scholarly journals KLEENE'S NORMAL FORM THEOREM FOR ARITHMETICAL PETRI NETS

Author(s):  
Z.R. K\"{o}nigsberg
1982 ◽  
Vol 47 (3) ◽  
pp. 605-624 ◽  
Author(s):  
Douglas N. Hoover

AbstractWe show that every formula of Lω1P is equivalent to one which is a propositional combination of formulas with only one quantifier. It follows that the complete theory of a probability model is determined by the distribution of a family of random variables induced by the model. We characterize the class of distribution which can arise in such a way. We use these results together with a form of de Finetti’s theorem to prove an almost sure interpolation theorem for Lω1P.


1979 ◽  
Vol 44 (3) ◽  
pp. 289-306 ◽  
Author(s):  
Victor Harnik

The central notion of this paper is that of a (conjunctive) game-sentence, i.e., a sentence of the formwhere the indices ki, ji range over given countable sets and the matrix conjuncts are, say, open -formulas. Such game sentences were first considered, independently, by Svenonius [19], Moschovakis [13]—[15] and Vaught [20]. Other references are [1], [3]—[5], [10]—[12]. The following normal form theorem was proved by Vaught (and, in less general forms, by his predecessors).Theorem 0.1. Let L = L0(R). For every -sentence ϕ there is an L0-game sentence Θ such that ⊨′ ∃Rϕ ↔ Θ.(A word about the notations: L0(R) denotes the language obtained from L0 by adding to it the sequence R of logical symbols which do not belong to L0; “⊨′α” means that α is true in all countable models.)0.1 can be restated as follows.Theorem 0.1′. For every-sentence ϕ there is an L0-game sentence Θ such that ⊨ϕ → Θ and for any-sentence ϕ if ⊨ϕ → ϕ and L′ ⋂ L ⊆ L0, then ⊨ Θ → ϕ.(We sketch the proof of the equivalence between 0.1 and 0.1′.0.1 implies 0.1′. This is obvious once we realize that game sentences and their negations satisfy the downward Löwenheim-Skolem theorem and hence, ⊨′α is equivalent to ⊨α whenever α is a boolean combination of and game sentences.


1991 ◽  
Vol 56 (1) ◽  
pp. 129-149 ◽  
Author(s):  
Gunnar Stålmarck

In this paper we prove the strong normalization theorem for full first order classical N.D. (natural deduction)—full in the sense that all logical constants are taken as primitive. We also give a syntactic proof of the normal form theorem and (weak) normalization for the same system.The theorem has been stated several times, and some proofs appear in the literature. The first proof occurs in Statman [1], where full first order classical N.D. (with the elimination rules for ∨ and ∃ restricted to atomic conclusions) is embedded in a system for second order (propositional) intuitionistic N.D., for which a strong normalization theorem is proved using strongly impredicative methods.A proof of the normal form theorem and (weak) normalization theorem occurs in Seldin [1] as an extension of a proof of the same theorem for an N.D.-system for the intermediate logic called MH.The proof of the strong normalization theorem presented in this paper is obtained by proving that a certain kind of validity applies to all derivations in the system considered.The notion “validity” is adopted from Prawitz [2], where it is used to prove the strong normalization theorem for a restricted version of first order classical N.D., and is extended to cover the full system. Notions similar to “validity” have been used earlier by Tait (convertability), Girard (réducibilité) and Martin-Löf (computability).In Prawitz [2] the N.D. system is restricted in the sense that ∨ and ∃ are not treated as primitive logical constants, and hence the deductions can only be seen to be “natural” with respect to the other logical constants. To spell it out, the strong normalization theorem for the restricted version of first order classical N.D. together with the well-known results on the definability of the rules for ∨ and ∃ in the restricted system does not imply the normalization theorem for the full system.


2019 ◽  
Vol 375 (3) ◽  
pp. 2089-2153 ◽  
Author(s):  
Luca Biasco ◽  
Jessica Elisa Massetti ◽  
Michela Procesi

1981 ◽  
Vol 46 (2) ◽  
pp. 240-248 ◽  
Author(s):  
Daniel Leivant

In classical arithmetic a natural measure for the complexity of relations is provided by the number of quantifier alternations in an equivalent prenex normal form. However, the proof of the Prenex Normal Form Theorem uses the following intuitionistically invalid rules for permuting quantifiers with propositional constants.Each one of these schemas, when added to Intuitionistic (Heyting's) Arithmetic IA, generates full Classical (Peano's) Arithmetic. Schema (3) is of little interest here, since one can obtain a formula intuitionistically equivalent to A ∨ ∀xBx, which is prenex if A and B are:For the two conjuncts on the r.h.s. (1) may be successively applied, since y = 0 is decidable.We shall readily verify that there is no way of similarly going around (1) or (2). This fact calls for counting implication (though not conjunction or disjunction) in measuring in IA the complexity of arithmetic relations. The natural implicational measure for our purpose is the depth of negative nestings of implication, defined as follows. I(F): = 0 if F is atomic; I(F ∧ G) = I(F ∨ G): = max[I(F), I(G)]; I(∀xF) = I(∃xF): = I(F); I(F → G):= max[I(F) + 1, I(G)].


2007 ◽  
Vol 17 (05n06) ◽  
pp. 951-961 ◽  
Author(s):  
L. A. BOKUT ◽  
V. V. CHAYNIKOV ◽  
K. P. SHUM

In this paper we will present the results of Artin–Markov on braid groups by using the Gröbner–Shirshov basis. As a consequence we can reobtain the normal form of Artin–Markov–Ivanovsky as an easy corollary.


2007 ◽  
Vol 17 (08) ◽  
pp. 1577-1592 ◽  
Author(s):  
JONATHAN D. H. SMITH

Evans defined quasigroups equationally, and proved a Normal Form Theorem solving the word problem for free extensions of partial Latin squares. In this paper, quasigroups are redefined as algebras with six basic operations related by triality, manifested as coupled right and left regular actions of the symmetric group on three symbols. Triality leads to considerable simplifications in the proof of Evans' Normal Form Theorem, and makes it directly applicable to each of the six major varieties of quasigroups defined by subgroups of the symmetric group. Normal form theorems for the corresponding varieties of idempotent quasigroups are obtained as immediate corollaries.


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