by the authors in which S was required to be function-linear, instead of predicate-linear.

Monadic datalog over finite structures of bounded treewidth

Bounded treewidth and monadic second-order (MSO) logic have proved to be key concepts in establishing fixed-parameter tractability results. Indeed, by Courcelle's Theorem we know that any property of finite structures, which is expressible by an MSO sentence, can be decided in linear time (data complexity) if the structures have bounded treewidth. In principle, Courcelle's Theorem can be applied directly to construct concrete algorithms by transforming the MSO evaluation problem into a tree language recognition problem. The latter can then be solved via a finite tree automaton (FTA). However, this approach has turned out to be problematical, since even relatively simple MSO formulae may lead to a “state explosion” of the FTA. In this work we propose monadic datalog (i.e., datalog where all intentional predicate symbols are unary) as an alternative method to tackle this class of fixed-parameter tractable problems. We show that if some property of finite structures is expressible in MSO then this property can also be expressed by means of a monadic datalog program over the decomposed structure : we mean by this that the original structure is augmented with new elements and new relations that encode one of its tree decompositions. In the first place, we thus compare the expressive power of two query languages. However, we also show that the resulting fragment of datalog can be evaluated in linear time (both with respect to the program size and with respect to the data size). Hence, our transformation of an MSO query into a monadic datalog program yields an alternative proof of Courcelle's Theorem. In order to actually construct efficient algorithms for problems whose tractability is due to Courcelle's Theorem, we propose to use a fragment of full (i.e., not necessarily monadic) datalog which allows for a succinct representation of the corresponding monadic datalog programs and for an efficient execution. This new approach is put to work by devising datalog programs for the 3-Colorability problem of graphs and for the PRIMALITY problem of relational schemas (i.e., testing if some attribute in a relational schema is part of a key). We also report on experimental results with a prototype implementation.

The two-cardinal problem for languages of arbitrary cardinality

AbstractLet ℒ be a first-order language of cardinality κ++ with a distinguished unary predicate symbol U. In this paper we prove, working on L, the two cardinal transfer theorem (κ+,κ) ⇒ (κ++, κ+) for this language. This problem was posed by Chang and Keisler more than twenty years ago.

Predicate transformers in the context of symbolic modeling of transition systems

In modelling attribute transition systems, classes of their states can be described in a given signature of functional and predicate symbols. A procedure is developed for transforming such formulas by assignment operators and the obtained formulas are proved to correspond to strongest postconditions. A peculiarity of this paper is that functional-type attributes can be used in specifying transition systems, in particular, their array-type attributes.

Transformation of a language L* specification of an FSM into an automata equivalent specification in the language L

A method is proposed to transform an FSM specification in the language L* into a specification in the language L. Using additional predicate symbols, the former specification is first transformed into a specification of an FSM with finite memory. Then this specification is transformed into an automata equivalent specification in the language L.

The Bernays-Schönfinkel-Ramsey class for set theory: semidecidability

AbstractAs is well-known, the Bernays-Schönfinkel-Ramsey class of all prenex ∃*∀*-sentences which are valid in classical first-order logic is decidable. This paper paves the way to an analogous result which the authors deem to hold when the only available predicate symbols are ∈ and =, no constants or function symbols are present, and one moves inside a (rather generic) Set Theory whose axioms yield the well-foundedness of membership and the existence of infinite sets. Here semi-decidability of the satisfiability problem for the BSR class is proved by following a purely semantic approach, the remaining part of the decidability result being postponed to a forthcoming paper.

The atomic theory of multiplication and division of semiring ideals

We consider two-sided ideals of semirings. More precisely, we study the theory of two-sided ideals in the signature consisting of the predicate symbol ⊆ and three function symbols that denote the multiplication, right division, and left division of ideals. We prove that the set of those atomic formulas in this signature that are valid for all semirings and all valuations is decidable.

