V. Krupski, Operational logic of proofs with functionality condition on proof predicate, Logical foundations of Computer Science '97, Yaroslavl' (S. Adian and A. Nerode, editors), Lecture Notes in Computer Science, vol. 1234, Springer-Verlag, 1997, pp. 167--177.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....only the end formula (sequent) of a proof tree. On the other hand, the same systems may be regarded as multi conclusion by assuming that a proof derives all formulas assigned to the nodes of the proof tree. The logic of strictly single conclusion proof systems was studied in [6] 15] and in [60], 61] where it received a complete axiomatization (system FLP) However, FLP is not compatible with any modal logic. For example, FLP derives : x : x : which has the forgetful projection : 2 2( The latter is false in any normal modal logic. Therefore, provability as a modal operator ....

....CS: Comment 6.8. The standard provability semantics for LP above may be characterized as a call by value semantics, since the evaluation F of a given LP formula F depends upon the value of participating functions. A call by name provability semantics for LP was introduced in [7] and then used in [60], 61] 90] 105] In the latter semantics F depends upon the particular programs for the functions participating in . x7. A sequent formulation of Logic of Proofs. By sequent we mean a pair Gamma ) Delta, where Gamma and Delta are finite multisets of LP formulas. For Gamma; F we mean ....

V. Krupski, Operational logic of proofs with functionality condition on proof predicate, Logical foundations of computer science' 97, Yaroslavl', SpringerVerlag, 1997, LNCS Vol. 1234 (S. Adian and A. Nerode, editors), pp. 167--177.

....derives only the end formula (sequent) of a proof tree. On the other hand, the same systems may be regarded as multiconclusion by assuming that a proof derives all formulas assigned to the nodes of the proof tree. The logic of strictly single conclusion proof systems was studied in [2] 3] and in [42], where it meets a complete axiomatization (system FLP) FLP is not compatible with any modal logic (cf. Comment 8.5) and thus is not directly relevant to the problem of finding an intended semantics for the modal logic of provability. Therefore, provability as a modal operator corresponds to ....

..... 5.8 Comment. The standard provability semantics for LP above may be characterized as a call by value semantics, since the evaluation F of a given LP formula F depends upon the value of participating functions. A call by name provability semantics for LP was introduced in [4] and then used in [42], 64] In the latter semantics F depends upon the particular programs for the functions participating in . In order to define the call by name provability semantics for LP we assume that PA has the standard set of tools to introduce terms. We use a new functional symbol z: z) for each ....

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V.N. Krupski, "Operational Logic of Proofs with Functionality Condition on Proof Predicate ", Lecture Notes in Computer Science, v. 1234, Logical Foundations of Computer Science' 97, Yaroslavl', pp. 167-177, 1997

....proofs s and t. The free fragment LP Gamma of the logic of proofs deserves separate attention as a neutral logic of proofs. LP Gamma does not specify the determinacy of a proof predicate. LP Gamma can by expanded to the operational logic of functional proofs (FLP) developed in [12] by adding a special functionality axiom which by means of unification captures on the propositional level the deterministic character of a proof predicate (cf. the system F from [1] Logic of (nondeterministic) proofs LP Logic of deterministic proofs FLP Neutral logic of proofs LP Gamma ....

....point of view, LP gives system independent sufficient conditions for a logic theory to contain definable terms. To represent the usual calculus it suffices to have application of proof terms operation only, proofs of certain propositional axioms, 2 In the Functional Logic of Proofs FLP ([12]) a proof term t has an exact type (a formula, proven by t) 16 no matter in what system, and to enjoy some trivial closure properties, like the deduction theorem. Acknowledgements. I am greatly indebted to Anil Nerode for his constant encouragement and support. It is a pleasure to acknowledge ....

V.N. Krupski, "Operational Logic of Proofs with Functionality Condition on Proof Predicate ", Lecture Notes in Computer Science, v. 1234, Logical Foundations of Computer Science' 97, Yaroslavl', pp. 167-177, 1997

....a proof derives only the end formula (sequent) of a proof tree. On the other hand, the same systems may be regarded as non deterministic by assuming that a proof derives all formulas assigned to the nodes of the proof tree. The logic of strictly deterministic proof systems was studied in [2] 3] [27], where it meets a complete axiomatization (system FLP) FLP is not compatible with any modal logic (cf. Comment 6.6) and thus is not directly relevant to the problem of finding an intended semantics for the modal logic of provability. Therefore, provability as a modal operator corresponds to ....

....2.8 Comment. The standard provability semantics for LP above may be characterized as a call by value semantics, since the evaluation F of a given LP formula F depends upon the value of participating functions. A call by name provability semantics for LP was introduced in [4] and then used in [27], 45] In the latter semantics F depends upon the particular programs for the functions participating in . In order to define the call by name provability semantics for LP we assume that PA has the standard set of tools to introduce so called terms. We use a new functional symbol z (z) ....

[Article contains additional citation context not shown here]

V.N. Krupski, "Operational Logic of Proofs with Functionality Condition on Proof Predicate ", Lecture Notes in Computer Science, v. 1234, Logical Foundations of Computer Science' 97, Yaroslavl', pp. 167-177, 1997

.... (by assuming that a proof derives only the end formula sequent of a proof tree) or as non deterministic (by assuming that a proof derives all intermediate formulas assigned to the nodes of the proof tree) The logic of strictly deterministic proof systems was studied in [1] 2] and in [15], where it meets a complete axiomatization (system FLP) 3 Realization of modal logic in LP It is easy to see that a forgetful projection of LP is correct with respect to S4. Let F o be the result of substituting 2X for all occurrences of t : X in F , and Gamma o = fF o j F 2 Gammag for ....

V.N. Krupski, "Operational Logic of Proofs with Functionality Condition on Proof Predicate ", Lecture Notes in Computer Science, v. 1234, Logical Foundations of Computer Science' 97, Yaroslavl', pp. 167-177, 1997

....derives only the end formula (sequent) of a proof tree. On the other hand, the same systems may be regarded as multi conclusion by assuming that a proof derives all formulas assigned to the nodes of the proof tree. The logic of strictly uni conclusion proof systems was studied in [2] 3] and in [26], where it meets a complete axiomatization (system FLP) FLP is not compatible with any modal logic (cf. 7] Therefore, provability as a modal operator corresponds to multi conclusion proof systems. No single operator t : in LP is a normal modality since none of them satisfies the property t ....

V.N. Krupski, "Operational Logic of Proofs with Functionality Condition on Proof Predicate ", Lecture Notes in Computer Science, v. 1234, Logical Foundations of Computer Science' 97, Yaroslavl', pp. 167-177, 1997

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V. Krupski, Operational logic of proofs with functionality condition on proof predicate, Logical foundations of Computer Science '97, Yaroslavl' (S. Adian and A. Nerode, editors), Lecture Notes in Computer Science, vol. 1234, Springer-Verlag, 1997, pp. 167--177.

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