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52
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 68 (27 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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Cited by 62 (14 self)
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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
Focusing and polarization in intuitionistic logic
 CSL 2007: Computer Science Logic, volume 4646 of LNCS
, 2007
"... dale.miller at inria.fr Abstract. A focused proof system provides a normal form to cutfree proofs that structures the application of invertible and noninvertible inference rules. The focused proof system of Andreoli for linear logic has been applied to both the proof search and the proof normaliza ..."
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Cited by 42 (17 self)
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dale.miller at inria.fr Abstract. A focused proof system provides a normal form to cutfree proofs that structures the application of invertible and noninvertible inference rules. The focused proof system of Andreoli for linear logic has been applied to both the proof search and the proof normalization approaches to computation. Various proof systems in literature exhibit characteristics of focusing to one degree or another. We present a new, focused proof system for intuitionistic logic, called LJF, and show how other 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 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. 1
Linear Logical Algorithms
"... Abstract. Bottomup logic programming can be used to declaratively specify many algorithms in a succinct and natural way, and McAllester and Ganzinger have shown that it is possible to define a cost semantics that enables reasoning about the running time of algorithms written as inference rules. Pre ..."
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Cited by 15 (5 self)
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Abstract. Bottomup logic programming can be used to declaratively specify many algorithms in a succinct and natural way, and McAllester and Ganzinger have shown that it is possible to define a cost semantics that enables reasoning about the running time of algorithms written as inference rules. Previous work with the programming language Lollimon demonstrates the expressive power of logic programming with linear logic in describing algorithms that have imperative elements or that must repeatedly make mutually exclusive choices. In this paper, we identify a bottomup logic programming language based on linear logic that is amenable to efficient execution and describe a novel cost semantics that can be used for complexity analysis of algorithms expressed in linear logic. Key words: Bottomup logic programming, forward reasoning, linear logic, deductive databases, cost semantics, abstract running time 1
On the specification of sequent systems
 IN LPAR 2005: 12TH INTERNATIONAL CONFERENCE ON LOGIC FOR PROGRAMMING, ARTIFICIAL INTELLIGENCE AND REASONING, NUMBER 3835 IN LNAI
, 2005
"... Recently, linear Logic has been used to specify sequent calculus proof systems in such a way that the proof search in linear logic can yield proof search in the specified logic. Furthermore, the metatheory of linear logic can be used to draw conclusions about the specified sequent calculus. For e ..."
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Cited by 13 (6 self)
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Recently, linear Logic has been used to specify sequent calculus proof systems in such a way that the proof search in linear logic can yield proof search in the specified logic. Furthermore, the metatheory of linear logic can be used to draw conclusions about the specified sequent calculus. For example, derivability of one proof system from another can be decided by a simple procedure that is implemented via bounded logic programmingstyle search. Also, simple and decidable conditions on the linear logic presentation of inference rules, called homogeneous and coherence, can be used to infer that the initial rules can be restricted to atoms and that cuts can be eliminated. In the present paper we introduce Llinda, a logical framework based on linear logic augmented with inference rules for definition (fixed points) and induction. In this way, the above properties can be proved entirely inside the framework. To further illustrate the power of Llinda, we extend the definition of coherence and provide a new, semiautomated proof of cutelimination for Girard’s Logic of Unicity (LU).
Focused Inductive Theorem Proving
"... Abstract. Focused proof systems provide means for reducing and structuring the nondeterminism involved in searching for sequent calculus proofs. We present a focused proof system for a firstorder logic with inductive and coinductive definitions in which the introduction rules are partitioned into ..."
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Cited by 13 (7 self)
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Abstract. Focused proof systems provide means for reducing and structuring the nondeterminism involved in searching for sequent calculus proofs. We present a focused proof system for a firstorder logic with inductive and coinductive definitions in which the introduction rules are partitioned into an asynchronous phase and a synchronous phase. These focused proofs allows us to naturally see proof search as being organized around interleaving intervals of computation and more general deduction. For example, entire Prologlike computations can be captured using a single synchronous phase and many modelchecking queries can be captured using an asynchronous phase followed by a synchronous phase. Leveraging these ideas, we have developed an interactive proof assistant, called Tac, for this logic. We describe its highlevel design and illustrate how it is capable of automatically proving many theorems using induction and coinduction. Since the automatic proof procedure is structured using focused proofs, its behavior is often rather easy to anticipate and modify. We illustrate the strength of Tac with several examples of proof developments, some achieved entirely automatically and others achieved with user guidance. 1
A Focused Approach to Combining Logics
, 2010
"... We present a compact sequent calculus LKU for classical logic organized around the concept of polarization. Focused sequent calculi for classical, intuitionistic, and multiplicativeadditive linear logics are derived as fragments of the host system by varying the sensitivity of specialized structura ..."
