## Higher-Order Quantification and Proof Search (2002)

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Venue: | In Proceedings of the AMAST confrerence, LNCS |

Citations: | 7 - 4 self |

### BibTeX

@INPROCEEDINGS{Miller02higher-orderquantification,

author = {Dale Miller},

title = {Higher-Order Quantification and Proof Search},

booktitle = {In Proceedings of the AMAST confrerence, LNCS},

year = {2002},

pages = {60--74},

publisher = {Springer}

}

### Years of Citing Articles

### OpenURL

### Abstract

Logical equivalence between logic programs that are firstorder logic formulas holds between few logic programs, partly because first-order logic does not allow auxiliary programs and data structures to be hidden. As a result of not having such abstractions, logical equivalence will force these auxiliaries to be present in any equivalence program.

### Citations

995 | D.: A calculus of mobile processes, i
- Milner, Parrow, et al.
- 1992
(Show Context)
Citation Context ...ched for), keys may extrude their scope and move freely around the network and into the memory of possible intruders. This dynamic notion of scope extrusion is similar to that found in the π-calculus=-= [MPW92] -=-and is modeled here in linear logic in a way similar to an encoding given in [Mil93] for an encoding of the π-calculus into linear logic. Example 2. As an example to illustrate the possible power of ... |

851 |
A formulation of the simple theory of types
- Church
- 1940
(Show Context)
Citation Context ...he sake of being concrete, we shall assume that the logics considered here are based on a simple type discipline, and that the type of logical formulas is o (following Church’s Simple Theory of Type=-=s [Chu40]). Given -=-this typing, if a quantifier binds a variable of type τ1 → . . . τn → τ0, where τ0 is a primitive type, then that quantifier binds a variable at predicate type if τ0 is o and binds a variable... |

375 | Uniform proofs as a foundation for logic programming
- Miller, Nadathur, et al.
- 1991
(Show Context)
Citation Context ...te that they prove the same goals: for example, if P1 ⊢ G and P2 ⊢ P1 then P2 ⊢ G. If provability of a goal from a program is described via cut-free proofs, as is often the case for logic progra=-=mming [MNPS91], -=-then cut-elimination for the underlying logic is needed to support this most basic of inferences. ⋆ This work was supported in part by NSF grants CCR-9912387, CCR-9803971, INT9815645, and INT-981573... |

332 | Logic programming with focusing proofs in linear logic
- Andreoli
- 1992
(Show Context)
Citation Context ... ⊥ n ) . . . Q1 · · · Qm. Such a formula is a nesting of synchronous connectives within asynchronous connectives, but no synchronous connective is in the scope of an asynchronous connective. Andr=-=eoli [And92] sho-=-wed that if we allow the addition of additional constants, arbitrary formulas of linear logic can be “compiled” to a collection of bipolar forms: basically, when the nesting alternates once, simpl... |

303 |
Higher-order abstract syntax
- Pfenning, Elliott
- 1988
(Show Context)
Citation Context ...fication over function symbols and enrich the term language with λ-terms provide support for the higherorder abstract syntax approach to representing syntactic expressions involving binders and scope=-= [PE88].-=- As is familiar with, say, implementations of λProlog, Isabelle, and Elf, higher-order unification is adequate for discovering substitutions for quantifiers during proof search in languages containin... |

302 | Logic programming in a fragment of intuitionistic linear logic
- Hodas, Miller
- 1994
(Show Context)
Citation Context ...oding of computational dynamics is possible. In linear logic, for example, it is possible to model a switch that is now on but later off and to assign and update imperative programming like variables =-=[HM94]-=- as well as model concurrent processes that evolve independently or synchronize among themselves [AP91,Mil96]. Here, counters, imperative variables, and processes are all represented as formulas and n... |

287 | A logic programming language with lambda-abstraction, function variables, and simple unification
- Miller
(Show Context)
Citation Context ...on, and search (see, for example, [NM99]). Higher-order unification [Hue75] is rather complicated, but much of higher-order abstract syntax can be maintained using a much weaker notion of unification =-=[Mil91]-=-. Quantification at predicate type is, however, a more complex problem. As is well known, cut-free proofs involving formulas with predicate quantification do not necessarily have the sub-formula prope... |

