Results 1  10
of
26
The monadic secondorder logic of graphs I. Recognizable sets of Finite Graphs
 Information and Computation
, 1990
"... The notion of a recognizable sef offinite graphs is introduced. Every set of finite graphs, that is definable in monadic secondorder logic is recognizable, but not vice versa. The monadic secondorder theory of a contextfree set of graphs is decidable. 0 19W Academic Press. Inc. This paper begins ..."
Abstract

Cited by 208 (14 self)
 Add to MetaCart
The notion of a recognizable sef offinite graphs is introduced. Every set of finite graphs, that is definable in monadic secondorder logic is recognizable, but not vice versa. The monadic secondorder theory of a contextfree set of graphs is decidable. 0 19W Academic Press. Inc. This paper begins an investigation of the monadic secondorder logic of graphs and of sets of graphs, using techniques from universal algebra, and the theory of formal languages. (By a graph, we mean a finite directed hyperedgelabelled hypergraph, equipped with a sequence of distinguished vertices.) A survey of this research can be found in Courcelle [ 111. An algebraic structure on the set of graphs (in the above sense) has been proposed by Bauderon and Courcelle [2,7]. The notion of a recognizable set of finite graphs follows, as an instance of the general notion of recognizability introduced by Mezei and Wright in [25]. A graph can also be considered as a logical structure of a certain type. Hence, properties of graphs can be written in firstorder logic or in secondorder logic. It turns out that monadic secondorder logic, where quantifications over sets of vertices and sets of edges are used, is a reasonably powerful logical language (in which many usual graph properties can be written), for which one can obtain decidability results. These decidability results do not hold for secondorder logic, where quantifications over binary relations can also be used. Our main theorem states that every definable set of finite graphs (i.e., every set that is the set of finite graphs satisfying a graph property expressible in monadic secondorder logic) is recognizable. * This work has been supported by the “Programme de Recherches Coordonntes: Mathematiques et Informatique.”
Fixed Point Characterization of Infinite Behavior of Finite State Systems
, 1996
"... Infinite behavior of nondeterministic finite state automata running over infinite trees or more generally over elements of an arbitrary algebraic structure is characterized by a calculus of fixed point terms interpreted in powerset algebras. These terms involve the least and greatest fixed point ope ..."
Abstract

Cited by 28 (5 self)
 Add to MetaCart
Infinite behavior of nondeterministic finite state automata running over infinite trees or more generally over elements of an arbitrary algebraic structure is characterized by a calculus of fixed point terms interpreted in powerset algebras. These terms involve the least and greatest fixed point operators and disjunction as the only logical operation. A tight correspondence is established between a hierarchy of Rabin indices of automata and a hierarchy induced by alternation of the least and greatest fixed point operators. It is shown that, in the powerset algebra of trees constructed from a set of functional symbols, the fixed point hierarchy is infinite unless all the symbols are unary (i.e. trees are words). It is also shown that an interpretation of a closed fixed point term in any powerset algebra can be factorized through the interpretation of this term in the powerset algebra of trees, from which it is deduced that the question whether a term denotes always ; can be answered in ...
Perfect Model Checking via Unfold/Fold Transformations
 In Computational Logic, LNCS 1861
, 2000
"... We show how program transformation rules and strategies may be used for proving the satisfiability of first order formulas in some classes of models. In particular, we propose a technique for showing that a closed first order formula ' holds in the perfect model M(P ) of a logic program P with local ..."
Abstract

Cited by 10 (7 self)
 Add to MetaCart
We show how program transformation rules and strategies may be used for proving the satisfiability of first order formulas in some classes of models. In particular, we propose a technique for showing that a closed first order formula ' holds in the perfect model M(P ) of a logic program P with locally stratified negation. This property is denoted by M(P ) j= '. For this purpose we consider a new version of the unfold/fold transformation rules and we show that this version preserves the perfect model semantics. Our proof method, called unfold/fold proof method, shows M(P ) j= ' by: (i) introducing a new predicate symbol f and constructing a conjunction F (f ; ') of clauses such that M(P ) j= ' i M(P ^ F (f ; ')) j= f , and then (ii) transforming the program P ^F (f ; ') into a new program of the form Q^f , for some conjunction Q of clauses. We also present a strategy for applying our unfold/fold rules in a semiautomatic way. Our strategy may or may not terminate, depending on t...
Expressiveness of Full First Order Constraints in the Algebra of Finite or Infinite Trees
, 2000
"... We are interested in the expressiveness of constraints represented by general first order formulae, with equality as unique relational symbol and functional symbols taken from an infinite set F. The chosen domain is the set of trees whose nodes, in possibly in nite number, are labeled by elements of ..."
Abstract

