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Least and greatest fixed points in linear logic Extended Version
, 2007
"... david.baelde at enslyon.org dale.miller at inria.fr Abstract. The firstorder theory of MALL (multiplicative, additive linear logic) over only equalities is an interesting but weak logic since it cannot capture unbounded (infinite) behavior. Instead of accounting for unbounded behavior via the addi ..."
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david.baelde at enslyon.org dale.miller at inria.fr Abstract. The firstorder theory of MALL (multiplicative, additive linear logic) over only equalities is an interesting but weak logic since it cannot capture unbounded (infinite) behavior. Instead of accounting for unbounded behavior via the addition of the exponentials (! and?), we add least and greatest fixed point operators. The resulting logic, which we call µMALL = , satisfies two fundamental proof theoretic properties. In particular, µMALL = satisfies cutelimination, which implies consistency, and has a complete focused proof system. This second result about focused proofs provides a strong normal form for cutfree proof structures that can be used, for example, to help automate proof search. We then consider applying these two results about µMALL = to derive a focused proof system for an intuitionistic logic extended with induction and coinduction. The traditional approach to encoding intuitionistic logic into linear logic relies heavily on using the exponentials, which unfortunately weaken the focusing discipline. We get a better focused proof system by observing that certain fixed points satisfy the structural rules of weakening and contraction (without using exponentials). The resulting focused proof system for intuitionistic logic is closely related to the one implemented in Bedwyr, a recent model checker based on logic programming. We discuss how our proof theory might be used to build a computational system that can partially automate induction and coinduction. 1
HigherOrder Horn Clauses
 JOURNAL OF THE ACM
, 1990
"... A generalization of Horn clauses to a higherorder logic is described and examined as a basis for logic programming. In qualitative terms, these higherorder Horn clauses are obtained from the firstorder ones by replacing firstorder terms with simply typed #terms and by permitting quantification ..."
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Cited by 61 (20 self)
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A generalization of Horn clauses to a higherorder logic is described and examined as a basis for logic programming. In qualitative terms, these higherorder Horn clauses are obtained from the firstorder ones by replacing firstorder terms with simply typed #terms and by permitting quantification over all occurrences of function symbols and some occurrences of predicate symbols. Several prooftheoretic results concerning these extended clauses are presented. One result shows that although the substitutions for predicate variables can be quite complex in general, the substitutions necessary in the context of higherorder Horn clauses are tightly constrained. This observation is used to show that these higherorder formulas can specify computations in a fashion similar to firstorder Horn clauses. A complete theorem proving procedure is also described for the extension. This procedure is obtained by interweaving higherorder unification with backchaining and goal reductions, and constitutes a higherorder generalization of SLDresolution. These results have a practical realization in the higherorder logic programming language called λProlog.
Unification of simply typed lambdaterms as logic programming
 In Eighth International Logic Programming Conference
, 1991
"... The unification of simply typed λterms modulo the rules of β and ηconversions is often called “higherorder ” unification because of the possible presence of variables of functional type. This kind of unification is undecidable in general and if unifiers exist, most general unifiers may not exist ..."
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Cited by 60 (3 self)
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The unification of simply typed λterms modulo the rules of β and ηconversions is often called “higherorder ” unification because of the possible presence of variables of functional type. This kind of unification is undecidable in general and if unifiers exist, most general unifiers may not exist. In this paper, we show that such unification problems can be coded as a query of the logic programming language Lλ in a natural and clear fashion. In a sense, the translation only involves explicitly axiomatizing in Lλ the notions of equality and substitution of the simply typed λcalculus: the rest of the unification process can be viewed as simply an interpreter of Lλ searching for proofs using those axioms. 1
Pure Pattern Type Systems
 In POPL’03
, 2003
"... We introduce a new framework of algebraic pure type systems in which we consider rewrite rules as lambda terms with patterns and rewrite rule application as abstraction application with builtin matching facilities. This framework, that we call “Pure Pattern Type Systems”, is particularly wellsuite ..."
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Cited by 54 (26 self)
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We introduce a new framework of algebraic pure type systems in which we consider rewrite rules as lambda terms with patterns and rewrite rule application as abstraction application with builtin matching facilities. This framework, that we call “Pure Pattern Type Systems”, is particularly wellsuited for the foundations of programming (meta)languages and proof assistants since it provides in a fully unified setting higherorder capabilities and pattern matching ability together with powerful type systems. We prove some standard properties like confluence and subject reduction for the case of a syntactic theory and under a syntactical restriction over the shape of patterns. We also conjecture the strong normalization of typable terms. This work should be seen as a contribution to a formal connection between logics and rewriting, and a step towards new proof engines based on the CurryHoward isomorphism.
