Unary Function Symbols Research Articles

PURE INDUCTIVE LOGIC WITH FUNCTIONS

AbstractWe consider the version of Pure Inductive Logic which obtains for the language with equality and a single unary function symbol giving a complete characterization of the probability functions on this language which satisfy Constant Exchangeability.

Uniform model-completeness for the real field expanded by power functions

AbstractWe prove that given any first order formula ϕ in the language L′ = {+, ·, <,(fi)iЄI, (ci)iЄI}, where the fi are unary function symbols and the ci are constants, one can find an existential formula Ψ such that φ and Ψ are equivalent in any L′-structure

Context unification with one context variable

The context unification problem is a generalization of standard term unification. It consists of finding a unifier for a set of term equations containing first-order variables and context variables. In this paper we analyze the special case of context unification where the use of at most one context variable is allowed and show that it is in NP. The motivation for investigating this subcase of context unification is interprocedural program analysis for programs described using arbitrary terms, generalizing the case where terms were restricted to using unary function symbols. Our results imply that the redundancy problem is in coNP, and that the finite redundancy property holds in this case. We also exhibit particular cases where one context unification is polynomial.

Tyrolean termination tool: Techniques and features

The Tyrolean Termination Tool ( T T T for short) is a powerful tool for automatically proving termination of rewrite systems. It incorporates several new refinements of the dependency pair method that are easy to implement, increase the power of the method, result in simpler termination proofs, and make the method more efficient. T T T employs polynomial interpretations with negative coefficients, like x − 1 for a unary function symbol or x − y for a binary function symbol, which are useful for extending the class of rewrite systems that can be proved terminating automatically. Besides a detailed account of these techniques, we describe the convenient web interface of T T T and provide some implementation details.

Dichotomies for classes of homomorphism problems involving unary functions

We study non-uniform constraint satisfaction problems where the underlying signature contains constant and function symbols as well as relation symbols. Amongst our results are the following. We establish a dichotomy result for the class of non-uniform constraint satisfaction problems over the signature consisting of one unary function symbol by showing that every such problem is either complete for L , via very restricted logical reductions, or trivial (depending upon whether the template function has a fixed point or not). We show that the class of non-uniform constraint satisfaction problems whose templates are structures over the signature λ 2 consisting of two unary function symbols reflects the full computational significance of the class of non-uniform constraint satisfaction problems over relational structures. We prove a dichotomy result for the class of non-uniform constraint satisfaction problems where the template is a λ 2-structure with the property that the two unary functions involved are the reverse of one another, in that every such problem is either solvable in polynomial-time or NP -complete. Finally, we extend some of our results to the situation where instances of non-uniform constraint satisfaction problems come equipped with lists of elements of the template structure which restrict the set of allowable homomorphisms.

Linear Time and the Power of One First-Order Universal Quantifier

The aim of this paper is to give two new logical characterizations of NLIN (nondeterministic linear time) improving significantly a previous characterization of E. Grandjean (1994, SIAM J. Comput. 23 , 573–597; and F. Olive, 1998, Comput. Complexity 7 , 54–97). It is known that NLIN coincides with the class of problems definable by formulas of the prenex form ∃ f 1 …∃ f k ϕ where the f i are unary function symbols and ϕ is first-order, prenex, with only one universal quantifier. We show that the characterization remains true in the two following cases: (a) Unary functions are replaced by a single binary relation whose outdegree is bounded by some fixed constant h . (b) When only ordered structures are considered, unary functions are restricted to be bijective.

Linear Time and the Power of One First-Order Universal Quantifier

The aim of this paper is to give two new logical characterizations of NLIN (nondeterministic linear time) improving significantly a previous characterization of E. Grandjean (1994, SIAM J. Comput. 23 , 573–597; and F. Olive, 1998, Comput. Complexity 7 , 54–97). It is known that NLIN coincides with the class of problems definable by formulas of the prenex form ∃ f 1 …∃ f k ϕ where the f i are unary function symbols and ϕ is first-order, prenex, with only one universal quantifier. We show that the characterization remains true in the two following cases: (a) Unary functions are replaced by a single binary relation whose outdegree is bounded by some fixed constant h . (b) When only ordered structures are considered, unary functions are restricted to be bijective.

The accepting power of unary string logic programs

The set of programs written in a small subset of pure Prolog called US is shown to accept exactly the class of regular languages. The language US contains only unary predicates and unary function symbols. Also, a subset of US called RUS is shown to be equivalent to US in its ability in accepting the class of regular languages. Every clause in RUS contains at most one function symbol in the head and at most one literal with no function symbol in the body. The result is very close to a theorem of Matos (TCS April 1997) but our proof is quite different. Though US and RUS have the same accepting power, their conciseness of expression is dramatically different: if we try to write an RUS program equivalent to a US program, the number of predicates in the RUS program could be O(2 2 N ) where N is the sum of the number of predicates and the number of functors in the US program.