Cited by 7 (0 self)
 Add to MetaCart
We are interested in the expressiveness of constraints represented by general first order formulae, with equality as unique relational symbol and functional symbols taken from an infinite set F. The chosen domain is the set of trees whose nodes, in possibly in nite number, are labeled by elements of F . The operation linked to each element f of F is the mapping (a1 , ..., an ) 7! b, where b is the tree whose initial node is labeled f and whose sequence of daughters is a1 , ..., an . We first consider constraints involving long alternated sequences of quantifiers 9898 . . . . We show how to express winning positions of twopartners games with such constraints and apply our results to two examples. We then construct a family of strongly expressive constraints, inspired by a constructive proof of a complexity result by Pawel Mielniczuk. This family involves the huge number (k), obtained by evaluating top down a power tower of 2's, of height k. With elements of this family, of sizes at most proportional to k, we de ne a nite tree having (k) nodes, and we express the result of a Prolog machine executing at most (k) instructions. By replacing the Prolog machine by a Turing machine we rediscover the following result of Sergei Vorobyov: the complexity of an algorithm, deciding whether a constraint without free variables is true, cannot be bounded above by a function obtained by nite composition of elementary functions including exponentiation. Finally, taking advantage of the fact that we have at our disposal an algorithm for solving such constraints in all their generality, we produce a set of benchmarks for separating feasible examples from purely speculative ones. Among others we solve constraints involving alternated sequences of more than 160 quantifiers.
Proving properties of constraint logic programs by eliminating existential variables
 In Proc. ICLP ’06, LNCS 4079
, 2006
"... Abstract. We propose a method for proving rst order properties of constraint logic programs which manipulate nite lists of real numbers. Constraints are linear equations and inequations over reals. Our method consists in converting any given rst order formula into a strati ed constraint logic progra ..."
Abstract

Cited by 5 (5 self)
 Add to MetaCart
Abstract. We propose a method for proving rst order properties of constraint logic programs which manipulate nite lists of real numbers. Constraints are linear equations and inequations over reals. Our method consists in converting any given rst order formula into a strati ed constraint logic program and then applying a suitable unfold/fold transformation strategy that preserves the perfect model. Our strategy is based on the elimination of existential variables, that is, variables which occur in the body of a clause and not in its head. Since, in general, the rst order properties of the class of programs we consider are undecidable, our strategy is necessarily incomplete. However, experiments show that it is powerful enough to prove several nontrivial program properties. 1
Rules + Strategies for Transforming Lazy Functional Logic Programs
 Theoretical Computer Science
, 2004
"... This work introduces a transformation methodology for functional logic programs based on needed narrowing, the optimal and complete operational principle for modern declarative languages which integrate the best features of functional and logic programming. We provide correctness results for the tra ..."
Abstract

Cited by 5 (4 self)
 Add to MetaCart
This work introduces a transformation methodology for functional logic programs based on needed narrowing, the optimal and complete operational principle for modern declarative languages which integrate the best features of functional and logic programming. We provide correctness results for the transformation system w.r.t. the set of computed values and answer substitutions and show that the prominent properties of needed narrowing—namely, the optimality w.r.t. the length of derivations and the number of computed solutions—carry over to the transformation process and the transformed programs. We illustrate the power of the system by taking on in our setting two wellknown transformation strategies (composition and tupling). We also provide an implementation of the transformation system which, by means of some experimental results, highlights the potentiality of our approach.
The Evaluation of FirstOrder Substitution is Monadic SecondOrder Compatible
"... We denote firstorder substitutions of finite and infinite terms by function symbols indexed by the sequences of firstorder variables to which substitutions are made. We consider the evaluation mapping from infinite terms to infinite terms that evaluates these substitution operations. This mapping ..."
Abstract

Cited by 5 (1 self)
 Add to MetaCart
We denote firstorder substitutions of finite and infinite terms by function symbols indexed by the sequences of firstorder variables to which substitutions are made. We consider the evaluation mapping from infinite terms to infinite terms that evaluates these substitution operations. This mapping may perform infinitely many nested substitutions, so that a term which has the structure of an infinite string can be transformed into one isomorphic to an infinite binary tree. We prove that this mapping is Monadic Secondorder compatible which means that, for all finite sets of function symbols and variables, a monadic secondorder formula expressing a property of the output term produced by the evaluation mapping can be translated into a monadic secondorder formula expressing this property over the input term. This implies that, deciding the monadic secondorder theory of the output term reduces to deciding that of the input term. As an application, we obtain another proof that the monadic secondorder properties of the algebraic trees, which represent the behaviours of recursive applicative program schemes, are decidable. This proof extends to hyperalgebraic trees. These infinite trees correspond to certain recursive program schemes with functional parameters of arbitrary high type.
Program Derivation = Rules + Strategies
 Computational Logic: Logic Programming and Beyond (Essays in honour of Bob Kowalski, Part I), Lecture Notes in Computer Science 2407
, 2001
"... In a seminal paper [38] Prof. Robert Kowalski advocated the paradigm Algorithm = Logic + Control which was intended to characterize program executions. Here we want to illustrate the corresponding paradigm Program Derivation = Rules + Strategies which is intended to characterize program derivations, ..."
Abstract

Cited by 4 (2 self)
 Add to MetaCart
In a seminal paper [38] Prof. Robert Kowalski advocated the paradigm Algorithm = Logic + Control which was intended to characterize program executions. Here we want to illustrate the corresponding paradigm Program Derivation = Rules + Strategies which is intended to characterize program derivations, rather than executions. During program execution, the Logic component guarantees that the computed results are correct, that is, they are true facts in the intended model of the given program, while the Control component ensures that those facts are derived in an efficient way. Likewise, during program derivation, the Rules component guarantees that the derived programs are correct and the Strategies component ensures that the derived programs are efficient.