On the Decidability of the U4 Logic System

Theoretical Computer Science, 2007

Computability logic is a formal theory of computational tasks and resources. Formulas in it represent interactive computational problems, and "truth" is understood as algorithmic solvability. Interactive computational problems, in turn, are defined as a certain sort games between a machine and its environment, with logical operators standing for operations on such games. Within the ambitious program of finding axiomatizations for incrementally rich fragments of this semantically introduced logic, the earlier article "From truth to computability I" proved soundness and completeness for system CL3, whose language has the so called parallel connectives (including negation), choice connectives, choice quantifiers, and blind quantifiers. The present paper extends that result to the significantly more expressive system CL4 with the same collection of logical operators. What makes CL4 expressive is the presence of two sorts of atoms in its language: elementary atoms, representing elementary computational problems (i.e. predicates, i.e. problems of zero degree of interactivity), and general atoms, representing arbitrary computational problems. CL4 conservatively extends CL3, with the latter being nothing but the general-atom-free fragment of the former. Removing the blind (classical) group of quantifiers from the language of CL4 is shown to yield a decidable logic despite the fact that the latter is still first-order.

2014

The problem of finding a mediator to compose secured services has been reduced in our former work to the problem of solving deducibility constraints similar to those employed for cryptographic protocol analysis. We extend in this paper the mediator synthesis procedure by a construction for expressing that some data is not accessible to the mediator. Then we give a decision procedure for verifying that a mediator satisfying this non-disclosure policy can be effectively synthesized. This procedure has been implemented in CL-AtSe, our protocol analysis tool. The procedure extends constraint solving for cryptographic protocol analysis in a significative way as it is able to handle negative deducibility constraints without restriction. In particular it applies to all subterm convergent theories and therefore covers several interesting theories in formal security analysis including encryption, hashing, signature and pairing.

Fundamenta Informaticae

We examine the issue of collecting and processing information from various sources, which involves handling incomplete and inconsistent information. Inspired by the framework first proposed by Belnap, we consider structures consisting of information sources which provide information about the values of formulas of classical propositional logic, and a processor which collects that information and extends it by deriving conclusions following from it according to the truth tables of classical logic, applied forward and backward. Our model extends Belnap's in allowing the sources to provide information also about complex formulas. As that framework cannot be captured by finite ordinary logical matrices, we use Nmatrices for that purpose. In opposition to the approach proposed in our earlier work, we assume that the information sources are reasonable, i.e. that they provide information consistent with certain coherence rules. We provide sound and complete sequent calculi admitting strong cut elimination for the logic of a single information source, and (several variants of) the logic generated by the source and processor structures described above. In doing this, we also provide new characterizations for some known logics. We prove that, in opposition to the variant with unconstrained information sources considered earlier, the latter logic cannot be generated by structures with any bounded number of sources.

1996

The language we consider is that of classical first order logic augmented with the unary modal operator □. Sentences of this language are regarded as true or false in a knowledge-base KB, which is any finite set of □-free formulas. Truth of □α in KB is understood as that α is true in all classical models of KB and this interpretation is intended to capture the intuition "we know that α" behind □α. The resulting logic is, in general, undecidable and not even semidecidable. However, there is a natural fragment of the above language, called the constructive language, which yields a decidable logic. The only syntactic constraint in the constructive language is that there exists x should always be followed by □. That is, we are not allowed to simply say "there is x such that ..." and we can only say "there is x for which we know that ...". Under this constraint, truth of there existsxα(x) will always imply that an object x for which α(x) holds not only exists, but can be effectively found. This is generally what we want of there exists in practical applications: knowing that "there exists a combination c that opens safe S" has no significance unless such a combination c can actually be found, which, in our semantics, will be equivalent to saying that there is c for which we know that c opens S. So, it is only truth of the sentence there existsc□OPENS(c,S) that really matters, and the latter, unlike there existsc□OPENS(c,S) is a perfectly legal formula of the constructive language. I introduce a decidable sequent system C K N in the constructive language and prove its soundness and completeness with respect to the above semantics.

Information and Computation, 2011

Verification problems can often be encoded as first-order validity or satisfiability problems. The availability of efficient automated theorem provers is a crucial prerequisite for automating various verification tasks as well as their cooperation with specialized decision procedures for selected theories, such as Presburger Arithmetic. In this paper, we investigate how automated provers based on a form of equational reasoning, called paramodulation, can be used in verification tools. More precisely, given a theory T axiomatizing some data structure, we devise a procedure to answer the following questions. Is the satisfiability problem of T decidable by paramodulation? Can a procedure based on paramodulation for T be efficiently combined with other specialized procedures by using the Nelson-Oppen schema? Finally, if paramodulation decides the satisfiability problem of two theories, does it decide satisfiability in their union? The procedure capable of answering all questions above is based on Schematic Saturation; an inference system capable of over-approximating the inferences of paramodulation when solving satisfiability problems in a given theory T. Clause schemas derived by Schematic Saturation describe all clauses derived by paramodulation so that the answers to the questions above are obtained by checking that only finitely many different clause schemas are derived or that certain clause schemas are not derived.