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A Linear Logical Framework
, 1996
"... We present the linear type theory LLF as the forAppeared in the proceedings of the Eleventh Annual IEEE Symposium on Logic in Computer Science  LICS'96 (E. Clarke editor), pp. 264275, New Brunswick, NJ, July 2730 1996. mal basis for a conservative extension of the LF logical framework. LLF c ..."
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Cited by 217 (44 self)
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We present the linear type theory LLF as the forAppeared in the proceedings of the Eleventh Annual IEEE Symposium on Logic in Computer Science  LICS'96 (E. Clarke editor), pp. 264275, New Brunswick, NJ, July 2730 1996. mal basis for a conservative extension of the LF logical framework. LLF combines the expressive power of dependent types with linear logic to permit the natural and concise representation of a whole new class of deductive systems, namely those dealing with state. As an example we encode a version of MiniML with references including its type system, its operational semantics, and a proof of type preservation. Another example is the encoding of a sequent calculus for classical linear logic and its cut elimination theorem. LLF can also be given an operational interpretation as a logic programming language under which the representations above can be used for type inference, evaluation and cutelimination. 1 Introduction A logical framework is a formal system desig...
Efficient Representation and Validation of Proofs
, 1998
"... This paper presents a logical framework derived from the Edinburgh Logical Framework (LF) [5] that can be used to obtain compact representations of proofs and efficient proof checkers. These are essential ingredients of any application that manipulates proofs as firstclass objects, such as a Proof ..."
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Cited by 61 (7 self)
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This paper presents a logical framework derived from the Edinburgh Logical Framework (LF) [5] that can be used to obtain compact representations of proofs and efficient proof checkers. These are essential ingredients of any application that manipulates proofs as firstclass objects, such as a ProofCarrying Code [11] system, in which proofs are used to allow the easy validation of properties of safetycritical or untrusted code. Our framework, which we call LF i , inherits from LF the capability to encode various logics in a natural way. In addition, the LF i framework allows proof representations without the high degree of redundancy that is characteristic of LF representations. The missing parts of LF i proof representations can be reconstructed during proof checking by an efficient reconstruction algorithm. We also describe an algorithm that can be used to strip the unnecessary parts of an LF representation of a proof. The experimental data that we gathered in the context of a Proof...
Unification via Explicit Substitutions: The Case of HigherOrder Patterns
 PROCEEDINGS OF JICSLP'96
, 1998
"... In [6] we have proposed a general higherorder unification method using a theory of explicit substitutions and we have proved its completeness. In this paper, we investigate the case of higherorder patterns as introduced by Miller. We show that our general algorithm specializes in a very convenient ..."
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Cited by 56 (14 self)
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In [6] we have proposed a general higherorder unification method using a theory of explicit substitutions and we have proved its completeness. In this paper, we investigate the case of higherorder patterns as introduced by Miller. We show that our general algorithm specializes in a very convenient way to patterns. We also sketch an efficient implementation of the abstract algorithm and its generalization to constraint simplification, which has yielded good experimental results at the core of a higherorder constraint logic programming language.
OracleBased Checking of Untrusted Software
, 2001
"... We present a variant of ProofCarrying Code (PCC) in which the trusted inference rules are represented as a higherorder logic program, the proof checker is replaced by a nondeterministic higherorder logic interpreter and the proof by an oracle implemented as a stream of bits that resolve the nondet ..."
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Cited by 55 (3 self)
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We present a variant of ProofCarrying Code (PCC) in which the trusted inference rules are represented as a higherorder logic program, the proof checker is replaced by a nondeterministic higherorder logic interpreter and the proof by an oracle implemented as a stream of bits that resolve the nondeterministic interpretation choices. In this setting, ProofCarrying Code allows the receiver of the code the luxury of using nondeterminism in constructing a simple yet powerful checking procedure. This oraclebased variant of PCC is able to adapt quite naturally to situations when the property being checked is simple or there is a fairly directed search procedure for it. As an example, we demonstrate that if PCC is used to verify type safety of assembly language programs compiled from Java source programs, the oracles that are needed are on the average just 12% of the size of the code, which represents an improvement of a factor of 30 over previous syntactic representations of PCC proofs. ...
