## Information and Computation 207 (2009) 258–283 Contents lists available at ScienceDirect

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@MISC{_informationand,

author = {},

title = {Information and Computation 207 (2009) 258–283 Contents lists available at ScienceDirect},

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}

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### Abstract

Information and Computation journal homepage: www.elsevier.com/locate/ic

### Citations

1461 | An axiomatic basis for computer programming
- Hoare
- 1969
(Show Context)
Citation Context ...08 Elsevier Inc. All rights reserved. 1. Introduction The connection between the use of fixpoints in denotational semantics [24] and the use of rule-based inductive definitions in axiomatic semantics =-=[15]-=- and structural operational semantics (SOS) [28,30,29] can be made by a generalization of inductive definitions [2] to include co-inductive definitions [11]. It is then possible to generalize natural ... |

1353 | A Structural Approach to Operational Semantics
- Plotkin
- 1981
(Show Context)
Citation Context ...ction The connection between the use of fixpoints in denotational semantics [24] and the use of rule-based inductive definitions in axiomatic semantics [15] and structural operational semantics (SOS) =-=[28,30,29]-=- can be made by a generalization of inductive definitions [2] to include co-inductive definitions [11]. It is then possible to generalize natural semantics describing finite input/output behaviors [17... |

663 | Systematic design of program analysis frameworks
- Cousot, Cousot
- 1979
(Show Context)
Citation Context ... domain 〈℘(S∞), ⊑, Sω , S +, ⊔, ⊓〉. 3. Abstraction We consider a simple form of abstraction based on a continuous abstraction function α [9], which includes the particular case of a Galois connection =-=[8]-=- (denoted 〈P, ≼〉 −−→ ←−− γ γ 〈Q, ⊑〉, or〈P, ≼〉 α −−→−→ ←−−− 〈Q, ⊑〉 when α is onto, where 〈P, ≼〉 and 〈Q, ⊑〉 α are posets, and ∀x ∈ P :∀y ∈ Q : α(x) ⊑ y ⇐⇒ x ≼ γ(y)). 6 But not necessarily ⊑-monotone. 7 ... |

246 | Abstract interpretation frameworks
- Cousot, Cousot
- 1992
(Show Context)
Citation Context ... Y � (X � ∪ Y � ) ∪ (Xω ∩ Y ω ). Similarly, for the bi-semantic domain 〈℘(S∞), ⊑, Sω , S +, ⊔, ⊓〉. 3. Abstraction We consider a simple form of abstraction based on a continuous abstraction function α =-=[9]-=-, which includes the particular case of a Galois connection [8] (denoted 〈P, ≼〉 −−→ ←−− γ γ 〈Q, ⊑〉, or〈P, ≼〉 α −−→−→ ←−−− 〈Q, ⊑〉 when α is onto, where 〈P, ≼〉 and 〈Q, ⊑〉 α are posets, and ∀x ∈ P :∀y ∈ ... |

240 | A tutorial on (co)algebras and (co)induction
- Jacobs, Rutten
- 1997
(Show Context)
Citation Context ...er an order-theoretic [6] or metric [35] fixpoint definition or else a categorical definition as a final coalgebra for a behaviour functor (modeling the transition relation) up to a weak bisimulation =-=[16,34,20]-=- or using an equational definition for recursion in an order-enriched category [19]. However, the description of execution traces by small steps may be impractical as compared to a compositional defin... |

142 | Towards a mathematical operational semantics
- Turi, Plotkin
- 1997
(Show Context)
Citation Context ...er an order-theoretic [6] or metric [35] fixpoint definition or else a categorical definition as a final coalgebra for a behaviour functor (modeling the transition relation) up to a weak bisimulation =-=[16,34,20]-=- or using an equational definition for recursion in an order-enriched category [19]. However, the description of execution traces by small steps may be impractical as compared to a compositional defin... |

131 |
Operational and algebraic semantics of concurrent processes
- Milner
- 1990
(Show Context)
Citation Context ...s ∪ and ⃗ S = ⋃ ⃗S[[a]]. a∈T □ Observe that the inductive definition of ⃗ S[[a]] should neither be understood as a structural induction [28] ona (since a[x ← v] ̸≺ (λ x. a) v) nor as action induction =-=[23]-=- (because of infinite traces). The definition could be split in inductive rules for termination and co-inductive rules for divergence, as shown in Theorem 14, but the above bi-inductive definition avo... |

123 |
The algebraic theory of context-free languages
- Chomsky, Schützenberger
- 1963
(Show Context)
Citation Context ... the call-by-value λ-calculus considered in Section 6.3. 5. Structural order-theoretic inductive definitions of the semantics of context-free grammars The Ginsburg-Rice/Chomsky-Schützenberger theorem =-=[4,14,31]-=- shows that the terminal language generated by a contextfree grammar can be expressed in ˙⊆-least fixpoint form. This was extended to the infinite language generated by a context-free grammar by Nivat... |

