## Collection analysis for Horn clause programs (2006)

Venue: | In Proceedings of PPDP 2006: 8th International ACM SIGPLAN Conference on Principles and Practice of Declarative Programming |

Citations: | 3 - 1 self |

### BibTeX

@INPROCEEDINGS{Miller06collectionanalysis,

author = {Dale Miller},

title = {Collection analysis for Horn clause programs},

booktitle = {In Proceedings of PPDP 2006: 8th International ACM SIGPLAN Conference on Principles and Practice of Declarative Programming},

year = {2006},

pages = {179--188}

}

### OpenURL

### Abstract

dale.miller [at] inria.fr

### Citations

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Abstract Interpretation: A Unified Lattice Model for Static Analysis of Programs By Construction or Approximation of Fixpoints
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Citation Context ...construct the mapping btree ↦→ λx.o emp ↦→ ⊥ bt ↦→ λxλyλz.itemA(x) . x . y that can, for example, approximate a binary tree with the multiset of the labels for internal nodes. Abstract interpretation =-=[8]-=- can associate to a program an approximation to its semantics. Such approximations can help to determine various kinds of properties of programs. It will be interesting to see how well the particular ... |

847 |
A formulation of the simple theory of types
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Citation Context ...g for (eigen)variables and non-logical constants within a sequent. Simple types are, formally speaking, also a simple class of untyped λ-terms: the type o is used to denote formulas (following Church =-=[7]-=-). In a sequent calculus proof, simple type expressions are global and admit no bindings. As a result, it is an easy matter to show that if one takes a proof with a type constant σ and replaces everyw... |

374 | Uniform Proofs as a Foundation for Logic Programming
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Citation Context ...be related simply in a setting where substitution into both has well understood properties is certainly one of the strengths of the proof theoretic foundations of logic programming (see, for example, =-=[22]-=-). 2.3 What good are atomic formulas? In proof theory, there is interesting problem of duality involving atomic formulas. The initial rule and the cut rule given as C − C Initial Γ1 − C,∆1 Γ2, C − ∆2 ... |

352 |
Proofs and Types
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Citation Context ...re is interesting problem of duality involving atomic formulas. The initial rule and the cut rule given as C − C Initial Γ1 − C,∆1 Γ2, C − ∆2 Cut Γ1,Γ2 − ∆1,∆2 can be seen as being dual to each other =-=[13]-=-. In particular, the initial rule states that an occurrence of a formula on the left is stronger than the same occurrence on the right, whereas the cut rule states the dual: an occurrence of a formula... |

331 | Logic programming with focusing proofs in linear logic
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Citation Context ...mal form theorem for the fragment of linear logic for which we are interested. The key result of linear logic surrounding the search for cut-free proofs is given by the completeness of focused proofs =-=[3]-=-. Focused proofs are a normal form that significantly generalizes standard completeness results in logic programming, including the completeness of SLD-resolution and uniform proofs as well as various... |

303 | D.: Logic programming in a fragment of intuitionistic linear logic
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Citation Context ...logic formula. 2.5 Linear logic underlies computational logic Linear logic [10] is able to explain the proof theory of usual Horn clause logic programming (and even richer logic programming languages =-=[15]-=-). It is also able to provide means to reason about resources, such as items in multisets and sets. Thus, linear logic will allow us to sit within one declarative framework to describe both usual logi... |

215 | Logical frameworks
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Citation Context ...oofs search using this style encoding places multiset on the left of the sequent arrow. This approach is favored when an intuitionistic subset of linear logic is used, such as in Lolli [15], LinearLF =-=[6]-=-, and MSR [5]. The dual encoding, the disjunctive encoding, uses ⊥ for the empty multiset and . to combine two multisets. Proofs search using this style encoding places multisets on the right of the s... |

207 | Linear Objects: logical processes with built-in inheritance
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Citation Context ... multiset and . to combine two multisets. Proofs search using this style encoding places multisets on the right of the sequent arrow. Multiple conclusion sequents are now required. Systems such as LO =-=[2]-=- and Forum [19] use this style of encoding. If negation is available, then the choice of which encoding one chooses is mostly a matter of style. We pick the disjunctive encoding for the rather shallow... |

143 | A meta-notation for protocol analysis
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Citation Context ...sing this style encoding places multiset on the left of the sequent arrow. This approach is favored when an intuitionistic subset of linear logic is used, such as in Lolli [15], LinearLF [6], and MSR =-=[5]-=-. The dual encoding, the disjunctive encoding, uses ⊥ for the empty multiset and . to combine two multisets. Proofs search using this style encoding places multisets on the right of the sequent arrow.... |

101 | The π-calculus as a theory in linear logic: Preliminary results - Miller - 1993 |

99 | An Overview of λProlog
- Nadathur, Miller
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Citation Context ...rification proof search, static analysis, Horn clauses, linear logic Static analysis of logic programs can provide useful information for programmers and compilers. Typing systems, such as in λProlog =-=[23, 24]-=-, have proved valuable during the development of code: type errors often represent program errors that are caught at compile time when they are easier to find and fix than at runtime when they are muc... |

89 | Decidability issues for Petri nets – a survey
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Citation Context ...e observation, it follows that determining if S0 −◦ T0 is provable from a set of multiset rewriting clauses is decidable, since this problem is contained within the reachability problem of Petri Nets =-=[9]-=-. Proving a multiset inclusion judgment ∃q(S0 . q −◦ T0) involves first instantiating this higher-order quantifier. In principle, this instantiation can be delayed until attempting to apply the sole i... |

