scholarly journals Theory of finite or infinite trees revisited

2008 ◽  
Vol 8 (04) ◽  
pp. 431-489 ◽  
Author(s):  
KHALIL DJELLOUL ◽  
THI-BICH-HANH DAO ◽  
THOM FRÜHWIRTH
Keyword(s):  
Constraint Satisfaction ◽  
Decision Procedure ◽  
Constraint Satisfaction Problem ◽  
Free Variable ◽  
Constraint Solver ◽  
First Order ◽  
Infinite Trees ◽  
Order Constraint ◽  
Infinite Set ◽  
Rewriting Rules

AbstractWe present in this paper a first-order axiomatization of an extended theoryTof finite or infinite trees, built on a signature containing an infinite set of function symbols and a relationfinite(t), which enables to distinguish between finite and infinite trees. We show thatThas at least one model and prove its completeness by giving not only a decision procedure, but a full first-order constraint solver that gives clear and explicit solutions for any first-order constraint satisfaction problem inT. The solver is given in the form of 16 rewriting rules that transform any first-order constraintinto an equivalent disjunction φ of simple formulas such that φ is either the formulatrueor the formulafalseor a formula having at least one free variable, being equivalent neither totruenor tofalseand where the solutions of the free variables are expressed in a clear and explicit way. The correctness of our rules implies the completeness ofT. We also describe an implementation of our algorithm in CHR (Constraint Handling Rules) and compare the performance with an implementation in C++ and that of a recent decision procedure for decomposable theories.

scholarly journals Permutation groups with small orbit growth

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Manuel Bodirsky ◽  
Bertalan Bodor
Keyword(s):  
Automorphism Group ◽  
Constraint Satisfaction ◽  
Constraint Satisfaction Problem ◽  
Permutation Groups ◽  
First Order ◽  
Finite Covers

Abstract Let K exp + \mathcal{K}_{{\operatorname{exp}}{+}} be the class of all structures 𝔄 such that the automorphism group of 𝔄 has at most c ⁢ n d ⁢ n cn^{dn} orbits in its componentwise action on the set of 𝑛-tuples with pairwise distinct entries, for some constants c , d c,d with d < 1 d<1 . We show that K exp + \mathcal{K}_{{\operatorname{exp}}{+}} is precisely the class of finite covers of first-order reducts of unary structures, and also that K exp + \mathcal{K}_{{\operatorname{exp}}{+}} is precisely the class of first-order reducts of finite covers of unary structures. It follows that the class of first-order reducts of finite covers of unary structures is closed under taking model companions and model-complete cores, which is an important property when studying the constraint satisfaction problem for structures from K exp + \mathcal{K}_{{\operatorname{exp}}{+}} . We also show that Thomas’ conjecture holds for K exp + \mathcal{K}_{{\operatorname{exp}}{+}} : all structures in K exp + \mathcal{K}_{{\operatorname{exp}}{+}} have finitely many first-order reducts up to first-order interdefinability.

scholarly journals Interpreting GPFCSP within the LΠ ½ logic framework

Filomat ◽  
2013 ◽  
Vol 27 (5) ◽  
pp. 889-897
Author(s):  
Aleksandar Takaci ◽  
Aleksandar Perovic ◽  
Aleksandar Jovanovic
Keyword(s):  
Constraint Satisfaction ◽  
Constraint Satisfaction Problem ◽  
Order Theory ◽  
First Order ◽  
First Order Theory ◽  
Complete Axiomatization ◽  
Interpretation Method

The aim of this paper is to interpret Generalized Priority Constraint Satisfaction Problem (GPFCSP) using the interpretational method. We will interpret the L? ? logic into the first order theory of the reals, in order to obtain alternative, simple-complete axiomatization of L? ? logic. A complete axiomatization using the interpretation method as a syntactical approach is given.

scholarly journals Tractability of quantified temporal constraints to the max

2014 ◽  
Vol 24 (08) ◽  
pp. 1141-1156 ◽  
Author(s):  
Manuel Bodirsky ◽  
Hubie Chen ◽  
Michał Wrona
Keyword(s):  
Constraint Satisfaction ◽  
Constraint Satisfaction Problem ◽  
Temporal Constraints ◽  
Temporal Constraint ◽  
First Order ◽  
Constraint Language ◽  
Constraint Languages

A temporal constraint language is a set of relations that are first-order definable over (ℚ;<). We show that several temporal constraint languages whose constraint satisfaction problem is maximally tractable are also maximally tractable for the more expressive quantified constraint satisfaction problem. These constraint languages are defined in terms of preservation under certain binary polymorphisms. We also present syntactic characterizations of the relations in these languages.

