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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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Cited by 35 (12 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
The Bedwyr system for model checking over syntactic expressions
 21th Conference on Automated Deduction, LNAI 4603, 391–397
, 2007
"... Bedwyr is a generalization of logic programming that allows model checking directly on syntactic expressions possibly containing bindings. This system, written in OCaml, is a direct implementation of two recent advances in the theory of proof search. The first is centered on the fact that both finit ..."
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Cited by 21 (12 self)
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Bedwyr is a generalization of logic programming that allows model checking directly on syntactic expressions possibly containing bindings. This system, written in OCaml, is a direct implementation of two recent advances in the theory of proof search. The first is centered on the fact that both finite success and finite failure can be captured in the sequent calculus by incorporating inference rules for definitions that allow fixed points to be explored. As a result, proof search in such a sequent calculus can capture simple model checking problems as well as may and must behavior in operational semantics. The second is that higherorder abstract syntax is directly supported using termlevel λbinders and the quantifier known as ∇. These features allow reasoning directly on expressions containing bound variables. 2
Proof search specifications of bisimulation and modal logics for the πcalculus
 ACM Trans. on Computational Logic
"... We specify the operational semantics and bisimulation relations for the finite πcalculus within a logic that contains the ∇ quantifier for encoding generic judgments and definitions for encoding fixed points. Since we restrict to the finite case, the ability of the logic to unfold fixed points allo ..."
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Cited by 8 (6 self)
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We specify the operational semantics and bisimulation relations for the finite πcalculus within a logic that contains the ∇ quantifier for encoding generic judgments and definitions for encoding fixed points. Since we restrict to the finite case, the ability of the logic to unfold fixed points allows this logic to be complete for both the inductive nature of operational semantics and the coinductive nature of bisimulation. The ∇ quantifier helps with the delicate issues surrounding the scope of variables within πcalculus expressions and their executions (proofs). We shall illustrate several merits of the logical specifications permitted by this logic: they are natural and declarative; they contain no sideconditions concerning names of variables while maintaining a completely formal treatment of such variables; differences between late and open bisimulation relations arise from familar logic distinctions; the interplay between the three quantifiers (∀, ∃, and ∇) and their scopes can explain the differences between early and late bisimulation and between various modal operators based on bound input and output actions; and proof search involving the application of inference rules, unification, and backtracking can provide complete proof systems for onestep transitions, bisimulation, and satisfaction in modal logic. We also illustrate how one can encode the πcalculus with replications, in an extended logic with induction and coinduction.
Representing and reasoning with operational semantics
 In: Proceedings of the Joint International Conference on Automated Reasoning
, 2006
"... The operational semantics of programming and specification languages is often presented via inference rules and these can generally be mapped into logic programminglike clauses. Such logical encodings of operational semantics can be surprisingly declarative if one uses logics that directly account ..."
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Cited by 7 (2 self)
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The operational semantics of programming and specification languages is often presented via inference rules and these can generally be mapped into logic programminglike clauses. Such logical encodings of operational semantics can be surprisingly declarative if one uses logics that directly account for termlevel bindings and for resources, such as are found in linear logic. Traditional theorem proving techniques, such as unification and backtracking search, can then be applied to animate operational semantic specifications. Of course, one wishes to go a step further than animation: using logic to encode computation should facilitate formal reasoning directly with semantic specifications. We outline an approach to reasoning about logic specifications that involves viewing logic specifications as theories in an objectlogic and then using a metalogic to reason about properties of those objectlogic theories. We motivate the principal design goals of a particular metalogic that has been built for that purpose.
A User Guide to Bedwyr
, 2006
"... Some recent theoretical work in proof search has illustrated that it is possible to combine the following two computational principles into one computational logic. 1. A symmetric treatment of finite success and finite failure. This allows capturing both aspects of may and must behavior in operation ..."
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Cited by 4 (2 self)
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Some recent theoretical work in proof search has illustrated that it is possible to combine the following two computational principles into one computational logic. 1. A symmetric treatment of finite success and finite failure. This allows capturing both aspects of may and must behavior in operational semantics and mixing model checking and logic programming. 2. Direct support for λtree syntax, as in λProlog, via termlevel λbinders, higherorder pattern unification, and the ∇quantifier. All these features have a clean proof theory. The combination of these features allow, for example, specifying rather declarative approaches to model checking syntactic expressions containing bindings. The Bedwyr system is intended as an implementation of these computational logic principles. Why the name Bedwyr? In the legend of King Arthur and the round table, several knights shared in the search for the holy grail. The name of one of them, Parsifal, is used for an INRIA team associated with the “Slimmer ” effort. Bedwyr was another one of those knights. Wikipedia (using the spelling “Bedivere”) mentions that Bedwyr appears in Monty Python and the Holy Grail where he is “portrayed as a master of the extremely odd logic in the ancient times, whom occasionally blunders. ” Bedwyr is a reimplementation and rethinking ∗ Support has been obtained for this work from the following sources: from INRIA through
A proof theoretic approach to operational semantics, in
 Proc. of the workshop on Algebraic Process Calculi: The First Twenty Five Years and Beyond
, 2005
"... Proof theory can be applied to the problem of specifying and reasoning about the operational semantics of process calculi. We overview some recent research in which λtree syntax is used to encode expressions containing bindings and sequent calculus is used to reason about operational semantics. The ..."
