G. Bellin. Mechanizing Proof Theory: Resource-Aware Logics and ProofTransformations to Extract Implicit Information. PhD thesis, Stanford University, 1990.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....38] has a highly developed rewriting component. The rewriter compiles contextual information and stores it as annotations to subexpressions in the form of rewrite rules. There is a language for expressing rewriting control strategies. EKL also incorporates a decision procedure for direct logic [37, 5] and some facility for representing terms as data and carrying out meta level reasoning. RRL [34, 35] represents all axioms as equations and views theories as congruence relations. The deductive machinery of RRL includes: a completion procedure; 4 rewriting; associative commutative theories; ....

G. Bellin. Mechanizing Proof Theory: Resource-Aware Logics and ProofTransformations to Extract Implicit Information. PhD thesis, Stanford University, 1990. Available as University of Edinburgh Department of Computer Science Report CST-80-91.

....Proof Nets (i.e. legal proofs) inside the universe of rough, ill typed proofs , called Proof Structures. In this paper, we shall provide a new soundness criterion for Proof Structures of Multiplicative Linear Logic with Mix rule, also known in the literature under the name of Direct Logic [Gi86, Da90, FR91, Ret90, Ret93, Be90, BK92]. Our correctness condition is essentially a distributed version of Girard s token trip [Gi86] It has been inspired by an original interpretation of logical formulae as distributed processes, and Proof Structures as concurrent systems. In particular, the flow of information (tokens) inside the ....

....from the trivial inductive characterisation) for recognising proof nets among all proof structures. Soundness criteria for (variant of) Multiplicative Linear Logic have been proposed by Girard [Gi86] the well known long trip condition) Danos and Regnier [DR89, Da90] Roorda [Ro90] and Bellin [Be90]. We briefly recall Danos and Regnier s acyclicity condition, since we shall use it to provide a first proof of the correctness of our condition. we shall be a bit informal here; see [Da90] for a detailed exposition) Definition 3.1 Let P be a proof structure. A correctness graph for P is a ....

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G. Bellin. Mechanizing Proof theory: Resource-Aware Logics and ProofTransformations to Extract Implicit Information. Ph.D Thesis, Report CST-80-91, Dpt. of Computer Science, Univ. of Edinburgh. June 1990.

....and constants, in addition to the positive and negative literals. The proof rules of mall are all of the rules in the Appendix B that are associated with these connectives and constants. This logic has been studied 8 Decision Problems for Propositional Linear Logic by Girard [Gir87a] and Bellin [Bel90]. While provability for the classical propositional logic is co NP complete, we show below that provability for mall is pspace complete. An important property of the sequent calculus formulation of mall is cutelimination. This property follows from the Theorem A.3 of Appendix A. Theorem 2.1 Any ....

....np completeness, we studied propositional multiplicative linear logic with unrestricted weakening, but without contraction. We will call this direct The Multiplicative Fragment 59 logic or dl, as it is similar to the direct logic of [KW84] dl is also considered in considerable detail in Bellin [Bel90]. The rules for this system are the identity and cut rules, the rules for the multiplicatives, constants, and the structural rule W of unrestricted weakening. W Sigma A; Sigma We first demonstrate cut elimination for dl, yielding consistency, which will facilitate our later proof of ....

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G. Bellin. Mechanizing Proof Theory: Resource-Aware Logics and ProofTransformations to Extract Implicit Information. PhD thesis, Stanford University, 1990.

....3 Partition, in mll, and show that our encoding is sound and complete. 8 The main idea is that the small proof property of mll allows us to encode resource distribution problems naturally. Since linear logic treats propositions as resources natively, it has been called resource consciousness [5]. Note that since full linear logic is conservative over mll, our encoding remains sound and complete even in larger fragments. This does not lead to new results, however, since the complexity of most larger linear logics have already been completely characterized [15] 2.1 3 Partition We use ....

G. Bellin. Mechanizing Proof Theory: Resource-Aware Logics and Proof-Transformations to Extract Implicit Information. PhD thesis, Stanford University, 1990.

.... area of study is directly relevant to the logic programming use of linear logic, where linear logic sequents are taken to be logic programs which execute by performing proof search [20, 5, 4, 19] This area of research is also directly relevant to the construction of linear logic theorem provers [40, 44, 35, 6, 11, 10, 41]. The results here also lead into the study of semantics of linear logic, pointing to deep connections between various fragments of linear logic and familiar structures from computer science [12, 29, 30] In particular, work has progressed in attempting to find viewpoints where the proof theory of ....

G. Bellin. Mechanizing Proof Theory: Resource-Aware Logics and ProofTransformations to Extract Implicit Information. PhD thesis, Stanford University, 1990.

.... Phi, the constants 0, 1, and , but excludes the modal storage operators and . The proof rules of mall are all of the rules in the Appendix A that are associated with these connectives and constants. mall has been studied by Girard and Bellin, and used by Andreoli, Pareschi, and Cerrito [16, 6, 2, 7]. In contrast to classical propositional logic, which is co NPcomplete, we show below that provability for mall is pspace complete. The cut elimination theorem for the mall fragment follows from the cut elimination theorem and the subformula property for linear logic. Lemma 2.1 The provability ....

G. Bellin. Mechanizing Proof Theory: Resource-Aware Logics and Proof-Transformations to Extract Implicit Information. PhD thesis, Stanford University, 1990.

