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Focusing and Polarization in Linear, Intuitionistic, and Classical Logics
, 2009
"... A focused proof system provides a normal form to cutfree proofs in which the application of invertible and noninvertible inference rules is structured. Within linear logic, the focused proof system of Andreoli provides an elegant and comprehensive normal form for cutfree proofs. Within intuitioni ..."
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Cited by 69 (28 self)
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A focused proof system provides a normal form to cutfree proofs in which the application of invertible and noninvertible inference rules is structured. Within linear logic, the focused proof system of Andreoli provides an elegant and comprehensive normal form for cutfree proofs. Within intuitionistic and classical logics, there are various different proof systems in the literature that exhibit focusing behavior. These focused proof systems have been applied to both the proof search and the proof normalization approaches to computation. We present a new, focused proof system for intuitionistic logic, called LJF, and show how other intuitionistic proof systems can be mapped into the new system by inserting logical connectives that prematurely stop focusing. We also use LJF to design a focused proof system LKF for classical logic. Our approach to the design and analysis of these systems is based on the completeness of focusing in linear logic and on the notion of polarity that appears in Girard’s LC and LU proof systems.
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
Canonical sequent proofs via multifocusing
 Fifth IFIP International Conference on Theoretical Computer Science, volume 273 of IFIP International Federation for Information Processing
, 2008
"... Abstract The sequent calculus admits many proofs of the same conclusion that differ only by trivial permutations of inference rules. In order to eliminate this “bureaucracy” from sequent proofs, deductive formalisms such as proof nets or natural deduction are usually used instead of the sequent calc ..."
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Abstract The sequent calculus admits many proofs of the same conclusion that differ only by trivial permutations of inference rules. In order to eliminate this “bureaucracy” from sequent proofs, deductive formalisms such as proof nets or natural deduction are usually used instead of the sequent calculus, for they identify proofs more abstractly and geometrically. In this paper we recover permutative canonicity directly in the cutfree sequent calculus by generalizing focused sequent proofs to admit multiple foci, and then considering the restricted class of maximally multifocused proofs. We validate this definition by proving a bijection to the wellknown proofnets for the unitfree multiplicative linear logic, and discuss the possibility of a similar correspondence for larger fragments. 1
Algorithmic specifications in linear logic with subexponentials
 In ACM SIGPLAN Conference on Principles and Practice of Declarative Programming (PPDP
, 2009
"... nigam at lix.polytechnique.fr, dale.miller at inria.fr The linear logic exponentials!, ? are not canonical: one can add to linear logic other such operators, say! l, ? l, which may or may not allow contraction and weakening, and where l is from some preordered set of labels. We shall call these add ..."
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nigam at lix.polytechnique.fr, dale.miller at inria.fr The linear logic exponentials!, ? are not canonical: one can add to linear logic other such operators, say! l, ? l, which may or may not allow contraction and weakening, and where l is from some preordered set of labels. We shall call these additional operators subexponentials and use them to assign locations to multisets of formulas within a linear logic programming setting. Treating locations as subexponentials greatly increases the algorithmic expressiveness of logic. To illustrate this new expressiveness, we show that focused proof search can be precisely linked to a simple algorithmic specification language that contains whileloops, conditionals, and insertion into and deletion from multisets. We also give some general conditions for when a focused proof step can be executed in constant time. In addition, we propose a new logical connective that allows for the creation of new subexponentials, thereby further augmenting the algorithmic expressiveness of logic.
The Focused Calculus of Structures
"... The focusing theorem identifies a complete class of sequent proofs that have no inessential nondeterministic choices and restrict the essential choices to a particular normal form. Focused proofs are therefore well suited both for the search and for the representation of sequent proofs. The calculus ..."
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Cited by 4 (2 self)
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The focusing theorem identifies a complete class of sequent proofs that have no inessential nondeterministic choices and restrict the essential choices to a particular normal form. Focused proofs are therefore well suited both for the search and for the representation of sequent proofs. The calculus of structures is a proof formalism that allows rules to be applied deep inside a formula. Through this freedom it can be used to give analytic proof systems for a wider variety of logics than the sequent calculus, but standard presentations of this calculus are too permissive, allowing too many proofs. In order to make it more amenable to proof search, we transplant the focusing theorem from the sequent calculus to the calculus of structures. The key technical contribution is an incremental treatment of focusing that avoids trivializing the calculus of structures. We give a direct inductive proof of the completeness of the focused calculus of structures with respect to a more standard unfocused form. We also show that any focused sequent proof can be represented in the focused calculus of structures, and, conversely, any proof in the focused calculus of structures corresponds to a focused sequent proof.
Compact Proof Certificates For Linear Logic
"... Abstract. Linear logic is increasingly being used as a tool for communicating reasoning agents in domains such as authorization, access control, electronic voting, etc., where proof certificates represent evidence that must be verified by proof consumers as part of higher protocols. Controlling the ..."
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Abstract. Linear logic is increasingly being used as a tool for communicating reasoning agents in domains such as authorization, access control, electronic voting, etc., where proof certificates represent evidence that must be verified by proof consumers as part of higher protocols. Controlling the size of these certificates is critical. We assume that the proof consumer is allowed to do some search to reconstruct details of the full proof that are omitted from the certificates. Because the decision problem for linear logic is unsolvable, the certificate must contain at least enough information to bound the search: we show how to use the sequence of contractions in the sequent proof for this bound. The remaining content of the proof, in particular the information about resource divisions, can then be omitted from the certificate. We also describe a technique for giving a variable amount of additional search hints to the proof consumer to limit its nondeterminism. 1
Focused Proof Search for Linear Logic in the Calculus of Structures
 in "Technical Communications of the 26th International Conference on Logic Programming (ICLP
, 2010
"... Abstract. The prooftheoretic approach to logic programming has benefited from the introduction of focused proof systems, through the nondeterminism reduction and control they provide when searching for proofs in the sequent calculus. However, this technique was not available in the calculus of str ..."