102 | Constructive design of a hierarchy of semantics of a transition system by abstract interpretation
- Cousot
(Show Context)
Citation Context ... Section 3 recalls a few elements of abstract interpretation, including soundness and completeness. Section 4 is a simple illustration of this approach to give a trace semantics to transition systems =-=[6]-=-. The semantics of context-free grammars in Section 5 combines the classical definitions of the finite and infinite languages generated by a grammar, which can be recovered by simple abstractions. Sec... |

95 |
Inductive definitions, semantics and abstract interpretation, in: 19th symposium Principles of Programming Languages
- Cousot, Cousot
- 1992
(Show Context)
Citation Context ...ve definitions [2] to include co-inductive definitions [11]. It is then possible to generalize natural semantics describing finite input/output behaviors [17] so as to also include infinite behaviors =-=[10]-=-. This is necessary since the definition of the infinite behaviors cannot be derived from the finite big-step SOS behaviors. 1.1. Motivating example Let us consider, for example, the choice operator E... |

80 |
A lattice theoretical fixed point theorem and its applications
- Tarski
- 1955
(Show Context)
Citation Context ... (X) � (F(S + ∪ X− )) − , S − = gfp ⊆ F− . Then S � S + ∪ S − = lfp ⊑ F. Proof. 〈℘(L), ⊆〉 is a complete lattice and F is ⊆-monotone when so are F + and F− proving that lfp ⊆ Tarski’s fixpoint theorem =-=[33]-=-. We first prove that S is a fixpoint of F. S = S + ∪ S − F + and gfp ⊆ F − exist by = F + (S + ) ∪ F − (S − ) by fixpoint definitionsS + � lfp ⊆ F + and S − � gfp ⊆ F − = (F(S + )) + ∪ (F(S + ∪ S − )... |

78 |
The theory and practice of transforming call-by-need into call-by- value
- Mycroft
- 1980
(Show Context)
Citation Context ...finitary rule-base form is (a ⇒ b stands for 〈a, b〉 ∈ � S and r ∈ V ∪ {⊥}) v ⇒ v, v ∈ V a −A b, b ⇒ r a ⇒ r ⊑ 7. Related work Divergence/nonterminating behaviors are needed in static program analysis =-=[25]-=- 10 or typing [5,22]. Such divergence information is part of the classical order-theoretic fixpoint denotational semantics [24] but not explicit in small-step/abstractmachine-based operational semanti... |

68 | Structural operational semantics
- Plotkin
- 1981
(Show Context)
Citation Context ...ction The connection between the use of fixpoints in denotational semantics [24] and the use of rule-based inductive definitions in axiomatic semantics [15] and structural operational semantics (SOS) =-=[28,30,29]-=- can be made by a generalization of inductive definitions [2] to include co-inductive definitions [11]. It is then possible to generalize natural semantics describing finite input/output behaviors [17... |

55 |
TYPOL: a formalism to implement natural semantics
- Despeyroux
- 1988
(Show Context)
Citation Context ...cs [17] will have its diverging behaviors undefined by the formal semantics hence determined by the behavior of the implementation. This is the case with left-toright evaluation Prolog implementation =-=[3,13]-=-, but the problem is general and concerns the class of all implementations that conform to the semantics, regardless of how they were produced. So the natural big-step convergence semantics is an abst... |

49 |
Semantics of interaction: an introduction to game semantics
- Abramsky
- 1996
(Show Context)
Citation Context ...S[[ℓ]] � S f [[ℓ]] = Se[[ℓ]] = Sr[[ℓ]]. If ∀ℓ ∈ L, F f [[ℓ]] is ⊑ℓ-monotone then S[[ℓ]] = Sc[[ℓ]] = Sp[[ℓ]]. This generalization of [2] could also include a game-theoretic version (the game semantics =-=[1]-=- being of quite different nature). The closure-condition version [2] is also easy to adapt. 2.12. Example: inductive definitions The classical inductive definition [2] of a subset S of a universe U by... |

47 | Constructive Versions of Tarskis Fixed Point Theorems
- Cousot, Cousot
- 1979
(Show Context)
Citation Context ...since λ X . i Fℓ (X, ∏ ℓ ′ −≺ℓ Sf [[ℓ ′ ]]) is monotone for all i ∈ �ℓ and �ℓ is monotone by hypothesis. It follows that the least fixpoint lfp ⊑ℓ Ff [[ℓ]] does exist in the dcpo 〈Dℓ, ⊑ℓ〉 as shown by =-=[7]-=- 3 (or [27] without the axiom of choice, see [18,21] for historical perspectives), proving that Sf [[ℓ]] is well defined. □ Definitions without fixpoint or join can nevertheless be encompassed as fixp... |

