Page 1. Implementation and Behavioural Equivalence: A Survey Fernando Orejas Dept. Leng. Sist. In... more Page 1. Implementation and Behavioural Equivalence: A Survey Fernando Orejas Dept. Leng. Sist. Inf., Univ. Polit. Catalufia Barcelona, SPAIN Marisa Navarro, Ana Sfinchez Dept. Leng. Sist. Inf., Univ. del Pals Vasco San Sebastian, SPAIN ...
Horn ⊃ is a logic programming language which extends usual Horn clauses by adding intuitionistic ... more Horn ⊃ is a logic programming language which extends usual Horn clauses by adding intuitionistic implication in goals and clause bodies. This extension can be seen as a form of structuring programs in logic programming. We are interested in finding correct and efficient translations from Horn ⊃ programs into some representation type that, preserving the signature, allow us suitable implementations of these kind of programs. In this paper we restrict to the propositional setting of Horn ⊃ and we study correct translations into Boolean circuits, i.e. graphs; into Boolean formulas, i.e. trees; and into conjunctions of propositional Horn clauses. Different results about the efficiency of the transformations are obtained in the three cases.
In this paper we try to shed some light on the similarities and differences in the different appr... more In this paper we try to shed some light on the similarities and differences in the different approaches denning the notions of implementation and implementation correctness. For obvious reasons, we do not discuss all existing approaches individually. Instead, a formal framework is introduced in order to discuss the most important ones. Additionally, we discuss some issues, which in our opinion are often misunderstood, concerning transitivity of implementation correctness and its role in the software development process. In particular, on the one hand, we show that for reasonable notions of implementation, it is almost impossible to prove transitivity of implementation correctness at the specification level. On the other hand, we show that this is not really important if the programming language satisfies the properties of horizontal and vertical composition.
In this paper we provide foundations to contextual rewriting. We present the class of LOG-specifi... more In this paper we provide foundations to contextual rewriting. We present the class of LOG-specifications, which may be seen, essentially, as a variety of conditional specifications with predicates. Semantically, the models of LOG-specifications are LOG-algebras, i.e. algebras satisfying that their boolean part coincides with the boolean algebra of two elements. With respect to this semantics, a proof-system called L is presented characterizing logical consequence proof-theoretically. Then, we show that, under adequate assumptions of confluence and finite termination, contextual rewriting is a complete method for this kind of specifications.
A strong (L) logic programming language ([14, 15]) is given by two subclasses of formulas (progra... more A strong (L) logic programming language ([14, 15]) is given by two subclasses of formulas (programs and goals) of the underlying logic L, provided that: firstly, any program P (viewed as a L-theory) has a canonical model MP which is initial in the category of all its L-models; secondly, the L-satisfaction of a goal G in MP is equivalent to the L-derivability of G from P , and finally, there exists an effective (computable) proof-subcalculus of the L-calculus which works out for derivation of goals from programs. In this sense, Horn clauses constitute a strong (first-order) logic programming language. Following the methodology suggested in [15] for designing logic programming languages, an extension of Horn clauses should be made by extending its underlying first-order logic to a richer logic which supports a strong axiomatization of the extended logic programming language. A well-known approach for extending Horn clauses with embedded implications is the static scope programming language presented in [8]. In this paper we show that such language can be seen as a strong FO ⊃ logic programming language, where FO ⊃ is a very natural extension of first-order logic with intuitionistic implication. That is, we present a new characterization of the language in [8] which shows that Horn clauses extended with embedded implications, viewed as FO ⊃-theories, preserves all the attractive mathematical and computational properties that Horn clauses satisfy as first-order-theories.