69 | Set Constraints: Results, Applications, and Future Directions
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Citation Context ... In the case of determining the validity of a set statement, the use of linear logic here appears to be rather weak when compared to the large body of results for solving set-based constraint systems =-=[1, 25]-=-. 9. Automation of deduction We describe how automation of proof for the linear logic translations of set and multiset statements given in Propositions 1 and 2 can be performed. In order to understand... |

61 | Cut-elimination for a logic with definitions and induction
- McDowell, Miller
- 2000
(Show Context)
Citation Context ... couple of recent proof theoretic approaches that try to eliminate them entirely. For example, in the work on definitions and fixed points by SchroederHeister [26], Girard [11], and McDowell & Miller =-=[17]-=-, atoms are defined to be other formulas. In this approach, the only primitive judgment involving terms is that of equality. In that setting, if definitions are stratified (no recursion through negati... |

56 | Rules of definitional reflection
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(Show Context)
Citation Context ...nce of atomic formulas, there have been a couple of recent proof theoretic approaches that try to eliminate them entirely. For example, in the work on definitions and fixed points by SchroederHeister =-=[26]-=-, Girard [11], and McDowell & Miller [17], atoms are defined to be other formulas. In this approach, the only primitive judgment involving terms is that of equality. In that setting, if definitions ar... |

47 | Asynchronous Communication Model based on Linear Logic
- Kobayashi, Yonezawa
- 1995
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Citation Context ...ut: this ordering is faithfully captured by proof search in linear logic. Such an encoding of asynchronous process calculi into linear logic has been explored in a number of papers: see, for example, =-=[16, 21]-=-. The example above suggests that lists and list equality can be captured directly in linear logic using the following encoding: nil ↦→ λl.⊥ cons ↦→ λxλRλl. item x ◦− (l ◦− (R l)) The encoding of the ... |

45 |
Integrated Program Debugging, Verification, and Optimization Using Abstract Interpretation (and The
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(Show Context)
Citation Context ... to know that a recursive program that she has just written has linear or quadratic runtime complexity, or that a relation she just specified actually defines a function. The Ciao system preprocessor =-=[14]-=- provides for such functionality by allowing a programmer to write various properties about code that the preprocessor attempts to verify.Having an open set of properties and analysis tools is an int... |

40 |
A fixpoint theorem in linear logic. An email posting to the mailing list linear@cs.stanford.edu
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(Show Context)
Citation Context ... formulas, there have been a couple of recent proof theoretic approaches that try to eliminate them entirely. For example, in the work on definitions and fixed points by SchroederHeister [26], Girard =-=[11]-=-, and McDowell & Miller [17], atoms are defined to be other formulas. In this approach, the only primitive judgment involving terms is that of equality. In that setting, if definitions are stratified ... |

27 | The type system of a higher-order logic programming language
- Nadathur, Pfenning
- 1992
(Show Context)
Citation Context ...rification proof search, static analysis, Horn clauses, linear logic Static analysis of logic programs can provide useful information for programmers and compilers. Typing systems, such as in λProlog =-=[23, 24]-=-, have proved valuable during the development of code: type errors often represent program errors that are caught at compile time when they are easier to find and fix than at runtime when they are muc... |

27 | Set constraints: a pearl in research on constraints
- Pacholski, Podelski
(Show Context)
Citation Context ... In the case of determining the validity of a set statement, the use of linear logic here appears to be rather weak when compared to the large body of results for solving set-based constraint systems =-=[1, 25]-=-. 9. Automation of deduction We describe how automation of proof for the linear logic translations of set and multiset statements given in Propositions 1 and 2 can be performed. In order to understand... |

20 |
Forum: A multiple-conclusion specification language
- Miller
- 1996
(Show Context)
Citation Context .... to combine two multisets. Proofs search using this style encoding places multisets on the right of the sequent arrow. Multiple conclusion sequents are now required. Systems such as LO [2] and Forum =-=[19]-=- use this style of encoding. If negation is available, then the choice of which encoding one chooses is mostly a matter of style. We pick the disjunctive encoding for the rather shallow reason that th... |

18 | Naive reverse can be linear
- Brisset, Ridoux
- 1991
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Citation Context ...odes the concatenation of two lists. Given concatenation, it is then easy to encode the judgments of prefix and suffix. To see other example of computing on difference lists described in fashion, see =-=[4]-=-. Lists can be encoded using the difference list notion with the following mapping into linear logic formulas. nil ↦→ λLλl. L l cons ↦→ λxλRλLλl. item x ◦− (l ◦− (R L l)) The encoding of the list, say... |

7 | Higher-order quantification and proof search
- Miller
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Citation Context ...me judgment about the data structures encoded in the terms t1, . . . , tm. A few simple examples of using higher-order instantiations of logic programs in order to help reasoning about them appear in =-=[20]-=-. 5. Encoding multisets as formulas We wish to encode multisets and sets and simple judgments about them (such as inclusion and equality) as linear logic formulas. We consider multisets first. Let tok... |

7 | Encryption as an abstract data-type: An extended abstract
- Miller
- 2003
(Show Context)
Citation Context ...ut: this ordering is faithfully captured by proof search in linear logic. Such an encoding of asynchronous process calculi into linear logic has been explored in a number of papers: see, for example, =-=[16, 21]-=-. The example above suggests that lists and list equality can be captured directly in linear logic using the following encoding: nil ↦→ λl.⊥ cons ↦→ λxλRλl. item x ◦− (l ◦− (R l)) The encoding of the ... |