The Classification of Polynomial Orderings on Monadic Terms

We investigate, for the case of unary function symbols, polynomial orderings on term algebras, that is reduction orderings determined by polynomial interpretations of the function symbols. Any total reduction ordering over unary function symbols can be characterised in terms of numerical invariants determined by the ordering alone: we show that for polynomial orderings these invariants, and in some cases the ordering itself, are essentially determined by the degrees and leading coefficients of the polynomial interpretations. Hence any polynomial ordering has a much simpler description, and thus the apparent complexity and variety of these orderings is less than it might seem at first sight.

Equational unification, word unification, and 2nd-order equational unification

For finite convergent term-rewriting systems it is shown that the equational unification problem is recursively independent of the equational matching problem, the word matching problem, and the 2nd- order equational matching problem. Apart from the latter these results are derived by considering term-rewriting systems on signatures that contain unary function symbols only (i.e., string-rewriting systems). Also for this special case 2nd-order equational matching is shown to be reducible to 1st-order equational matching. In addition, we present some new decidability results for simultaneous equational matching and unification. Finally, we compare the word unification problem to the 2nd- order equational unification problem.

The order types of termination orderings on monadic terms, strings and multisets

AbstractWe consider total well-founded orderings on monadic terms satisfying the replacement and full invariance properties. We show that any such ordering on monadic terms in one variable and two unary function symbols must have order typeω,ω2orωω. We show that a familiar construction gives rise to continuum many such orderings of order typeω. We construct a new family of such orderings of order typeω2, and show that there are continuum many of these. We show that there are only four such orderings of order typeωω, the two familiar recursive path orderings and two closely related orderings. We consider also total well-founded orderings onNnwhich are preserved under vector addition. We show that any such ordering must have order typeωkfor some 1 ≤k≤n. We show that ifk<nthere are continuum many such orderings, and ifk=nthere are onlyn!, then! lexicographic orderings.

First-order spectra with one binary predicate

The spectrum, Sp(ϑ), of a sentence ϑ is the set of cardinalities of finite structures which satisfy ϑ. We prove that any set of integers which is in ≻ Func 1 ∞, i.e. in the class of spectra of first-order sentences of type containing only unary function symbols, is also in BIN 1, i.e. in the class of spectra of first-order sentences of type involving only a single binary relation. We give similar results for generalized spectra and some corollaries: in particular, from the fact that the large complexity class ∪ c NTIME RAM( cn) is included in ≻ Func 1 ∞ for unary languages ( n denotes the input integer), we deduce that the set of primes and many “natural” sets belong to BIN 1. We also give some consequences for the image of spectra under polynomials of Q[x] .

EXPRESSIBILITY OF THE SEMANTICS OF SEQUENTIAL PROGRAMS IN FIRST-ORDER LOGIC

The notion of an expressive interpretation was originally introduced by Cook in order to formulate completeness results for Hoare-style proof systems. An interpretation is called expressive for a programming language if the input/output relation of every program can be expressed by a first-order formula. Various necessary or sufficient conditions for an interpretation to be expressive are known. In this paper a characterization is given which improves on these results. It holds for a wide spectrum of programming languages. An interpretation is expressive iff it is either 1. uniformly locally finite or 2. weakly arithmetic and it has evaluation predicates (which return the truth value of a boolean expression, given its godel number and the values of its free variables). Besides this characterization, it is investigated which first-order signatures allow nontrivial (i.e. non uniformly locally finite) expressive interpretations. Signatures which have (besides constants) only one unary function symbol and only unary relation symbols, have only trivial expressive interpretations. All other signatures allow nontrivial expressive interpretations as well.

Remark on Kreisel's conjecture

At the end of the 60's, Kreisel conjectured that formal arithmetics admit infinite induction rules if the lengths of proofs of their premises are uniformly bounded by the same number. By the length of a proof we mean the number of applications of axioms and rules of inference. In this article we construct a theoryR*with a finite number of specific axioms. The language ofR* contains a constant 0, a unary function symbol, equality, and ternary predicates for addition and multiplication. It is proved that for any consistent axiomatizable extension R* it is possible to find a formula A(a) that satisfies the following conditions: a) ∀xA(x) cannot be derived in ; b) for any n the length of the proof of the formula A(0(n)) is no greater than c1[log2 (n+1)]+c2, where the constants c1 and c2 are independent of n. Here the expression 0(n) indicates 0 with n primes.

On subsumption and semiunification in feature algebras

We consider a generalization of term subsumption, or matching, to a class of mathematical structures which we call feature algebras. We show how these generalize both first-order terms and the feature structures used in computational linguistics. The notion of term subsumption generalizes to a natural notion of algebra homomorphism. In the setting of feature algebras, unification, corresponds naturally to solving constraints involving equalities between strings of unary function symbols, and semiunification also allows inequalities representing subsumption constraints. Our generalization allows us to show that the semiunification problem for finite feature algebras is undecidable. This implies that the corresponding problem for rational trees (cyclic terms) is also undecidable.