Resolution is a well-known proof method for classical logics that is well suited for mechanizatio... more Resolution is a well-known proof method for classical logics that is well suited for mechanization. The most fruitful approach in the literature on temporal logic, which was started with the seminal paper of M. Fisher, deals with Propositional Linear-time Temporal Logic (PLTL) and requires to generate invariants for performing resolution on eventualities. The methods and techniques developed in that approach have also been successfully adapted in order to obtain a clausal resolution method for Computation Tree Logic (CTL), but invariant handling seems to be a handicap for further extension to more general branching temporal logics. In this paper, we present a new approach to applying resolution to PLTL. The main novelty of our approach is that we do not generate invariants for performing resolution on eventualities. Hence, we say that the approach presented in this paper is invariantfree. Our method is based on the dual methods of tableaux and sequents for PLTL that we presented in a previous paper. Our resolution method involves translation into a clausal normal form that is a direct extension of classical CNF. We first show that any PLTL-formula can be transformed into this clausal normal form. Then, we present our temporal resolution method, called TRS-resolution, that extends classical propositional resolution. Finally, we This work has been partially supported by the Spanish Project TIN2007-66523 and the Basque Project LoRea GIU07/35.
We present a prototype that implements a set of logical rules to prove the satisfiability for a c... more We present a prototype that implements a set of logical rules to prove the satisfiability for a class of specifications on XML documents. Specifications are given by means of constrains built on Boolean XPath patterns. The main goal of this tool is to test whether a given specification is satisfiable or not, and justify the decision showing the execution history. It can also be used to test whether a given document is a model of a given specification and, as a by-product, it permits to look for all the relations (monomorphisms) between two patterns and to combine patterns in different ways. The results of these operations are visually shown and therefore the tool makes these operations more understandable. The implementation of the algorithm has been written in Prolog but the prototype has a Java interface for an easy and friendly use. In this paper we show how to use this interface in order to test all the desired properties.
In this paper we first present three sorts of constraints (positive, negative and conditional one... more In this paper we first present three sorts of constraints (positive, negative and conditional ones) to specify that certain patterns must be satisfied in a XML document. These constraints are built on boolean XPath patterns. We define a specification as a set of clauses whose literals are constraints.Then, to reason about these specifications, we study some sound rules which permit to infer, subsume or simplify clauses. The main goal is to design a refutation procedure (based on these rules) to test if a given specification is satisfiable or not. We have formally proven the soundness of our procedure and we study the completeness and termination of the proposed method.
One of the main advantages of the Logic of Nested Conditions, defined by Habel and Pennemann, for... more One of the main advantages of the Logic of Nested Conditions, defined by Habel and Pennemann, for reasoning about graphs, is its generality: this logic can be used in the framework of many classes of graphs and graphical structures. It is enough that the category of these structures satisfies certain basic conditions. In a previous paper [14], we extended this logic to be able to deal with graph properties including paths, but this extension was only defined for the category of untyped directed graphs. In addition it seemed difficult to talk about paths abstractly, that is, independently of the given category of graphical structures. In this paper we approach this problem. In particular, given an arbitrary category of graphical structures, we assume that for every object of this category there is an associated edge relation that can be used to define a path relation. Moreover, we consider that edges have some kind of labels and paths can be specified by associating them to a set of label sequences. Then, after the presentation of that general framework, we show how it can be applied to several classes of graphs. Moreover, we present a set of sound inference rules for reasoning in the logic.
In the context of software model-driven development, artifacts are specified by several models de... more In the context of software model-driven development, artifacts are specified by several models describing different aspects, e.g., different views, dynamic behavior, structure, distributed information, etc. Then, maintaining and repairing consistency of the whole specification are crucial issues if the models can be separately developed and updated. Model Synchronization is the process of restoring consistency after the update of one or several of the models. In the present work, we approach the case when conflicts may arise due to concurrently updating different models. Specifically, based on the Triple Graph Grammar approach, we propose an incremental algorithm CSynch for solving conflicts and repairing consistency. In addition, we identify and formalize when a synchronizing solution can be considered adequate and show that our procedure CSynch is sound and complete.