Efficient Representation and Validation of Logical Proofs
, 1997
"... This report describes a framework for representing and validating formal proofs in various axiomatic systems. The framework is based on the Edinburgh Logical Framework (LF) but is optimized for minimizing the size of proofs and the complexity of proof validation, by removing redundant representation ..."
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Cited by 45 (6 self)
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This report describes a framework for representing and validating formal proofs in various axiomatic systems. The framework is based on the Edinburgh Logical Framework (LF) but is optimized for minimizing the size of proofs and the complexity of proof validation, by removing redundant representation components. Several variants of representation algorithms are presented with the resulting representations being a factor of 15 smaller than similar LF representations. The validation algorithm is a reconstruction algorithm that runs about 7 times faster than LF typechecking. We present a full proof of correctness of the reconstruction algorithm and hints for the efficient implementation using explicit substitutions. We conclude with a quantitative analysis of the algorithms. This research was sponsored in part by the Advanced Research Projects Agency CSTO under the title "The Fox Project: Advanced Languages for Systems Software," ARPA Order No. C533, issued by ESC/ENS under Contract No. F1...
A Linear Spine Calculus
 Journal of Logic and Computation
, 2003
"... We present the spine calculus S ##&# as an efficient representation for the linear #calculus # ##&# which includes unrestricted functions (#), linear functions (#), additive pairing (&), and additive unit (#). S ##&# enhances the representation of Church's simply typed #calculus by enforcing ..."
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Cited by 33 (5 self)
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We present the spine calculus S ##&# as an efficient representation for the linear #calculus # ##&# which includes unrestricted functions (#), linear functions (#), additive pairing (&), and additive unit (#). S ##&# enhances the representation of Church's simply typed #calculus by enforcing extensionality and by incorporating linear constructs. This approach permits procedures such as unification to retain the efficient head access that characterizes firstorder term languages without the overhead of performing #conversions at run time. Applications lie in proof search, logic programming, and logical frameworks based on linear type theories. It is also related to foundational work on term assignment calculi for presentations of the sequent calculus. We define the spine calculus, give translations of # ##&# into S ##&# and viceversa, prove their soundness and completeness with respect to typing and reductions, and show that the typable fragment of the spine calculus is strongly normalizing and admits unique canonical, i.e. ##normal, forms.
Hybridizing a logical framework
 In International Workshop on Hybrid Logic 2006 (HyLo 2006), Electronic Notes in Computer Science
, 2006
"... The logical framework LF is a constructive type theory of dependent functions that can elegantly encode many other logical systems. Prior work has studied the benefits of extending it to the linear logical framework LLF, for the incorporation linear logic features into the type theory affords good r ..."
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Cited by 20 (1 self)
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The logical framework LF is a constructive type theory of dependent functions that can elegantly encode many other logical systems. Prior work has studied the benefits of extending it to the linear logical framework LLF, for the incorporation linear logic features into the type theory affords good representations of state change. We describe and argue for the usefulness of an extension of LF by features inspired by hybrid logic, which has several benefits. For one, it shows how linear logic features can be decomposed into primitive operations manipulating abstract resource labels. More importantly, it makes it possible to realize a metalogical framework capable of reasoning about stateful deductive systems encoded in the style familiar from prior work with LLF, taking advantage of familiar methodologies used for metatheoretic reasoning in LF.Acknowledgments From the very first computer science course I took at CMU, Frank Pfenning has been an exceptional teacher and mentor. For his patience, breadth of knowledge, and mathematical good taste I am extremely thankful. No less do I owe to the other two major contributors to my programming languages
Implementation Considerations for HigherOrder Features in Logic Programming
, 1993
"... This paper examines implementation problems that arise from providing for aspects of higherorder programming within and enhancing the metalanguage abilities of logic programming. One issue of concern is a representation for the simplytyped lambda terms that replace the usual firstorder terms as ..."