Decomposable theories

2007 ◽  
Vol 7 (5) ◽  
pp. 583-632 ◽  
Author(s):  
KHALIL DJELLOUL
Keyword(s):  
General Algorithm ◽  
Boolean Combination ◽  
First Order ◽  
Free Variables ◽  
Infinite Trees ◽  
Decomposable Theory ◽  
Rewriting Rules

AbstractWe present in this paper a general algorithm for solving first-order formulas in particular theories called decomposable theories. First of all, using special quantifiers, we give a formal characterization of decomposable theories and show some of their properties. Then, we present a general algorithm for solving first-order formulas in any decomposable theory T. The algorithm is given in the form of five rewriting rules. It transforms a first-order formula ϕ, which can possibly contain free variables, into a conjunction φ of solved formulas easily transformable into a Boolean combination of existentially quantified conjunctions of atomic formulas. In particular, if ϕ has no free variables then φ is either the formula true or ¬true. The correctness of our algorithm proves the completeness of the decomposable theories. Finally, we show that the theory ${\cal T}$ of finite or infinite trees is a decomposable theory and give some benchmarks realized by an implementation of our algorithm, solving formulas on two-partner games in ${\cal T}$ with more than 160 nested alternated quantifiers.

On deciding the provability of certain fixed point statements

1977 ◽  
Vol 42 (2) ◽  
pp. 191-193
Author(s):  
George Boolos
Keyword(s):  
Fixed Point ◽  
Fixed Points ◽  
Decision Procedure ◽  
Free Variable ◽  
Peano Arithmetic ◽  
First Order ◽  
Order Arithmetic ◽  
Functional Combination

Terminology. PA is Peano Arithmetic, classical first-order arithmetic with induction. ⌈A⌉ is the formal numeral in PA for the Gödel number of A. – A is the negation of A, (A&B) is the conjunction of A and B, and Bew(x) is the usual provability predicate for PA. neg(x), conj(x, y), bicond(x, y), and bew(x) are terms of PA such that for all sentences A and B of PA ⊢PA, neg(˹A˺) = ˹−A˺ ⊢PA Conj(˹A˺, ˹B˺)= ˹(A&B)˺ ⊢PA bicond(˹A˺, ˹B˺)= ˹(A ↔ B)˺, and ⊢PA bew(˹A˺) = ˹Bew(˹A˺)˺. T is the sentence ‘0 = 0’ and Con is the usual sentence expressing the consistency of PA. If A (x) is any formula of PA, then a fixed point of A(x) is a sentence S such that ⊢PAS ↔ A(˹S˺). (It is well known that every formula of PA with one free variable has a fixed point.) The P-terms are defined inductively by: the variable x is a P-term; if t(x) and u(x) are P-terms, so are neg(t(x)), conj(t(x), u(x)), and bew(t(x)). A basic P-formula is a formula Bew(t(x)), where t(x) is a P-term; and a P-formula is a truth-functional combination of basic P-formulas. An F-sentence is a member of the smallest class that contains Con and contains −A, (A&B), and −Bew(˹−A˺) whenever it contains A and B. In [B] we gave a decision procedure for the class of true F-sentences.−Bew(x), Bew(x), and Bew(neg(x)) are examples of P-formulas, and fixed points of these particular P-formulas have been studied by Gödel, Henkin [H] and Löb [L], and Jeroslow [J], respectively. In this note we show how to decide whether or not a fixed point of any given P-formula is provable in PA.

Sign in / Sign up

Export Citation Format

Share Document