One of the main advantages of the Logic of Nested Conditions, defined by Habel and Pennemann, for... more One of the main advantages of the Logic of Nested Conditions, defined by Habel and Pennemann, for reasoning about graphs, is its generality: this logic can be used in the framework of many classes of graphs and graphical structures. It is enough that the category of these structures satisfies certain basic conditions. In a previous paper [14], we extended this logic to be able to deal with graph properties including paths, but this extension was only defined for the category of untyped directed graphs. In addition it seemed difficult to talk about paths abstractly, that is, independently of the given category of graphical structures. In this paper we approach this problem. In particular, given an arbitrary category of graphical structures, we assume that for every object of this category there is an associated edge relation that can be used to define a path relation. Moreover, we consider that edges have some kind of labels and paths can be specified by associating them to a set of label sequences. Then, after the presentation of that general framework, we show how it can be applied to several classes of graphs. Moreover, we present a set of sound inference rules for reasoning in the logic.
In this paper we first present three sorts of constraints (positive, negative and conditional one... more In this paper we first present three sorts of constraints (positive, negative and conditional ones) to specify that certain patterns must be satisfied in an XML document. These constraints are built on boolean XPath patterns. We define a specification as a set of clauses whose literals are constraints. Then, to reason about these specifications, we study some sound rules which permit to infer, subsume or simplify clauses. The main goal is to design a refutation procedure (based on these rules) to test if a given specification is satisfiable or not. We have formally proven the soundness of our procedure and we study the completeness and termination of the proposed method.
Jose Meseguer is one of the earliest contributors in the area of Algebraic Specification. In this... more Jose Meseguer is one of the earliest contributors in the area of Algebraic Specification. In this paper, which we are happy to dedicate to him on the occasion of his 65th birthday, we use ideas and methods coming from that area with the aim of presenting an approach for the specification of the structure of classes of XML documents and for reasoning about them. More precisely, we specify the structure of documents using sets of constraints that are based on XPath and we present inference rules that are shown to define a sound and complete refutation procedure for checking satisfiability of a given specification using tableaux.
We show that a Navigational Logic, i.e., a logic to express properties about graphs and about pat... more We show that a Navigational Logic, i.e., a logic to express properties about graphs and about paths in graphs is a semi-exact institution. In this way, we can use a number of operations to structure and modularize our specifications. Moreover, using the properties of our institution, we also show how to structure single formulas, which in our formalism could be quite complex.
Page 1. Classifying the Problem of Learning Extended Horn Clauses∗ Joxe Gaintzarain Montserrat He... more Page 1. Classifying the Problem of Learning Extended Horn Clauses∗ Joxe Gaintzarain Montserrat Hermo Marisa Navarro EUITI de Bilbao Facultad de Informática Facultad de Informática Universidad del Paıs Vasco Universidad del Paıs Vasco Universidad del Paıs Vasco ...
There are mainly two approaches for structuring logic programs. The first one is based on definin... more There are mainly two approaches for structuring logic programs. The first one is based on defining some notion of program unit or module and on providing a number of composition operators. The second approach consists in enriching logic programming with a mechanism of abstraction and scoping rules that are frequently found, for instance, in procedural programming. More precisely, this approach has been advocated by Miller and others using implications embedded in the goals of the given program as a structuring mechanism. However, as Giordano, Martelli and Rossi pointed out, we can associate two different visibility rules (static and dynamic) to this kind of structuring mechanism where, obviously, the semantics of the given program depends on the chosen rule. In this paper we consider normal constraint logic programs (with constructive negation á la Drabent as operational semantics) extended with embedded implications with a static visibility rule. This class of programs combines the expressive power of normal programs with the capability to organize and to enhance dinamically their sets of clauses. In particular, first, we introduce an operational semantics based on constructive negation for this class of programs, taking into account the static visibility rule. Then, we present an alternative semantics in terms of a transformation of the given structured program into a flat one. Finally, we prove the adequacy of this transformation by showing that it preserves the computed answers of the given program. Obviously, this transformation semantics can be used as the basis for an implementation of this structuring mechanism.
... cation Goals. Theoretical Computer Science 95: 4374,(1992). 10. Kearns, M. and Valiant, L. C... more ... cation Goals. Theoretical Computer Science 95: 4374,(1992). 10. Kearns, M. and Valiant, L. Criptographic Limitations on Learning Boolean For-mulae and Finite Automata. Journal of the ACM 41 (1): 6795,(1994). 11. Lucio ...
