17,456 research outputs found
Categorical Comprehensions and Recursion
A new categorical setting is defined in order to characterize the
subrecursive classes belonging to complexity hierarchies. This is achieved by
means of coercion functors over a symmetric monoidal category endowed with
certain recursion schemes that imitate the bounded recursion scheme. This gives
a categorical counterpart of generalized safe composition and safe recursion.Comment: Comments are welcom
Provably Total Primitive Recursive Functions: Theories with Induction
A natural example of a function algebra is R (T), the class of provably total computable functions (p.t.c.f.) of a theory T in the language of first order Arithmetic. In this paper a simple characterization of that kind of function algebras is obtained. This provides a useful tool for studying the class of primitive recursive functions in R (T). We prove that this is the class of p.t.c.f. of the theory axiomatized by the induction scheme restricted to (parameter free) Δ1(T)–formulas (i.e. Σ1–formulas which are equivalent in T to Π1–formulas).
Moreover, if T is a sound theory and proves that exponentiation is a total function, we characterize the class of primitive recursive functions in R (T) as a function algebra described in terms of bounded recursion (and composition). Extensions of this result are related to open problems on complexity classes. We also discuss an application to the problem on the equivalence between (parameter free) Σ1–collection and (uniform) Δ1–induction schemes in Arithmetic.
The proofs lean upon axiomatization and conservativeness properties of the scheme of Δ1(T)–induction and its parameter free version
The weak pigeonhole principle for function classes in S^1_2
It is well known that S^1_2 cannot prove the injective weak pigeonhole
principle for polynomial time functions unless RSA is insecure. In this note we
investigate the provability of the surjective (dual) weak pigeonhole principle
in S^1_2 for provably weaker function classes.Comment: 11 page
Restricted Space Algorithms for Isomorphism on Bounded Treewidth Graphs
The Graph Isomorphism problem restricted to graphs of bounded treewidth or
bounded tree distance width are known to be solvable in polynomial time
[Bod90],[YBFT99]. We give restricted space algorithms for these problems
proving the following results: - Isomorphism for bounded tree distance width
graphs is in L and thus complete for the class. We also show that for this kind
of graphs a canon can be computed within logspace. - For bounded treewidth
graphs, when both input graphs are given together with a tree decomposition,
the problem of whether there is an isomorphism which respects the
decompositions (i.e. considering only isomorphisms mapping bags in one
decomposition blockwise onto bags in the other decomposition) is in L. - For
bounded treewidth graphs, when one of the input graphs is given with a tree
decomposition the isomorphism problem is in LogCFL. - As a corollary the
isomorphism problem for bounded treewidth graphs is in LogCFL. This improves
the known TC1 upper bound for the problem given by Grohe and Verbitsky
[GroVer06].Comment: STACS conference 2010, 12 page
The parameterized space complexity of model-checking bounded variable first-order logic
The parameterized model-checking problem for a class of first-order sentences
(queries) asks to decide whether a given sentence from the class holds true in
a given relational structure (database); the parameter is the length of the
sentence. We study the parameterized space complexity of the model-checking
problem for queries with a bounded number of variables. For each bound on the
quantifier alternation rank the problem becomes complete for the corresponding
level of what we call the tree hierarchy, a hierarchy of parameterized
complexity classes defined via space bounded alternating machines between
parameterized logarithmic space and fixed-parameter tractable time. We observe
that a parameterized logarithmic space model-checker for existential bounded
variable queries would allow to improve Savitch's classical simulation of
nondeterministic logarithmic space in deterministic space .
Further, we define a highly space efficient model-checker for queries with a
bounded number of variables and bounded quantifier alternation rank. We study
its optimality under the assumption that Savitch's Theorem is optimal
A Formalization of Polytime Functions
We present a deep embedding of Bellantoni and Cook's syntactic
characterization of polytime functions. We prove formally that it is correct
and complete with respect to the original characterization by Cobham that
required a bound to be proved manually. Compared to the paper proof by
Bellantoni and Cook, we have been careful in making our proof fully contructive
so that we obtain more precise bounding polynomials and more efficient
translations between the two characterizations. Another difference is that we
consider functions on bitstrings instead of functions on positive integers.
This latter change is motivated by the application of our formalization in the
context of formal security proofs in cryptography. Based on our core
formalization, we have started developing a library of polytime functions that
can be reused to build more complex ones.Comment: 13 page
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