207 | Linear Objects: logical processes with built-in inheritance - Andreoli, Pareschi - 1990 |

147 | A logical analysis of modules in logic programming - Miller - 1989 |

144 | Ametanotation for protocol analysis - Cervesato, Lincoln, et al. - 1999 |

114 |
A unification algorithm for typed λ-calculus
- Huet
- 1975
(Show Context)
Citation Context ...gy has gone into the effective implementation of such systems, including treatments of higher-order unification, explicit substitution, and search (see, for example, [NM99]). Higher-order unification =-=[Hue75]-=- is rather complicated, but much of higher-order abstract syntax can be maintained using a much weaker notion of unification [Mil91]. Quantification at predicate type is, however, a more complex probl... |

102 | The pi-Calculus as a Theory in Linear Logic: Preliminary Results
- Miller
- 1992
(Show Context)
Citation Context ... introduce multiset rewriting and how that can be encoded in proof search. To model multiset rewriting we shall use a subset of linear logic similar to the process clauses introduced by the author in =-=[Mil93]. Such-=- clauses are simply described as follows: Let G and H be formulas composed of ⊥, . , and ∀. (Think of the . connective as the multiset constructor and ⊥ as the empty multiset.) Process clauses a... |

65 | Lexical scoping as universal quantification
- Miller
- 1989
(Show Context)
Citation Context ...r encryption algorithm is, of course, abstracted away at this point. Building data objects using higher-type eigenvariables is a standard approach in logic programming for modeling abstract datatypes =-=[Mil89a]. In or-=-der to place this higher-type object within data, we introduce a new constructor · ◦ of type (data → data) → data that explicitly coerces an encryption function into data.sExplicit quantificati... |

61 | Higher-Order Horn Clauses
- Nadathur, Miller
(Show Context)
Citation Context ...der quantification (used for higher-order programming), but such uses are restricted so that computing necessary predicate substitutions can be done using (essentially) higher-order (pre) unification =-=[NM90]-=-. This can only be done, however, for rather serious restrictions on the use of higher-order predicate substitutions (such restrictions do, however, also have a natural operational meaning within the ... |

49 | Relating strands and multiset rewriting for security protocol analysis - Cervesato, Durgin, et al. - 2000 |

34 |
System description: Teyjus—a compiler and abstract machine based implementation of Lambda Prolog
- Nadathur, Mitchell
- 1999
(Show Context)
Citation Context ...nction types. A great deal of energy has gone into the effective implementation of such systems, including treatments of higher-order unification, explicit substitution, and search (see, for example, =-=[NM99]-=-). Higher-order unification [Hue75] is rather complicated, but much of higher-order abstract syntax can be maintained using a much weaker notion of unification [Mil91]. Quantification at predicate typ... |

24 | programming in linear logic is NP-complete - Horn - 1992 |

21 | The complexity of horn fragments of linear logic - Kanovich - 1994 |

20 |
Forum: A multiple-conclusion specification language
- Miller
- 1996
(Show Context)
Citation Context ...ple of how higher-order quantification can be used to form abstractions and to enhance reasoning about code, consider the two different specifications E1 and E2 of a simple counter object in Figure 2 =-=[Mil96]. Ea-=-ch of these specifications specify a counter using notions similar to the encapsulation of state (via linear logic) that responds to two “methods”, namely, get and inc, for getting and incrementin... |

17 | A complete proof synthesis method for the cube of type systems - Dowek - 1993 |

16 | A maximal method for set variables in automatic theoremproving - Bledsoe - 1979 |

3 | The calculus of constructions as a framework for proof search with set variable instantiation - Felty - 2000 |

2 |
Overview of Linear Logic Programming, to appear
- Miller
(Show Context)
Citation Context ...conventional programs. In this section we consider the linear logic specification for the reverse of lists and formally show, by direct reasoning on the specification, that it is a symmetric relation =-=[Mil]. -=-Let the constants nil and (· :: ·) denote the two constructors for lists. To compute the reverse of two lists, make a place for two piles on a table. Initialize one pile to the list you wish to reve... |