Page 1. Implementation and Behavioural Equivalence: A Survey Fernando Orejas Dept. Leng. Sist. In... more Page 1. Implementation and Behavioural Equivalence: A Survey Fernando Orejas Dept. Leng. Sist. Inf., Univ. Polit. Catalufia Barcelona, SPAIN Marisa Navarro, Ana Sfinchez Dept. Leng. Sist. Inf., Univ. del Pals Vasco San Sebastian, SPAIN ...
Horn ⊃ is a logic programming language which extends usual Horn clauses by adding intuitionistic ... more Horn ⊃ is a logic programming language which extends usual Horn clauses by adding intuitionistic implication in goals and clause bodies. This extension can be seen as a form of structuring programs in logic programming. We are interested in finding correct and efficient translations from Horn ⊃ programs into some representation type that, preserving the signature, allow us suitable implementations of these kind of programs. In this paper we restrict to the propositional setting of Horn ⊃ and we study correct translations into Boolean circuits, i.e. graphs; into Boolean formulas, i.e. trees; and into conjunctions of propositional Horn clauses. Different results about the efficiency of the transformations are obtained in the three cases.
In this paper we try to shed some light on the similarities and differences in the different appr... more In this paper we try to shed some light on the similarities and differences in the different approaches denning the notions of implementation and implementation correctness. For obvious reasons, we do not discuss all existing approaches individually. Instead, a formal framework is introduced in order to discuss the most important ones. Additionally, we discuss some issues, which in our opinion are often misunderstood, concerning transitivity of implementation correctness and its role in the software development process. In particular, on the one hand, we show that for reasonable notions of implementation, it is almost impossible to prove transitivity of implementation correctness at the specification level. On the other hand, we show that this is not really important if the programming language satisfies the properties of horizontal and vertical composition.
In this paper we provide foundations to contextual rewriting. We present the class of LOG-specifi... more In this paper we provide foundations to contextual rewriting. We present the class of LOG-specifications, which may be seen, essentially, as a variety of conditional specifications with predicates. Semantically, the models of LOG-specifications are LOG-algebras, i.e. algebras satisfying that their boolean part coincides with the boolean algebra of two elements. With respect to this semantics, a proof-system called L is presented characterizing logical consequence proof-theoretically. Then, we show that, under adequate assumptions of confluence and finite termination, contextual rewriting is a complete method for this kind of specifications.
A strong (L) logic programming language ([14, 15]) is given by two subclasses of formulas (progra... more A strong (L) logic programming language ([14, 15]) is given by two subclasses of formulas (programs and goals) of the underlying logic L, provided that: firstly, any program P (viewed as a L-theory) has a canonical model MP which is initial in the category of all its L-models; secondly, the L-satisfaction of a goal G in MP is equivalent to the L-derivability of G from P , and finally, there exists an effective (computable) proof-subcalculus of the L-calculus which works out for derivation of goals from programs. In this sense, Horn clauses constitute a strong (first-order) logic programming language. Following the methodology suggested in [15] for designing logic programming languages, an extension of Horn clauses should be made by extending its underlying first-order logic to a richer logic which supports a strong axiomatization of the extended logic programming language. A well-known approach for extending Horn clauses with embedded implications is the static scope programming language presented in [8]. In this paper we show that such language can be seen as a strong FO ⊃ logic programming language, where FO ⊃ is a very natural extension of first-order logic with intuitionistic implication. That is, we present a new characterization of the language in [8] which shows that Horn clauses extended with embedded implications, viewed as FO ⊃-theories, preserves all the attractive mathematical and computational properties that Horn clauses satisfy as first-order-theories.
