Theoretic Strength Research Articles

Intuitionistic Choice and Restricted Classical Logic

Recently, Coquand and Palmgren considered systems of intuitionistic arithmetic in all finite types together with various forms of the axiom of choice and a numerical omniscience schema (NOS) which implies classical logic for arithmetical formulas. Feferman subsequently observed that the proof theoretic strength of such systems can be determined by functional interpretation based on a non-constructive μ-operator and his well-known results on the strength of this operator from the 70’s. In this note we consider a weaker form LNOS (lesser numerical omniscience schema) of NOS which suffices to derive the strong form of binary Konig’s lemma studied by Coquand/Palmgren and gives rise to a new and mathematically strong semi-classical system which, nevertheless, can proof theoretically be reduced to primitive recursive arithmetic PRA. The proof of this fact relies on functional interpretation and a majorization technique developed in a previous paper. ∗Basic Research in Computer Science, Centre of the Danish National Research Foundation.

On the strength of Ramsey's theorem for pairs

AbstractWe study the proof–theoretic strength and effective content of the infinite form of Ramsey's theorem for pairs. Let RTkndenote Ramsey's theorem fork–colorings ofn–element sets, and let RT<∞ndenote (∀k)RTkn. Our main result on computability is: For anyn≥ 2 and any computable (recursive)k–coloring of then–element sets of natural numbers, there is an infinite homogeneous setXwithX″ ≤T0(n). LetIΣnandBΣndenote the Σninduction and bounding schemes, respectively. Adapting the casen= 2 of the above result (whereXis low2) to models of arithmetic enables us to show that RCA0+IΣ2+ RT22is conservative over RCA0+IΣ2for Π11statements and that RCA0+IΣ3+ RT<∞2is Π11-conservative over RCA0+IΣ3. It follows that RCA0+ RT22does not implyBΣ3. In contrast, J. Hirst showed that RCA0+ RT<∞2does implyBΣ3, and we include a proof of a slightly strengthened version of this result. It follows that RT<∞2is strictly stronger than RT22overRCA0.

Infinite Versions of Some Problems from Finite Complexity Theory

Recently, several authors have explored the connections between NP-complete problems for finite objects and the complexity of their analogs for infinite objects. In this paper, we will categorize infinite versions of several problems arising from finite complexity theory in terms of their recursion theoretic complexity and proof theoretic strength. These infinite analogs can behave in a variety of unexpected ways. Startling parallels exist between the computational complexity of certain graph theoretic problems and the recursion theoretic complexity and proof theoretic strength of their infi- nite analogs. For example, the problem of deciding which finite graphs have an Euler path is known to be P-time computable (9), and Beigel and Gasarch (4) have shown that the problem of deciding which infinite recursive graphs have an Euler path is arithmetical. By contrast, the problem of deciding which finite graphs have Hamilton paths is NP-complete (8), and Harel (6) has shown that the problem of deciding which infinite recursive graphs have a Hamilton graph is � 1 complete. Thus, the possibly greater computational complex- ity is paralleled by a demonstrable increase in recursion theoretic complexity. This pattern can also be seen through an application of the techniques of reverse mathematics. The

Evaluation of impact resistance and brittleness of structural ceramics

Abstract In this paper, a definition and calculation method of impact resistance and brittleness and impact modulus are presented, and the impact resistances of different ceramics are calculated. It is pointed out that the impact fracture energy includes two parts, i.e. the cracking energy and crack extension energy. The criterion for the toughening effect of brittle materials, as well as the relationship between impact strength and theoretic strength, are discussed. A series of experiments was carried out to prove the validity of these theories.

Some general incompleteness results for partial correctness logics

It is known that incompleteness of Hoare's logic relative to certain data type specifications can occur due to the ability of partial correctness assertions to code unsolvable problems; cf. Andréka, Németi, and Sain (1979, Lecture Notes in Computer Science Vol. 74, pp. 208–218, Springer-Verlag, New York/Berlin) and Bergstra and Tucker (1982, Theoret. Comput. Sci. 17 , 303–315). We improve what we think are the main known theorems of this kind, showing that they depend only on very weak assumptions on the data type specification (ensuring the ability to simulate arbitrarily long finite initial segments of the natural numbers with successor), and pointing out that the recursion theoretic strength of the obtained results can be increased.