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Abstract. The prooftheoretic approach to logic programming has benefited from the introduction of focused proof systems, through the nondeterminism reduction and control they provide when searching for proofs in the sequent calculus. However, this technique was not available in the calculus of structures, known for inducing even more nondeterminism than other logical formalisms. This work in progress aims at translating the notion of focusing into the presentation of linear logic in this setting, and use some of its specific features, such as deep application of rules and fine granularity, in order to improve proof search procedures. The starting point for this research line is the multiplicative fragment of linear logic, for which a simple focused proof system can be built. 1.
Magically Constraining the Inverse Method Using Dynamic Polarity Assignment
, 2010
"... Abstract. Given a logic program that is terminating and modecorrect in an idealized Prolog interpreter (i.e., in a topdown logic programming engine), a bottomup logic programming engine can be used to compute exactly the same set of answers as the topdown engine for a given modecorrect query by ..."
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Abstract. Given a logic program that is terminating and modecorrect in an idealized Prolog interpreter (i.e., in a topdown logic programming engine), a bottomup logic programming engine can be used to compute exactly the same set of answers as the topdown engine for a given modecorrect query by rewriting the program and the query using the Magic Sets Transformation (MST). In previous work, we have shown that focusing can logically characterize the standard notion of bottomup logic programming if atomic formulas are statically given a certain polarity assignment. In an analogous manner, dynamically assigning polarities can characterize the effect of MST without needing to transform the program or the query. This gives us a new proof of the completeness of MST in purely logical terms, by using the general completeness theorem for focusing. As the dynamic assignment is done in a general logic, the essence of MST can potentially be generalized to larger fragments of logic. 1
A MultiFocused Proof System Isomorphic to Expansion Proofs
, 2013
"... The sequent calculus is often criticized for requiring proofs to contain large amounts of lowlevel syntactic details that can obscure the essence of a given proof. Because each inference rule introduces only a single connective, sequent proofs can separate closely related steps—such as instantiatin ..."
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The sequent calculus is often criticized for requiring proofs to contain large amounts of lowlevel syntactic details that can obscure the essence of a given proof. Because each inference rule introduces only a single connective, sequent proofs can separate closely related steps—such as instantiating a block of quantifiers—by irrelevant noise. Moreover, the sequential nature of sequent proofs forces proof steps that are syntactically noninterfering and permutable to nevertheless be written in some arbitrary order. The sequent calculus thus lacks a notion of canonicity: proofs that should be considered essentially the same may not have a common syntactic form. To fix this problem, many researchers have proposed replacing the sequent calculus with proof structures that are more parallel or geometric. Proofnets, matings, and atomic flows are examples of such revolutionary formalisms. We propose, instead, an evolutionary approach to recover canonicity within the sequent calculus, which we illustrate for classical firstorder logic. The essential element of our approach is the use of a multifocused sequent calculus as the means for abstracting away lowlevel details from classical cutfree sequent proofs. We show that, among the multifocused proofs, the maximally multifocused proofs that collect together all possible parallel foci are canonical. Moreover, if we start with a certain focused sequent proof system, such proofs are isomorphic to expansion proofs—a well known, minimalistic, and parallel generalization of Herbrand disjunctions—for classical firstorder logic. This technique appears to be a systematic way to recover the “essence of proof ” from within sequent calculus proofs. 1
The Isomorphism Between Expansion Proofs and MultiFocused Sequent Proofs
, 2012
"... The sequent calculus is often criticized for requiring proofs to contain large amounts of lowlevel syntactic details that can obscure the essence of a given proof. Because each inference rule introduces only a single connective, sequent proofs can separate closely related steps—such as instantiatin ..."
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The sequent calculus is often criticized for requiring proofs to contain large amounts of lowlevel syntactic details that can obscure the essence of a given proof. Because each inference rule introduces only a single connective, sequent proofs can separate closely related steps—such as instantiating a block of quantifiers—by irrelevant noise. Moreover, the sequential nature of sequent proofs forces proof steps that are syntactically noninterfering and permutable to nevertheless be written in some arbitrary order. The sequent calculus thus lacks a notion of canonicity: proofs that should be considered essentially the same may not have a common syntactic form. To fix this problem, many researchers have proposed replacing the sequent calculus with proof structures that are more parallel or geometric. Proofnets, matings, and atomic flows are examples of such revolutionary formalisms. We propose, instead, an evolutionary approach to recover canonicity within the sequent calculus, which we illustrate for classical firstorder logic. The essential element of our approach is the use of a multifocused sequent calculus as the means for abstracting away lowlevel details from classical cutfree sequent proofs. We show that, among the multifocused proofs, the maximally multifocused proofs that collect together all possible parallel foci are canonical. Moreover, if we start with a certain focused sequent proof system, such proofs are isomorphic to expansion proofs—a well known, minimalistic, and parallel generalization of Herbrand disjunctions—for classical firstorder logic. This technique appears to be a systematic way to recover the essence of sequent calculus proofs. 1