F. Pfenning and H. C. Wong, On a modal lambda-calculus for S4, Electronic Notes in Computer Science, vol. 1 (1995). Home/Search Document Details and Download
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Explicit Provability and Constructive Semantics - Artemov (2000) (1 citation) (Correct)
....with abstraction. Since modal logic S4 and all standard term constructors can be represented by proof polynomials, the Logic of Proofs can also emulate modal calculi. As it was shown in [8] 11] the intuitionistic version of LP naturally realizes the modal calculus for IS4 ( 23] 68] [84], cf. also [27] and thus supplies modal terms with standard provability semantics. This EXPLICIT PROVABILITY 31 result may be considered as a more general abstract version of the CurryHoward isomorphism which relates terms types with proofs formulas. x11. First order case. Theories based on the ....
F. Pfenning and H.C. Wong, On a modal lambda-calculus for s4, Electronic Notes in Computer Science, vol. 1 (1995).
Uniform Provability Realization of Intuitionistic Logic, Modality .. - Artemov (1999) (Correct)
....Roughly speaking, LP is an advanced system of combinatory logic that accommodates not only the application operation, but also proof checker and choice . These operations subsume the simply typed calculus together with the modal logic S4, and thus the entire modal calculus ( 4] [18]) In particular, LP creates an environment where modality and terms are objects of the same nature, namely proof polynomials. Another way to look at it: modal logic is a forgetful projection of a combinatory logic with dependent types enriched by the operations proof checker and choice . ....
F. Pfenning and H.C. Wong, On a modal lambda-calculus for S4, \Electronic Notes in Computer Science", v.1, 1995.
Deep Isomorphism of Modal Derivations and λ-Terms - Artemov (1999) (Correct)
....can also be realized as admissible rules in LPGi (cf. 2] 3] Since both modal logic IS4 and all standard term constructors can be emulated by proof polynomials, LPi can also emulate modal calculi. As it was shown in [2] 3] LPGi naturally realizes the modal calculus for IS4 ( 4] 6] [7], cf. also [5] 6 Deep realization of modalities by combinatory ( terms Realization algorithm from Section 4 recovers combinatory terms for every occurrence of modalities in any IS4 derivation. Natural fragments of S4 may be be now regarded as implicit description of the corresponding ....
....is a typical example. Consider the sequent 2F ) 22F derivable in IS4. There is no IS4 derivation of this sequent that ends with the necessitation rule F ) 2F 2F ) 22F ; since F ) 2F is not derivable in IS4. Hence there is no modal calculus realization of 2F ) 22F in the sense of [4] 6] [7], i.e. there is no modal term t(x) such that the modal calculus derives x : F ) t(x) 2F . On the other hand, the formula 2F ) 22F admits a relization in LPi, namely x : F ) x : x : F where x is the proof checker polynomial. 7 Standard provability interpretation of LPi Within this section ....
F. Pfenning and H.C. Wong, \On a modal lambda-calculus for S4", Electronic Notes in Computer Science 1, 1995.
Explicit Provability And Constructive Semantics - Artemov (2001) (1 citation) (Correct)
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F. Pfenning and H. C. Wong, On a modal lambda-calculus for S4, Electronic Notes in Computer Science, vol. 1 (1995).
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