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Cited by 2 (0 self)
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Proof theory can be applied to the problem of specifying and reasoning about the operational semantics of process calculi. We overview some recent research in which λtree syntax is used to encode expressions containing bindings and sequent calculus is used to reason about operational semantics. There are various benefits of this proof theoretic approach for the πcalculus: the treatment of bindings can be captured with no side conditions; bisimulation has a simple and natural specification in which the difference between bound input and bound output is characterized using difference quantifiers; various modal logics for mobility can be specified declaratively; and simple logic programminglike deduction involving subsets of secondorder unification provides immediate implementations of symbolic bisimulation. These benefits should extend to other process calculi as well. As partial evidence of this, a simple λtree syntax extension to the tyft/tyxt rule format for namebinding and namepassing is possible that allows one to conclude that (open) bisimilarity is a congruence. Key words: operational semantics, proof theoretic specifications, λtree syntax, rule formats, πcalculus A number of frameworks have been used to formalize the semantics of process calculi and, more generally, programming languages. For example, algebra, category theory, and I/O automata have been used to provide formal settings for not only specifying but also reasoning about the operational semantics of calculi and languages. In this note, we overview recent results in making use of proof theory to encode and reason about such operational semantics. By the term “proof theory ” we refer the study of proofs for logics, particularly in the style initiated by Gentzen. 1 Support for this work comes from INRIA through the “Equipes Associées ” Slimmer and from the ACI grants GEOCAL and Rossignol.
On the proof theory of regular fixed points
"... Abstract. We consider encoding finite automata as least fixed points in a prooftheoretical framework equipped with a general induction scheme, and study automata inclusion in that setting. We provide a coinductive characterization of inclusion that yields a natural bridge to prooftheory. This leads ..."
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Cited by 1 (0 self)
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Abstract. We consider encoding finite automata as least fixed points in a prooftheoretical framework equipped with a general induction scheme, and study automata inclusion in that setting. We provide a coinductive characterization of inclusion that yields a natural bridge to prooftheory. This leads us to generalize these observations to regular formulas, obtaining new insights about inductive theorem proving and cyclic proofs in particular. 1
The Australian National University
"... We specify the operational semantics and bisimulation relations for the finite πcalculus within a logic that contains the ∇ quantifier for encoding generic judgments and definitions for encoding fixed points. Since we restrict to the finite case, the ability of the logic to unfold fixed points allo ..."
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We specify the operational semantics and bisimulation relations for the finite πcalculus within a logic that contains the ∇ quantifier for encoding generic judgments and definitions for encoding fixed points. Since we restrict to the finite case, the ability of the logic to unfold fixed points allows this logic to be complete for both the inductive nature of operational semantics and the coinductive nature of bisimulation. The ∇ quantifier helps with the delicate issues surrounding the scope of variables within πcalculus expressions and their executions (proofs). We illustrate several merits of the logical specifications permitted by this logic: they are natural and declarative; they contain no sideconditions concerning names of variables while maintaining a completely formal treatment of such variables; differences between late and open bisimulation relations arise from familar logic distinctions; the interplay between the three quantifiers (∀, ∃, and ∇) and their scopes can explain the differences between early and late bisimulation and between various modal operators based on bound input and output actions; and proof search involving the application of inference rules, unification, and backtracking can provide complete proof systems for onestep transitions, bisimulation, and satisfaction in modal logic. We also illustrate how one can encode
3.5. Deep Inference and Categorical Axiomatizations 5 3.6. Proof Nets and Combinatorial Characterization of Proofs 5
"... c t i v it y e p o r t ..."
Author manuscript, published in "Automated Reasoning with Analytic Tableaux and Related Methods, 18th International Conference, TABLEAUX 2009 (2009)" On the proof theory of regular fixed points
"... Abstract. We consider encoding finite automata as least fixed points in a prooftheoretical framework equipped with a general induction scheme, and study automata inclusion in that setting. We provide a coinductive characterization of inclusion that yields a natural bridge to prooftheory. This leads ..."
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Abstract. We consider encoding finite automata as least fixed points in a prooftheoretical framework equipped with a general induction scheme, and study automata inclusion in that setting. We provide a coinductive characterization of inclusion that yields a natural bridge to prooftheory. This leads us to generalize these observations to regular formulas, obtaining new insights about inductive theorem proving and cyclic proofs in particular. hal00772502, version 1 10 Jan 2013 1