Resolution is a well-known proof method for classical logics that is well suited for mechanizatio... more Resolution is a well-known proof method for classical logics that is well suited for mechanization. The most fruitful approach in the literature on temporal logic, which was started with the seminal paper of M. Fisher, deals with Propositional Linear-time Temporal Logic (PLTL) and requires to generate invariants for performing resolution on eventualities. The methods and techniques developed in that approach have also been successfully adapted in order to obtain a clausal resolution method for Computation Tree Logic (CTL), but invariant handling seems to be a handicap for further extension to more general branching temporal logics. In this paper, we present a new approach to applying resolution to PLTL. The main novelty of our approach is that we do not generate invariants for performing resolution on eventualities. Hence, we say that the approach presented in this paper is invariantfree. Our method is based on the dual methods of tableaux and sequents for PLTL that we presented in a previous paper. Our resolution method involves translation into a clausal normal form that is a direct extension of classical CNF. We first show that any PLTL-formula can be transformed into this clausal normal form. Then, we present our temporal resolution method, called TRS-resolution, that extends classical propositional resolution. Finally, we This work has been partially supported by the Spanish Project TIN2007-66523 and the Basque Project LoRea GIU07/35.
We present a prototype that implements a set of logical rules to prove the satisfiability for a c... more We present a prototype that implements a set of logical rules to prove the satisfiability for a class of specifications on XML documents. Specifications are given by means of constrains built on Boolean XPath patterns. The main goal of this tool is to test whether a given specification is satisfiable or not, and justify the decision showing the execution history. It can also be used to test whether a given document is a model of a given specification and, as a by-product, it permits to look for all the relations (monomorphisms) between two patterns and to combine patterns in different ways. The results of these operations are visually shown and therefore the tool makes these operations more understandable. The implementation of the algorithm has been written in Prolog but the prototype has a Java interface for an easy and friendly use. In this paper we show how to use this interface in order to test all the desired properties.
In this paper we first present three sorts of constraints (positive, negative and conditional one... more In this paper we first present three sorts of constraints (positive, negative and conditional ones) to specify that certain patterns must be satisfied in a XML document. These constraints are built on boolean XPath patterns. We define a specification as a set of clauses whose literals are constraints.Then, to reason about these specifications, we study some sound rules which permit to infer, subsume or simplify clauses. The main goal is to design a refutation procedure (based on these rules) to test if a given specification is satisfiable or not. We have formally proven the soundness of our procedure and we study the completeness and termination of the proposed method.
One of the main advantages of the Logic of Nested Conditions, defined by Habel and Pennemann, for... more One of the main advantages of the Logic of Nested Conditions, defined by Habel and Pennemann, for reasoning about graphs, is its generality: this logic can be used in the framework of many classes of graphs and graphical structures. It is enough that the category of these structures satisfies certain basic conditions. In a previous paper [14], we extended this logic to be able to deal with graph properties including paths, but this extension was only defined for the category of untyped directed graphs. In addition it seemed difficult to talk about paths abstractly, that is, independently of the given category of graphical structures. In this paper we approach this problem. In particular, given an arbitrary category of graphical structures, we assume that for every object of this category there is an associated edge relation that can be used to define a path relation. Moreover, we consider that edges have some kind of labels and paths can be specified by associating them to a set of label sequences. Then, after the presentation of that general framework, we show how it can be applied to several classes of graphs. Moreover, we present a set of sound inference rules for reasoning in the logic.
In the context of software model-driven development, artifacts are specified by several models de... more In the context of software model-driven development, artifacts are specified by several models describing different aspects, e.g., different views, dynamic behavior, structure, distributed information, etc. Then, maintaining and repairing consistency of the whole specification are crucial issues if the models can be separately developed and updated. Model Synchronization is the process of restoring consistency after the update of one or several of the models. In the present work, we approach the case when conflicts may arise due to concurrently updating different models. Specifically, based on the Triple Graph Grammar approach, we propose an incremental algorithm CSynch for solving conflicts and repairing consistency. In addition, we identify and formalize when a synchronizing solution can be considered adequate and show that our procedure CSynch is sound and complete.
