A reader who is mainly interested in the reflection principles themselves can skip $5 2-5 and 5 8 (which deal with the complexity of classical first order axiomatic systems). Conversely, these latter sections are self contained: t o make them so, we have occasionally repeated definitions from earlier sections. 3 2 contains the general theorem by one of us promised by the other in [27], p. 50.Scctions 6-7, which presuppose some knowledge of "delicate" proof theory as contained in [44], establish sharp results on (extensions of) first order arithmetic and on subsystems of classical analysis, again with applications to questions on the complexity of axiomatisations.We emphasize that the methods and results of this paper are not confined t o recursively enumerable axiom systems based on classical logic. On the contrary we consider non-recursively enumerable sets of axioms (e.g., 5 2), infinite proof figures and thus non-recursive proof relations (in $5 6-7), and both finite and infinite proof figures based on intuitionistic rules of proof ( § 10). These different cases present not only common features, but quite systematic differences, so that it seems reasonable to look for a general proof theory which covers (at least) the cases here considered.We do not know such a theory, but remark occasionally on the implications of our results for such a theory, and also on the kind of information that one should expect from it.One such desideratum is an exact analysis of the intuitive idea: the formula P together with given rules of proof is a natural or canonical description (of the relation defined by F in the intended interpretation of the system considered). For a discussion of the importance of this idea applied t o definitions of syntactic relations, see [6]. Our interim solution of this problem is to state derivability conditions on P , following [13] in connection with GODEL'S two incompleteness theorems. As in [13], our conditions are in fact satisfied by the natural definition for the syntactic relations of the axiomatic systems here considered. The matter is discussed in $ 6 .Note that the prcsentation of the formal system and the (canonical) definition of its proof relation enters into the very formulation of a reflection principle; in contrast to the results on the complexity of axiomatic systems which refer only to the set of theorems of the axiomatic system (and ncither to the presentation of the axioms nor to the rules of deduction). For this reason we shall identify an axiomatic system with its set of theorems in QQ 2-5 and Q 8 of this paper, but not elsewhere.l) The work Introduction Throughout most of the paper S will denote a formal system in which arithmetic can be developed to the extent that one can define all primitive recursive functions and prove those of their properties which are provable in primitive recursive arithmetic, and their consequences in first order classical or intuitionistic logic. Indeed, throughout almost the whole discussion the reader can think of S as containing firit order arithm...
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When termination of a program is observable by an adversary, confidential information may be leaked by terminating accordingly. While this termination covert channel has limited bandwidth for sequential programs, it is a more dangerous source of information leakage in concurrent settings. We address concurrent termination and timing channels by presenting a dynamic information-flow control system that mitigates and eliminates these channels while allowing termination and timing to depend on secret values. Intuitively, we leverage concurrency by placing such potentially sensitive actions in separate threads. While termination and timing of these threads may expose secret values, our system requires any thread observing these properties to raise its information-flow label accordingly, preventing leaks to lower-labeled contexts. We implement this approach in a Haskell library and demonstrate its applicability by building a web server that uses information-flow control to restrict untrusted web applications.
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