44 |
Fixed point theorems and semantics: A folk tale
- LASSEZ, NGUYEN, et al.
- 1982
(Show Context)
Citation Context ...onotone for all i ∈ �ℓ and �ℓ is monotone by hypothesis. It follows that the least fixpoint lfp ⊑ℓ Ff [[ℓ]] does exist in the dcpo 〈Dℓ, ⊑ℓ〉 as shown by [7] 3 (or [27] without the axiom of choice, see =-=[18,21]-=- for historical perspectives), proving that Sf [[ℓ]] is well defined. □ Definitions without fixpoint or join can nevertheless be encompassed as fixpoints such as � F ℓ i∈�ℓ i ℓ (Sf [[ℓ1]], ..., Sf [[ℓ... |

42 |
Two families of languages related to ALGOL
- Ginsburg, Rice
- 1962
(Show Context)
Citation Context ... the call-by-value λ-calculus considered in Section 6.3. 5. Structural order-theoretic inductive definitions of the semantics of context-free grammars The Ginsburg-Rice/Chomsky-Schützenberger theorem =-=[4,14,31]-=- shows that the terminal language generated by a contextfree grammar can be expressed in ˙⊆-least fixpoint form. This was extended to the infinite language generated by a context-free grammar by Nivat... |

41 |
Introduction to Lattices and
- Davey, Priestley
- 2002
(Show Context)
Citation Context ...er-theoretic inductive definitions We introduce different forms of structural order-theoretic inductive definitions and prove their equivalence. 2.1. Dcpos and complete lattices Let 〈S, ⊑〉 be a poset =-=[12]-=-. A chain in the poset 〈S, ⊑〉 is a subset of S such that any two elements in the chain are comparable by ⊑. A directed complete partial order (dcpo) is a poset such that any chain has a least upper bo... |

40 | Types as abstract interpretations, invited paper
- Cousot
- 1997
(Show Context)
Citation Context ... form is (a ⇒ b stands for 〈a, b〉 ∈ � S and r ∈ V ∪ {⊥}) v ⇒ v, v ∈ V a −A b, b ⇒ r a ⇒ r ⊑ 7. Related work Divergence/nonterminating behaviors are needed in static program analysis [25] 10 or typing =-=[5,22]-=-. Such divergence information is part of the classical order-theoretic fixpoint denotational semantics [24] but not explicit in small-step/abstractmachine-based operational semantics [28,29,30] and ab... |

35 | Coinductive Big-step Operational Semantics
- Leroy, Grall
(Show Context)
Citation Context ... gfp ˙⊆ S[[S]]. 6. Structural order-theoretic inductive definitions of the semantics of the call-by-value λ-calculus The next example of structural order-theoretic inductive definition is inspired by =-=[29,22]-=-. We introduce a maximal trace semantics describing terminating and diverging computations. The trace semantics is then abstracted into a sound and complete relational semantics. In turn this relation... |

25 | Compositional and inductive semantic definitions in fixpoint, equational, constraint, closure-condition, rule-based and game-theoretic form
- Cousot, Cousot
- 1995
(Show Context)
Citation Context ...nductive definitions in axiomatic semantics [15] and structural operational semantics (SOS) [28,30,29] can be made by a generalization of inductive definitions [2] to include co-inductive definitions =-=[11]-=-. It is then possible to generalize natural semantics describing finite input/output behaviors [17] so as to also include infinite behaviors [10]. This is necessary since the definition of the infinit... |

21 |
On a theorem of R
- Schützenberger
- 1962
(Show Context)
Citation Context ... the call-by-value λ-calculus considered in Section 6.3. 5. Structural order-theoretic inductive definitions of the semantics of context-free grammars The Ginsburg-Rice/Chomsky-Schützenberger theorem =-=[4,14,31]-=- shows that the terminal language generated by a contextfree grammar can be expressed in ˙⊆-least fixpoint form. This was extended to the infinite language generated by a context-free grammar by Nivat... |

12 |
Sur les ensembles de mots infinis engendrés par une grammaire algébrique
- Nivat
- 1978
(Show Context)
Citation Context ...shows that the terminal language generated by a contextfree grammar can be expressed in ˙⊆-least fixpoint form. This was extended to the infinite language generated by a context-free grammar by Nivat =-=[26]-=- using ˙⊆-greatest fixpoints. To illustrate bi-inductive structural definition on a simple example, we define the bifinite semantics of grammars mixing the least fixpoint for finite sentences and the ... |