One of the main advantages of the Logic of Nested Conditions, defined by Habel and Pennemann, for... more One of the main advantages of the Logic of Nested Conditions, defined by Habel and Pennemann, for reasoning about graphs, is its generality: this logic can be used in the framework of many classes of graphs and graphical structures. It is enough that the category of these structures satisfies certain basic conditions. In a previous paper [14], we extended this logic to be able to deal with graph properties including paths, but this extension was only defined for the category of untyped directed graphs. In addition it seemed difficult to talk about paths abstractly, that is, independently of the given category of graphical structures. In this paper we approach this problem. In particular, given an arbitrary category of graphical structures, we assume that for every object of this category there is an associated edge relation that can be used to define a path relation. Moreover, we consider that edges have some kind of labels and paths can be specified by associating them to a set of label sequences. Then, after the presentation of that general framework, we show how it can be applied to several classes of graphs. Moreover, we present a set of sound inference rules for reasoning in the logic.
In this paper we first present three sorts of constraints (positive, negative and conditional one... more In this paper we first present three sorts of constraints (positive, negative and conditional ones) to specify that certain patterns must be satisfied in an XML document. These constraints are built on boolean XPath patterns. We define a specification as a set of clauses whose literals are constraints. Then, to reason about these specifications, we study some sound rules which permit to infer, subsume or simplify clauses. The main goal is to design a refutation procedure (based on these rules) to test if a given specification is satisfiable or not. We have formally proven the soundness of our procedure and we study the completeness and termination of the proposed method.
Jose Meseguer is one of the earliest contributors in the area of Algebraic Specification. In this... more Jose Meseguer is one of the earliest contributors in the area of Algebraic Specification. In this paper, which we are happy to dedicate to him on the occasion of his 65th birthday, we use ideas and methods coming from that area with the aim of presenting an approach for the specification of the structure of classes of XML documents and for reasoning about them. More precisely, we specify the structure of documents using sets of constraints that are based on XPath and we present inference rules that are shown to define a sound and complete refutation procedure for checking satisfiability of a given specification using tableaux.
We show that a Navigational Logic, i.e., a logic to express properties about graphs and about pat... more We show that a Navigational Logic, i.e., a logic to express properties about graphs and about paths in graphs is a semi-exact institution. In this way, we can use a number of operations to structure and modularize our specifications. Moreover, using the properties of our institution, we also show how to structure single formulas, which in our formalism could be quite complex.
Page 1. Classifying the Problem of Learning Extended Horn Clauses∗ Joxe Gaintzarain Montserrat He... more Page 1. Classifying the Problem of Learning Extended Horn Clauses∗ Joxe Gaintzarain Montserrat Hermo Marisa Navarro EUITI de Bilbao Facultad de Informática Facultad de Informática Universidad del Paıs Vasco Universidad del Paıs Vasco Universidad del Paıs Vasco ...
There are mainly two approaches for structuring logic programs. The first one is based on definin... more There are mainly two approaches for structuring logic programs. The first one is based on defining some notion of program unit or module and on providing a number of composition operators. The second approach consists in enriching logic programming with a mechanism of abstraction and scoping rules that are frequently found, for instance, in procedural programming. More precisely, this approach has been advocated by Miller and others using implications embedded in the goals of the given program as a structuring mechanism. However, as Giordano, Martelli and Rossi pointed out, we can associate two different visibility rules (static and dynamic) to this kind of structuring mechanism where, obviously, the semantics of the given program depends on the chosen rule. In this paper we consider normal constraint logic programs (with constructive negation á la Drabent as operational semantics) extended with embedded implications with a static visibility rule. This class of programs combines the expressive power of normal programs with the capability to organize and to enhance dinamically their sets of clauses. In particular, first, we introduce an operational semantics based on constructive negation for this class of programs, taking into account the static visibility rule. Then, we present an alternative semantics in terms of a transformation of the given structured program into a flat one. Finally, we prove the adequacy of this transformation by showing that it preserves the computed answers of the given program. Obviously, this transformation semantics can be used as the basis for an implementation of this structuring mechanism.
... cation Goals. Theoretical Computer Science 95: 4374,(1992). 10. Kearns, M. and Valiant, L. C... more ... cation Goals. Theoretical Computer Science 95: 4374,(1992). 10. Kearns, M. and Valiant, L. Criptographic Limitations on Learning Boolean For-mulae and Finite Automata. Journal of the ACM 41 (1): 6795,(1994). 11. Lucio ...
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