We use cookies to distinguish you from other users and to provide you with a better experience on our websites. Close this message to accept cookies or find out how to manage your cookie settings.
We use cookies to distinguish you from other users and to provide you with a better experience on our websites. Close this message to accept cookies or find out how to manage your cookie settings.
Published online by Cambridge University Press: 02 December 2009
In this chapter we shall discuss the complexity of Frege systems without any restrictions on the depth. There is some nontrivial information, in particular nontrivial upper bounds, but no nontrivial lower bounds are known at present (only bounds from Lemma 4.4.12).
Counting in Frege systems
Theorems 9.1.5 and 9.1.6 are useful sufficient conditions guaranteeing the existence of the polynomial size EF-proofs and of quasipolynomial size F-proofs, respectively. For example, U11 proves the pigeonhole principle PHP(R) and hence there are quasipolynomial size F-proofs of PHPn. A subtheory of corresponding to the polynomial size F-proofs, based on a version of inductive definitions, was considered by Arai (1991); see Section 9.6. Its axiomatization however, stresses a logical construction, whereas we would like a theory based on a more combinatorial principle.
The most important property of a Frege system relevant for the upper bounds is that it can count. We shall make this precise by showing that F simulates an extension of I△0(α) by counting functions, and that F p-simulates a propositional proof system cutting planes.
Definition 13.1.1.
(a) Let L0 be the language of the second order bounded arithmetic but without the symbol #.
To save this book to your Kindle, first ensure no-reply@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about saving to your Kindle.
Note you can select to save to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be saved to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.
Find out more about the Kindle Personal Document Service.
To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Dropbox.
To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Google Drive.