Almost everywhere elimination of probability quantifiers

AbstractWe obtain an almost everywhere quantifier elimination for (the noncritical fragment of) the logic with probability quantifiers, introduced by the first author in [10]. This logic has quantifiers like ∃≥3/4y which says that “for at least 3/4 of all y”. These results improve upon the 0-1 law for a fragment of this logic obtained by Knyazev [11]. Our improvements are:1. We deal with the quantifier ∃≥ry, where y is a tuple of variables.2. We remove the closedness restriction, which requires that the variables in y occur in all atomic subformulas of the quantifier scope.3. Instead of the unbiased measure where each model with universe n has the same probability, we work with any measure generated by independent atomic probabilities PR for each predicate symbol R.4. We extend the results to parametric classes of finite models (for example, the classes of bipartite graphs, undirected graphs, and oriented graphs).5. We extend the results to a natural (noncritical) fragment of the infinitary logic with probability quantifiers.6. We allow each PR, as well as each r in the probability quantifier (∃≥ry), to depend on the size of the universe.

Constructing infinite models represented by tree automata

We propose a method to use finite model builders in order to construct infinite models of first-order formulae. The constructed models are Herbrand interpretations, in which the interpretation of the predicate symbols is specified by tree tuple automata (Comon et al. 1997). Our approach is based on formula transformation: a formula ? is transformed into a formula Δ(?) s.t. ? has a model representable by a term tuple automaton iff Δ(?) has a finite model. This paper is an extended version of Peltier (2008).

A Diagrammatic Logic for Object-Oriented Visual Modeling

Formal generalized sketches is a graph-based specification format that borrows its main ideas from categorical and ordinary first-order logic, and adapts them to software engineering needs. In the engineering jargon, it is a modeling language design pattern that combines mathematical rigor and appealing graphical appearance. The paper presents a careful motivation and justification of the applicability of generalized sketches for formalizing practical modeling notations. We extend the sketch formalism by dependencies between predicate symbols and develop new semantic notions based on the Instances-as-typed-structures idea. We show that this new framework fits in the general patterns of the institution theory and is well amenable to algebraic manipulations.

AXIOMS FOR DETERMINATENESS AND TRUTH

A new formal theory DT of truth extending PA is introduced, whose language is that of PA together with one new unary predicate symbol T (x), for truth applied to Gödel numbers of suitable sentences in the extended language. Falsity of x, F(x), is defined as truth of the negation of x; then, the formula D(x) expressing that x is the number of a determinate meaningful sentence is defined as the disjunction of T(x) and F(x). The axioms of DT are those of PA extended by (I) full induction, (II) strong compositionality axioms for D, and (III) the recursive defining axioms for T relative to D. By (II) is meant that a sentence satisfies D if and only if all its parts satisfy D; this holds in a slightly modified form for conditional sentences. The main result is that DT has a standard model. As an improvement over earlier systems developed by the author, DT meets a number of leading criteria for formal theories of truth that have been proposed in the recent literature and comes closer to realizing the informal view that the domain of the truth predicate consists exactly of the determinate meaningful sentences.

The atomic theory of division of semiring ideals

We consider two-sided ideals of semirings. More precisely, we study the theory of two-sided ideals in the signature consisting of the predicate symbol ⊆ and two function symbols that denote the right and left division of ideals. We prove that the set of those atomic formulas in this signature that are valid for all semirings and all valuations is decidable.

An Improvement of Herbrand's Theorem and Its Application to Model Generation Theorem Proving

This paper presents an improvement of Herbrand's theorem. We propose a method for specifying a subuniverse of the Herbrand universe of a clause set S for each argument of predicate symbols and function symbols in S. We prove that a clause set S is unsatisfiable if and only if there is a finite unsatisfiable set of ground instances of clauses of S that are derived by only instantiating each variable, which appears as an argument of predicate symbols or function symbols, in S over its corresponding argument's sub-universe of the Herbrand universe of S. Because such sub-universes are usually smaller (sometimes considerably) than the Herbrand universe of S, the number of ground instances may decrease considerably in many cases. We present an algorithm for automatically deriving the sub-universes for arguments in a given clause set, and show the correctness of our improvement. Moreover, we introduce an application of our approach to model generation theorem proving for non-range-restricted problems, show the range-restriction transformation algorithm based on our improvement and provide examples on benchmark problems to demonstrate the power of our approach.