Les beaux automorphismes

Assume that the class of partial automorphisms of the monster model of a complete theory has the amalgamation property. The beautiful automorphisms are the automorphisms of models ofT which: 1. are strong, i.e. leave the algebraic closure (inTeq) of the empty set pointwise fixed, 2. are obtained by the Fraisse construction using the amalgamation property that we have just mentioned. We show that all the beautiful automorphisms have the same theory (in the language ofT plus one unary function symbol for the automorphism), and we study this theory. In particular, we examine the question of the saturation of the beautiful automorphisms. We also prove that in some cases (in particular if the theory is ω-stable andG-trivial), almost all (in the sense of Baire categoricity) automorphisms of the saturated countable model are beautiful and conjugate.

On ground-confluence of term rewriting systems

It is undecidable whether or not a finite Noetherian term rewriting system is ground-confluent. This undecidability result holds even when systems involving only unary function symbols and one constant are being considered, or when left-linear or right-linear monadic systems are being considered. On the other hand, if all right-hand sides of rules of a finite Noetherian rewriting system R are either ground terms or proper subterms of the corresponding left-hand sides, then it is decidable whether or not R is ground-confluent.

First-order spectra with one variable

Define a (∀1 unary)-sentence to be a prenex first-order sentence of unary type (i.e., a type which only contains unary relation and function symbols and constant symbols) with only one (universal) quantifier. A successor structure is a structure 〈 B, S〉 such that S is a function which is a permutation of the basis B with only one cycle. We exhibit a (∀1, unary)-sentence φ of type { S, U 1, …, U p } such that if B is finite then 〈 B, S〉 is a successor structure if 〈 B, S〉 satisfies ∃ U 1, …, ∃ U p ϕ. It implies that ⋃ NRAM( cn)=SPECTRA(∀1, unary), c⩾1 where NRAM(cn) denotes the class of sets of positive integers accepted by a nondeterministic random access machine in time cn (where n is the input integer) and SPECTRA(∀1, unary) is the class of finite spectra of (∀1, unary)-sentences. Another consequence is that some graph properties (hamiltonicity, connectedness) can be characterised by sentences with unary function symbols and constant symbols and only one variable. This contrasts with the result (by Fagin and De Rougemont) that these two graph properties are not definable by monadic generalized spectra (without function symbols) even in the presence of an underlying successor relation.

Decidability in elementary analysis, II

We continue the study of the first order properties of real functions. More precisely, we consider sentences of the form (∀f∈#7B-F)φ, where #7B-F is a family of functions (continuous, C 1 , analytic, monotone, etc.) from R to R or from I=[0, 1] to I and φ is a first order formula. φ involves the ordering of R, quantification over R, and the unary function symbol F

A note on unification type zero

Let E be an equational theory and =E be the equality of terms induced by E. A substitution 8 is called an E-unifier of the pair of terms s, t iff sf3 =E tt9. The set of all E-unifiers of s, ? is denoted by UE(s, t). We are mostly interested in complete sets of E-unifiers, i.e., sets of E-unifiers from which UE(s, t) may be generated by instantation. More formally, we extend =E to UE(s, t) and define a quasi-ordering sE on UE(s, t) by l 13 =E r iff xt9 =E xr for all variables x occurring in s or t, l 7 <E B iff there exists a substitution X such that 8 =E T 0 X. A complete set cU,(s, t) of E-unifiers of s, t is defined as (I) cG(s9 t) c UE(& t), (2) for all 0 E U&s, t) there exists a u E cU,(s, t) such that u =& 8. A set of most general E-unifiers &(s, t) is a complete set of E-unifiers of s, t satisfying the minimality condition: (3) for all 6, T in pU,(s, t), u dE r implies u = 7. This notion was introduced by Plotkin [a]. In the same paper he conjectured that there exists an equational theory E and terms s, t such that UE(s, t) does not exist. We say that such a theory is of unification type zero. The first example of a type zero theory is due to Fages and Huet [4,5]. They constructed a theory E and terms s, t such that UE(s, t) contains a strictly decreasing chain with respect to Q~ which is a complete set of E-unifiers of s, t. Under these circumstances, no complete set of E-unifiers of s, t satisfies the minimality condition (see [4,5] or Section 3 where we use Fages-Huet’s method to show that the respective theory has type zero). Schmidt-Schauss [7] and the present author [l] showed that the theory of idempotent semigroups is of unification type zero and in (21 we have proved that almost all varieties of idempotent semigroups are defined by type zero theories. In this case, complete sets of unifiers are not simply chains but we also used a decreasing chain of unifiers without lower bound -which was not complete but satisfied another additional condition-to establish our result. In the present paper we show that these additional conditions are not superfluous, i.e., that it is not sufficient to find a strictly decreasing chain without lower bound in UE(s, t) for some terms s,? to prove that E has unification type zero. But note that-by Zom’s lemma-the existence of such a chain is a necessary condition for E to be type zero. An example which establishes the same result can be found in [3]. The authors of [3] use a theory which has only unary function symbols.