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Two distinct proofs of an exponential separation between regular resolution and unrestricted resolution are given. The previous best known separation between these systems was quasi-polynomial.

An exponential lower bound for the size of tree-like cutting planes refutations of a certain family of conjunctive normal form (CNF) formulas with polynomial size resolution refu-tations is proved. This implies an exponential separation between the tree-like versions and the dag-like versions of resolution and cutting planes. In both cases only… (More)

The problem of converting deterministic finite automata into (short) regular expressions is considered. It is known that the required expression size is 2 Θ(n) in the worst case for infinite languages, and for finite languages it is n Ω(log log n) and n O(log n) , if the alphabet size grows with the number of states n of the given automaton. A new lower… (More)

We define theories of Bounded Arithmetic characterizing classes of functions computable by constant-depth threshold circuits of polynomial and quasipoly-nomial size. Then we define certain second-order theories and show that they characterize the functions in the Counting Hierarchy. Finally we show that the former theories are isomorphic to the latter via… (More)

A typed lambda calculus with recursion in all finite types is defined such that the first order terms exactly characterize the parallel complexity class NC. This is achieved by use of the appropriate forms of recursion (concatenation recursion and logarithmic recursion), a ramified type structure and imposing of a linearity constraint.

A fragment of second-order lambda calculus (System <i>F</i>) is defined that characterizes the elementary recursive functions. Type quantification is restricted to be noninterleaved and stratified, that is, the types are assigned levels, and a quantified variable can only be instantiated by a type of smaller level, with a slightly liberalized treatment of… (More)

The systems K of transsnite cumulative types up to are extended to systems K 1 that include a natural innnitary inference rule, the so-called limit rule. For countable a semantic completeness theorem for K 1 is proved by the method of reduction trees, and it is shown that every model of K 1 is equivalent to a cumulative hierarchy of sets. This is used to… (More)

An exponential lower bound is proved for tree-like Cutting Planes refutations of a set of clauses which has polynomial size resolution refutations. This implies an exponential separation between tree-like and dag-like proofs for both Cutting Planes and resolution; in both cases only superpolynomial separations were known before 33, 22, 11]. In order to… (More)

Resolution refinements called w-resolution trees with lemmas (WRTL) and with input lemmas (WRTI) are introduced. Dag-like resolution is equivalent to both WRTL and WRTI when there is no regularity condition. For regular proofs, an exponential separation between regular dag-like resolution and both regular WRTL and regular WRTI is given. It is proved that… (More)

We deene a property of substructures of models of arithmetic, that of being length-initial , and show that sharply bounded formulae are absolute between a model and its length-initial submodels. We use this to prove independence results for some weak fragments of bounded arithmetic by constructing appropriate models as length-initial submodels of some given… (More)