Partial Horn logic and cartesian categories

A logic is developed in which function symbols are allowed to represent partial functions. It has the usual rules of logic (in the form of a sequent calculus) except that the substitution rule has to be modified. It is developed here in its minimal form, with equality and conjunction, as “partial Horn logic”. Various kinds of logical theory are equivalent: partial Horn theories, “quasi-equational” theories (partial Horn theories without predicate symbols), cartesian theories and essentially algebraic theories. The logic is sound and complete with respect to models in Set , and sound with respect to models in any cartesian (finite limit) category. The simplicity of the quasi-equational form allows an easy predicative constructive proof of the free partial model theorem for cartesian theories: that if a theory morphism is given from one cartesian theory to another, then the forgetful (reduct) functor from one model category to the other has a left adjoint. Various examples of quasi-equational theory are studied, including those of cartesian categories and of other classes of categories. For each quasi-equational theory T another, Cart ϖ T , is constructed, whose models are cartesian categories equipped with models of T . Its initial model, the “classifying category” for T , has properties similar to those of the syntactic category, but more precise with respect to strict cartesian functors.

Efficient solving of quantified inequality constraints over the real numbers

Let a quantified inequality constraint over the reals be a formula in the first-order predicate language over the structure of the real numbers, where the allowed predicate symbols are ≤ and <. Solving such constraints is an undecidable problem when allowing function symbols such sin or cos. In this article, we give an algorithm that terminates with a solution for all, except for very special, pathological inputs. We ensure the practical efficiency of this algorithm by employing constraint programming techniques.

Efficient solving of quantified inequality constraints over the real numbers

Let a quantified inequality constraint over the reals be a formula in the first-order predicate language over the structure of the real numbers, where the allowed predicate symbols are $\leq$ and $<$. Solving such constraints is an undecidable problem when allowing function symbols such $\sin$ or $\cos$. In the paper we give an algorithm that terminates with a solution for all, except for very special, pathological inputs. We ensure the practical efficiency of this algorithm by employing constraint programming techniques.

Learning of erasing primitive formal systems from positive examples

The present paper considers the learning problem of erasing primitive formal systems, PFSs for short, in view of inductive inference in Gold framework from positive examples. A PFS is a kind of logic program over strings called regular patterns, and consists of exactly two axioms of the forms p ( π ) ← and p ( τ ) ← p ( x 1 ) , … , p ( x n ) , where p is a unary predicate symbol, π and τ are regular patterns, and x i s are distinct variables. A PFS is erasing or nonerasing according to allowing the empty string substitution for some variables or not. We investigate the learnability of the class P F S L of languages generated by the erasing PFSs satisfying a certain condition. We first show that the class P F S L has M-finite thickness. Moriyama and Sato showed that a language class with M-finite thickness is learnable if and only if there is a finite tell tale set for each language in the class. We then introduce a particular type of finite set of strings for each erasing PFS, and show that the set is a finite tell tale set of the language. These imply that the class P F S L is learnable from positive examples.

Index conditions of resolution

In this paper, the following results are proved: (1) Using both deletion strategy and lock strategy, resolution is complete for a clause set where literals with the same predicate or proposition symbol have the same index. (2) Using deletion strategy, both positive unit lock resolution and input lock resolution are complete for a Horn set where the indexes of positive literals are greater than those of negative literals. (3) Using deletion strategy, input half-lock resolution is complete for a Horn set.