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Cited by 14 (10 self)
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This paper examines implementation problems that arise from providing for aspects of higherorder programming within and enhancing the metalanguage abilities of logic programming. One issue of concern is a representation for the simplytyped lambda terms that replace the usual firstorder terms as data structures; this representation must support an efficient realization of ...conversion operations on these terms. Another issue is the handling of higherorder unification that becomes an integral part of the computational model. An implementation must cater to the branching nature of this operation and also provide a means for temporarily suspending the solution of a unification problem. A final issue concerns the treatment of goals whose structure is not statically apparent. These problems are discussed in detail and solutions to them are described. A representation for lambda terms is presented that uses the de Bruijn "nameless" notation and also permits reduction substitutions to be performed lazily. This notation obviates ...conversion and also supports an efficient implementation of ...reduction. Branching in unification is implemented by using a depthfirst search strategy with backtracking. A structure that is called a branch point record and is akin to the choice point record of the Warren Abstract Machine (WAM) is described for remembering alternatives in unification. An explicit representation for unification problems is presented that permits sharing and also supports the rapid reinstatement of earlier versions of the problem. The implementation of unification is tuned to yield an efficient solution to firstorder like problems, in fact through the use of compiled code as in the WAM. A compilation method is also discussed for goals whose structure changes during execution. Th...
Practical higherorder pattern unification with onthefly raising
 In ICLP 2005: 21st International Logic Programming Conference, volume 3668 of LNCS
, 2005
"... Abstract. Higherorder pattern unification problems arise often in computations carried out within systems such as Twelf, λProlog and Isabelle. An important characteristic of such problems is that they are given by equations appearing under a prefix of alternating universal and existential quantifie ..."
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Cited by 8 (4 self)
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Abstract. Higherorder pattern unification problems arise often in computations carried out within systems such as Twelf, λProlog and Isabelle. An important characteristic of such problems is that they are given by equations appearing under a prefix of alternating universal and existential quantifiers. Existing algorithms for solving these problems assume that such prefixes are simplified to a ∀∃ ∀ form by an a priori application of a transformation known as raising. There are drawbacks to this approach. Mixed quantifier prefixes typically manifest themselves in the course of computation, thereby requiring a dynamic form of preprocessing that is difficult to support in lowlevel implementations. Moreover, raising may be redundant in many cases and its effect may have to be undone by a subsequent pruning transformation. We propose a method to overcome these difficulties. In particular, a unification algorithm is described that proceeds by recursively descending through the structures of terms, performing raising and other transformations onthefly and only as needed. This algorithm also exploits an explicit substitution notation for lambda terms. 1
Elf: A MetaLanguage for Deductive Systems (System Description)
 In 12th International Conference on Automated Deduction
, 1994
"... ce describing the Elf language is [10]. Gentler introductions can be found in [12] and [6]. Elf has also been used in a graduate course on the theory of programming languages. A draft of the course notes may be available from the author upon request. Below we provide a brief overview of how specific ..."
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Cited by 7 (0 self)
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ce describing the Elf language is [10]. Gentler introductions can be found in [12] and [6]. Elf has also been used in a graduate course on the theory of programming languages. A draft of the course notes may be available from the author upon request. Below we provide a brief overview of how specification, implementation, and metatheory tasks are supported in the Elf language. The subsequent sections list some case studies and describe the implementation of Elf. Object Language Specification. LF generalizes firstorder terms by allowing objects from a dependently typed calculus to represent object language expressions. This allows variables in the object language to be represented by variables in the metalanguage, using the technique of higherorder abstract syntax. Common operations ? Internet address: fp@cs.cmu.edu (e.g., renaming of bound variables or substitution) and sideconditions on infer