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Cited by 11 (9 self)
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We present a compact sequent calculus LKU for classical logic organized around the concept of polarization. Focused sequent calculi for classical, intuitionistic, and multiplicativeadditive linear logics are derived as fragments of the host system by varying the sensitivity of specialized structural rules to polarity information. We identify a general set of criteria under which cut elimination holds in such fragments. From cut elimination we derive a unified proof of the completeness of focusing. Furthermore, each sublogic can interact with other fragments through cut. We examine certain circumstances, for example, in which a classical lemma can be used in an intuitionistic proof while preserving intuitionistic provability. We also examine the possibility of defining classicallinear hybrid logics.
A focusing inverse method theorem prover for firstorder linear logic
 In Proceedings of CADE20
, 2005
"... Abstract. We present the theory and implementation of a theorem prover forfirstorder intuitionistic linear logic based on the inverse method. The central prooftheoretic insights underlying the prover concern resource management andfocused derivations, both of which are traditionally understood in ..."
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Cited by 10 (8 self)
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Abstract. We present the theory and implementation of a theorem prover forfirstorder intuitionistic linear logic based on the inverse method. The central prooftheoretic insights underlying the prover concern resource management andfocused derivations, both of which are traditionally understood in the domain of backward reasoning systems such as logic programming. We illustrate how resource management, focusing, and other intrinsic properties of linear connectives affect the basic forward operations of rule application, contraction, and forwardsubsumption. We also present some preliminary experimental results obtained with our implementation.
On focusing and polarities in linear logic and intuitionistic logic
, 2006
"... There are a number of cutfree sequent calculus proof systems known that are complete for firstorder intuitionistic logic. Proofs in these different systems can vary a great deal from one another. We are interested in providing a flexible and unifying framework that can collect together important a ..."
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Cited by 8 (3 self)
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There are a number of cutfree sequent calculus proof systems known that are complete for firstorder intuitionistic logic. Proofs in these different systems can vary a great deal from one another. We are interested in providing a flexible and unifying framework that can collect together important aspects of many of these proof systems. First, we suggest that one way to unify these proof systems is to first translate intuitionistic logic formulas into linear logic formulas, then assign a bias (positive or negative) to atomic formulas, and then examine the nature of focused proofs in the resulting linear logic setting. Second, we provide a single focusing proof system for intuitionistic logic and show that these other intuitionistic proof systems can be accounted for by assigning bias to atomic formulas and by inserting certain markers that halt focusing on formulas. Using either approach, we are able to account for proof search mechanisms that allow for forwardchaining (programdirected search), backwardchaining (goaldirected search), and combinations of these two. The keys to developing this kind of proof system for intuitionistic logic involves using Andreoli’s completeness result for focusing proofs and Girard’s notion of polarity used in his LC and LU proof systems. 1
Imogen: Focusing the Polarized Inverse Method for Intuitionistic Propositional Logic
"... Abstract. In this paper we describe Imogen, a theorem prover for intuitionistic propositional logic using the focused inverse method. We represent finegrained control of the search behavior by polarizing the input formula. In manipulating the polarity of atoms and subformulas, we can often improve ..."
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Cited by 8 (4 self)
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Abstract. In this paper we describe Imogen, a theorem prover for intuitionistic propositional logic using the focused inverse method. We represent finegrained control of the search behavior by polarizing the input formula. In manipulating the polarity of atoms and subformulas, we can often improve the search time by several orders of magnitude. We tested our method against seven other systems on the propositional fragment of the ILTP library. We found that our prover outperforms all other provers on a substantial subset of the library. 1