10 |
Adding recursive constructs to bialgebraic semantics
- Klin
- 2004
(Show Context)
Citation Context ...tion as a final coalgebra for a behaviour functor (modeling the transition relation) up to a weak bisimulation [16,34,20] or using an equational definition for recursion in an order-enriched category =-=[19]-=-. However, the description of execution traces by small steps may be impractical as compared to a compositional definition using big steps. Moreover, execution traces are not always at an appropriate ... |

9 | Incremental evaluation of natural semantics speci cations
- Attali, Chazarain, et al.
- 1992
(Show Context)
Citation Context ...cs [17] will have its diverging behaviors undefined by the formal semantics hence determined by the behavior of the implementation. This is the case with left-toright evaluation Prolog implementation =-=[3,13]-=-, but the problem is general and concerns the class of all implementations that conform to the semantics, regardless of how they were produced. So the natural big-step convergence semantics is an abst... |

7 | The mathematical import of Zermelo’s well-ordering theorem
- Kanamori
- 1997
(Show Context)
Citation Context ...onotone for all i ∈ �ℓ and �ℓ is monotone by hypothesis. It follows that the least fixpoint lfp ⊑ℓ Ff [[ℓ]] does exist in the dcpo 〈Dℓ, ⊑ℓ〉 as shown by [7] 3 (or [27] without the axiom of choice, see =-=[18,21]-=- for historical perspectives), proving that Sf [[ℓ]] is well defined. □ Definitions without fixpoint or join can nevertheless be encompassed as fixpoints such as � F ℓ i∈�ℓ i ℓ (Sf [[ℓ1]], ..., Sf [[ℓ... |

5 | Bialgebraic methods in structural operational semantics
- Klin
- 2007
(Show Context)
Citation Context ...er an order-theoretic [6] or metric [35] fixpoint definition or else a categorical definition as a final coalgebra for a behaviour functor (modeling the transition relation) up to a weak bisimulation =-=[16,34,20]-=- or using an equational definition for recursion in an order-enriched category [19]. However, the description of execution traces by small steps may be impractical as compared to a compositional defin... |

3 |
An introduction to inductive definitions, in: J. Barwise (Ed
- Aczel
- 1977
(Show Context)
Citation Context ...mantics [24] and the use of rule-based inductive definitions in axiomatic semantics [15] and structural operational semantics (SOS) [28,30,29] can be made by a generalization of inductive definitions =-=[2]-=- to include co-inductive definitions [11]. It is then possible to generalize natural semantics describing finite input/output behaviors [17] so as to also include infinite behaviors [10]. This is nece... |

1 |
Leeuwen (Ed.), Formal Models and Semantics
- Mosses, semantics, et al.
- 1990
(Show Context)
Citation Context ...by-value λ-calculus (for which co-induction is shown to be inadequate). © 2008 Elsevier Inc. All rights reserved. 1. Introduction The connection between the use of fixpoints in denotational semantics =-=[24]-=- and the use of rule-based inductive definitions in axiomatic semantics [15] and structural operational semantics (SOS) [28,30,29] can be made by a generalization of inductive definitions [2] to inclu... |

1 |
A constructive proof of Tarski’s fixed-point theorem for dcpo’s, 65th Peripatetic Seminar on Sheaves and Logic
- Pataria
- 1997
(Show Context)
Citation Context .... i Fℓ (X, ∏ ℓ ′ −≺ℓ Sf [[ℓ ′ ]]) is monotone for all i ∈ �ℓ and �ℓ is monotone by hypothesis. It follows that the least fixpoint lfp ⊑ℓ Ff [[ℓ]] does exist in the dcpo 〈Dℓ, ⊑ℓ〉 as shown by [7] 3 (or =-=[27]-=- without the axiom of choice, see [18,21] for historical perspectives), proving that Sf [[ℓ]] is well defined. □ Definitions without fixpoint or join can nevertheless be encompassed as fixpoints such ... |

1 |
Selective thunkification, in: B. Le Charlier, (Ed
- Steckler, Wand
- 1994
(Show Context)
Citation Context ...and at various levels of abstractions for trace/relational/ operational semantics. The lattice of abstractions of the big-step bifinite trace semantics is the following 10 For example, the authors of =-=[32]-=- claim that their “work is the first provably correct strictness analysis and call-by-name to call-by-value transformation for an untyped higher-order language” but since the considered big-step seman... |

1 |
Breugel, An introduction to metric semantics: operational and denotational models for programming and specification languages, Theoretical Computer Science 258
- van
- 2001
(Show Context)
Citation Context ...tics [17]. A standard approach is therefore to generate an execution trace semantics from a (labeled) transition system/small-step operational semantics, using either an order-theoretic [6] or metric =-=[35]-=- fixpoint definition or else a categorical definition as a final coalgebra for a behaviour functor (modeling the transition relation) up to a weak bisimulation [16,34,20] or using an equational defini... |