....the role of the units in all that follows. For simplicity, we also consider only atomic axiom links. Much of the proof theory used in this paper is standard; further details for the case of linear logic are contained in Girard s original paper [13] Troelstra s recent book [32] as well as in [6, 7], 12] etc. 9 3.1 The Abramsky Translation: the multiplicatives Logical Rule translation x : A; y : A Ixy = x(a)yhai . F w : Gamma; x : A . G v : Delta; y : B w : Gamma; v : Delta; z : A Omega B Omega x;y O z (F; G) w vz = xy(zhxyi(F wx k ....

....if there is no ambiguity, and refer to it as the D R graph of R. For further information, cf. 12, 6] Occasionally (e.g. in Section 5. 3 below) we will consider the system of sequent calculus for multiplicative linear logic, with the additional structural rule of Mix, also called direct logic DL [7, 11, 31, 4, 10]: mix : Gamma Delta Gamma; Delta Definition. A proof structure R is a proof net for Direct Logic DL if for every switching s of R, the graph G s (R) is acyclic (but not necessarily connected. The following fundamental result (Girard [13] relates sequent calculus and proof nets for ....

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G. Bellin. Mechanizing Proof Theory: Resource-Aware Logics and Proof-Transformations to Extract Implicit Information, Phd Thesis, Stanford University. Available as: Report CST-80-91, June 1990, Dept. of Computer Science, Univ. of Edinburgh. 72

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G. Bellin. Mechanizing Proof Theory: Resource-Aware Logics and ProofTransformations to Extract Implicit Information, Phd Thesis, Stanford University. Available as: Report CST-80-91, June 1990, Dept. of Computer Science, Univ. of Edinburgh.

....criterion characterizes those proofstructures which correspond to proofs in the sequent calculus. Girard s condition ( there are no short trip ) is exponential in time on the size of the proof structure, while other quadratic criteria are known (among others, one was given in the author s thesis [7]) Thus it is natural to ask what additional information is contained in the construction of Girard s trips other than the correctness of a proof structure. The beginning of an answer came in 1992, when Jacques van de Wiele and the author, inspired by Danos notion of a pure net, defined the trip ....

G. Bellin. Mechanizing Proof Theory: Resource-Aware Logics and ProofTransformations to Extract Implicit Information, Phd Thesis, Stanford University. Available as: Report CST-80-91, June 1990, Dept. of Computer Science, Univ. of Edinburgh.

....the role of the units in all that follows. For simplicity, we also consider only atomic axiom links. Much of the proof theory used in this paper is standard; further details for the case of linear logic are contained in Girard s original paper [13] Troelstra s recent book [32] as well as in [6, 7], 12] etc. 9 3.1 The Abramsky Translation: the multiplicatives Logical Rule translation x : A; y : A Ixy = x(a)yhai . F w : 0; x : A . G v : 1; y : B w : 0; v : 1; z : A Omega B Omega x;y O z (F; G) w vz = xy(zhxyi(F wx k G vy) ....

....if there is no ambiguity, and refer to it as the D R graph of R. For further information, cf. 12, 6] Occasionally (e.g. in Section 5. 3 below) we will consider the system of sequent calculus for multiplicative linear logic, with the additional structural rule of Mix, also called direct logic DL [7, 11, 31, 4, 10]: mix : 0 1 0; 1 Definition. A proof structure R is a proof net for Direct Logic DL if for every switching s of R, the graph G s (R) is acyclic (but not necessarily connected. The following fundamental result (Girard [13] relates sequent calculus and proof nets for MLL. 19 Theorem 4 ....

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G. Bellin. Mechanizing Proof Theory: Resource-Aware Logics and Proof-Transformations to Extract Implicit Information, Phd Thesis, Stanford University. Available as: Report CST-80-91, June 1990, Dept. of Computer Science, Univ. of Edinburgh. 72

....of X. Remark. The computational complexity of Girard s no short trip condition and of Danos Regnier s requirement that all D R graphs be acyclic and con Subnets of proof nets in MLL Gamma 13 nected is clearly exponential on the size of the given proof structure. It is known (see, e.g. [3, 4, 1]) that there are procedures to decide whether or not a proof structure R for MLL Gamma is a proof net in time quadratic over the cardinality of R. 2.5 Permutability of Inferences in the Sequent Calculus Given a derivation D and two formula occurrences X 1 and X 2 in some sequents of D, if X 1 ....

G. Bellin. Mechanizing Proof Theory: Resource-Aware Logics and ProofTransformations to Extract Implicit Information, Phd Thesis, Stanford University. Available as: Report CST-80-91, June 1990, Dept. of Computer Science, Univ. of Edinburgh.

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G. Bellin. Mechanizing Proof Theory: Resource-Aware Logics and ProofTransformations to Extract Implicit Information. PhD thesis, Stanford University, 1990.

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G. Bellin. Mechanizing Proof Theory: Resource-Aware Logics and ProofTransformations to Extract Implicit Information. PhD thesis, Stanford University, 1990.

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G. Bellin. Mechanizing Proof Theory: Resource-Aware Logics and ProofTransformations to Extract Implicit Information. PhD thesis, Stanford University, 1990.

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G. Bellin. Mechanizing Proof Theory: Resource-Aware Logics and Proof-Transformations to Extract Implicit Information. PhD thesis, Stanford University, 1990.

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G. Bellin. Mechanizing Proof Theory: Resource-Aware Logics and ProofTransformations to Extract Implicit Information. PhD thesis, Stanford University, 1990.

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