To Clark . ii
نویسنده
چکیده
In this thesis we study a formal first order system T in the standard language L, of Gentzen’s LK, see [Tak87]. T(tind) extends LK by the following valid first-order inference rule (A is quantifier-free). Γ,A(a), Λ→ ∆,A(s(a)), Θ Γ,A(0), Λ→ ∆,A(sn(0)), Θ (tind) This rule is called term induction, it derives a restricted term built from successor s and the constant 0. We call such terms numerals. To characterise the difference between T(tind) and pure logic, we employ proof theoretic methods. Firstly we establish a variant of Herbrand’s Theorem for T(tind). Let ∃x̄F (x̄) be a Σ1 formula; provable by Π. Then there exists a disjunction ∨N i1 · · · ∨N il M1(s1(0), . . . , sil(0)) ∨ · · · ∨Mm(s1(0), . . . , sil(0)), denoted by H that is valid for some N ∈ IN, furthermore the Mi are instances of F (ā). In T(tind) it is not possible to bound the length of Herbrand disjunctions in terms of proof length and logical complexity of the end-formula as usual. The main result is that we can bound the length of the {s, 0}-matrix of the above disjunctions in this way. The {s, 0}-matrix M1(a1, . . . , al) ∨ · · · ∨Mm(a1, . . . , al) is defined by replacing the indicated numerals in H by pairwise distinct free variables. Secondly we study the possibility to generalise a given proof deriving an endformula A(sn(0)) such that its logical form is preserved. If the initial proof admits only propositional cuts and all term induction inferences are non-trivial, then we can solve this problem satisfactorily. Assume Π fulfills the restrictions and Σ denotes a suitable partial proof description of the logical form of Π. Assume furthermore that n is large relative to the size of Σ, the term depth of A(a) and parameters from the language L. Then there exist generalisations of Π proving A(sn+r·k(0)) for some r ∈ IN and all k ∈ IN. If the initial proof Π does not fulfill the restriction, we can always transform Π to a proof Π ′ of the same endformula fulfilling the necessary properties. Thirdly we investigated the validity problem for disjunctions H of the form above. We see that already the presence of two ’big’ disjunction in H is sufficient to reduce the problem to the general halting problem. However if we restrict our attention to disjunction of a particular simple form H with parameter sn(0). Then the problem whether H(sn(0)) is valid for all n, remains decidable. Furthermore we establish (with restrictions) the reverse implication for the variant of Herbrand’s Theorem established earlier.
منابع مشابه
Detection of power oscillation and simultaneous faults using Clark transform
Distance relays are widely used to protect transmission lines. Sometimes, in these lines due to the occurrence of the oscillation of the power, the impedance calculated in the distance relay enters into its functional zones and leads to the cutting off of the lines. This issue can cause global power outages. Accordingly, in this paper, a Clark-based method for detecting the oscillation of power...
متن کاملA Novel Heuristic Algorithm Based on Clark and Wright Algorithm for Green Vehicle Routing Problem
A significant portion of Gross Domestic Production (GDP) in any country belongs to the transportation system. Transportation equipment, in the other hand, is supposed to be great consumer of oil products. Many attempts have been assigned to the vehicles to cut down Greenhouse Gas (GHG). In this paper a novel heuristic algorithm based on Clark and Wright Algorithm called Green Clark and Wright (...
متن کاملSimple Sentences , Substitutions , and
Many competent speakers initially judge that (i) is true and (ii) is false, though they know that (iii) is true. (i) Superman leaps more tall buildings than Clark Kent. (ii) Superman leaps more tall buildings than Superman. (iii) Superman is identical with Clark Kent. Semantic explanations of these intuitions say that (i) and (ii) really can differ in truthvalue. Pragmatic explanations deny thi...
متن کاملارزیابی روایی ابعاد الگوی سه بخشی اضطراب و افسردگی
AbstractObjectives: The purpose of the present study was to assess the validity of Clark and Watson’s tripartite theory of anxiety and depression and to assess the differences between the two genders with regard to social anxiety, general anxiety, and depression in a sample of adolescents. Method: In a descriptive cross-sectional study, 403 secondary students (225 girls, 178 boys) who were s...
متن کاملThe effect of bicarbonate on iron (Fe) and zinc (Zn) uptakes by soybean varieties
Bicarbonate anion, has a very high concentration in the calcareous soil, and in fact, is the main cause of chlorosis in calcareous soils. This study was conducted to evaluate eight soybean cultivars with iron deficiency and bicarbonate presence. The present research was carried out with the aim of assessing the effect of irrigation with water containing bicarbonate on enzyme activities and Fe a...
متن کاملکارآیی آبنمود واحد لحظه ای Clark در بازسازی آبنمود واحد سیل
تحقیق حاضر به منظور تعیین میزان کارایی آبنمود واحد بدست آمده براساس روش Clarkدر بازسازی آبنمود واحد مشاهدهای ایستگاه مرغک واقع در خروجی حوزه آبخیز بازفت در استان چهار محال و بختیاری انجام گردید. نتایج ارزیابی مقایسهای آبنمود واحد 2 ساعته متوسط حوزه با آبنمود واحد 2 ساعته بدست آمده از روش Clark با کمک آمارههای خطای نسبی، مجذور میانگین مربعات خطا، ضریب کارآیی و میزان انحراف دلالت بر دقت مناسب ...
متن کامل