Possible Worlds and Resources: The Semantics of BI
 THEORETICAL COMPUTER SCIENCE
, 2003
"... The logic of bunched implications, BI, is a substructural system which freely combines an additive (intuitionistic) and a multiplicative (linear) implication via bunches (contexts with two combining operations, one which admits Weakening and Contraction and one which does not). BI may be seen to a ..."
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Cited by 52 (19 self)
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The logic of bunched implications, BI, is a substructural system which freely combines an additive (intuitionistic) and a multiplicative (linear) implication via bunches (contexts with two combining operations, one which admits Weakening and Contraction and one which does not). BI may be seen to arise from two main perspectives. On the one hand, from prooftheoretic or categorical concerns and, on the other, from a possibleworlds semantics based on preordered (commutative) monoids. This semantics may be motivated from a basic model of the notion of resource. We explain BI's prooftheoretic, categorical and semantic origins. We discuss in detail the question of completeness, explaining the essential distinction between BI with and without ? (the unit of _). We give an extensive discussion of BI as a semantically based logic of resources, giving concrete models based on Petri nets, ambients, computer memory, logic programming, and money.
Focusing the inverse method for linear logic
 Proceedings of CSL 2005
, 2005
"... 1.1 Quantification and the subformula property.................. 3 1.2 Ground forward sequent calculus......................... 5 1.3 Lifting to free variables............................... 10 ..."
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Cited by 51 (15 self)
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1.1 Quantification and the subformula property.................. 3 1.2 Ground forward sequent calculus......................... 5 1.3 Lifting to free variables............................... 10
Proof Search in the Intuitionistic Sequent Calculus
 11th International Conference on Automated Deduction
, 1991
"... The use of Herbrand functions (more popularly known as Skolemization) plays an important role in classical theorem proving and logic programming. We define a notion of Herbrand functions for the full intuitionistic predicate calculus. The definition is based on the view that the prooftheoretic role ..."
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Cited by 46 (1 self)
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The use of Herbrand functions (more popularly known as Skolemization) plays an important role in classical theorem proving and logic programming. We define a notion of Herbrand functions for the full intuitionistic predicate calculus. The definition is based on the view that the prooftheoretic role of Herbrand functions (to replace universal quantifiers), and of unification (to find instances corresponding to existential quantifiers), is to ensure that the eigenvariable conditions on a sequent proof are respected. The propositional impermutabilities that arise in the intuitionistic but not the classical sequent calculus motivate a generalization of the classical notion of Herbrand functions. Proof search using generalized Herbrand functions also provides a framework for generalizing logic programming to subsets of intuitionistic logic that are larger than Horn clauses. The search procedure described here has been implemented and observed to work effectively in practice. The generaliza...
The Uniform Prooftheoretic Foundation of Linear Logic Programming (Extended Abstract)
 PROCEEDINGS OF THE INTERNATIONAL LOGIC PROGRAMMING SYMPOSIUM
, 1991
"... We present a prooftheoretic analysis of a natural notion of logic programming for Girard's linear logic. This analysis enables us to identify a suitable notion of uniform proof. This in turn enables us to identify choices of classes of definite and goal formulae for which uniform proofs are c ..."
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Cited by 45 (7 self)
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We present a prooftheoretic analysis of a natural notion of logic programming for Girard's linear logic. This analysis enables us to identify a suitable notion of uniform proof. This in turn enables us to identify choices of classes of definite and goal formulae for which uniform proofs are complete and so to obtain the appropriate formulation of resolution proof for such choices. Resolution proofs in linear logic are somewhat difficult to define. This difficulty arises from the need to decompose definite formulae into a form suitable for the use of the linear resolution rule, a rule which requires the selected clause to be deleted after use, and from the presence of the modality ! (of course). We consider a translation  resembling ...
Programming in Lygon: An Overview
 ALGEBRAIC METHODOLOGY AND SOFTWARE TECHNOLOGY
, 1996
"... Recently, there has been much interest in the derivation of logic programming languages based on linear logic, a logic of resourceconsumption. Such languages provide a notion of resourceoriented programming, often leading to programs that are more elegant and concise than their equivalents in la ..."
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Cited by 44 (18 self)
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Recently, there has been much interest in the derivation of logic programming languages based on linear logic, a logic of resourceconsumption. Such languages provide a notion of resourceoriented programming, often leading to programs that are more elegant and concise than their equivalents in languages, such as Prolog, based on classical logics. We discuss, with examples, the design, implementation and applications of Lygon, a linear logic programming language. Lygon is based on a prooftheoretic analysis of which occurrences of the linear connectives provide an adequate basis for programming. In common with other linear logic programming languages, Lygon allows clauses to be used exactly once in a computation, thereby avoiding the need for the explicit resourcecounting often necessary in Prologlike languages. Indeed, it appears that resourcesensitivity leads to significant differences between the natural programming methodologies in Lygon and Prolog. Just as linear logic...