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....of the theory NLg. Otter proves this much in the same way as in Section 3, the relevant les match bezem NL NLg. 5 Proof procedure for geometric theories The following proof procedure can be seen as a specialization of the proof of the completeness theorem as presented for instance in [7]. We present it informally rst, and make it precise in the next section. We divide a geometric theory in two parts: one part collects the axioms C D with D essentially one single atom; the remaining part collects the axioms where D is 0, a proper disjunction or an existential formula. Let us ....

....axioms (1) 6) form the Horn part. Proving conclusion (10) starts with instantiating (10) with two new parameters, say a 1 and a 2 , and adding the facts S a 0 a 1 ; S a 0 a 2 . The proof obligation is then At each step in the proof procedure, only a nite number of parameters are alive (as in [7]) We then collect all facts that we can derive using the Horn part Such a separation of the theory appears both in [12] and [3] In [12] this is used to prove that Desargues Theorem is independent of the theory of projective geometry presented above. 10 of the theory. In the example of the ....

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S.C. Kleene, Mathematical logic, John Wiley & Sons, Inc., New YorkLondon -Sydney, 1967.

....of e#ective computability, respectively by identifying the notion with that of computability by a Turing Machine and in the lambda calculus, both methods are equivalent. After this proof of equivalence, Kleene introduced the term Church Turing thesis to refer to any of the two equivalent theses ([48], page 232) Church Turing thesis: The intuitive notion of an algorithm equals the Turing Machine algorithm or (equivalent) the calculable functions of lambdacalculus There are a number of misunderstandings of the Church Turing thesis, collected in [16] Turing did not show that . Any problem ....

S.C. Kleene. Mathematical Logic. New York, 1967.

....problem . We can estimate the minimal code size, but with enormous computational e orts. 4 Incompleteness of Formal Methods Following our approach, we argue that formal methods have the property of incompleteness in the same sense that G odel proved in 1931 the incompleteness of arithmetics [13, 19]. This conclusion is based on the work of Greg Chaitin about the incompleteness of formal systems [2, 4 6] Chaitin constructed an information theoretic version of G odel s proof, showing that a theorem with more Kolmogorov complexity than the systems s axioms can t be proved (Chaitin s theorem) ....

Kleene, S. Mathematical Logic, New York, Wiley, 1967.

....1 For instance the subalgorithms QF and NF of the Euclidean algorithm from above could be modeled by external functions. 2 See [Boe95] or http: www.eecs.umich.edu gasm for an overview. 10 Chapter 2 The Mizar System Mizar [Rud92] is a theorem prover based on natural deduction (see [Kle67]) Starting from the axioms of set theory. 1 and some axioms of the real numbers, up to now about 20,000 theorems from such different fields of mathematics as topology, algebra, category theory and many more have been proven and stored in a library. From our point of view the main contribution ....

Stephen C. Kleene, Mathematical Logic, Wiley, New York, 1967.

...., P n be the environment propositions of E 0 and let R 0 be the GTS rewrite system of E 0 . Then: P R 0 1 , P R 0 2 , P R 0 n j= P R 0 5.2.1 Predicate Calculus To prove Theorem 79 we rst recall the appropriate facts of predicate calculus 13 . Note that we follow Kleene[Kle67] in using the notation j= V 1 ; V m to keep track of variables to which the generalization inference rule has been applied, and we have adapted the statement of predicate calculus facts below to our situation. Notation 80 In the below we shall use P to denote a set of propositions and V to ....

....P R is the set of all propositions P R for every proposition P 2 P. The notation [V 1 7 E 1 ; V 2 7 E 2 ; V n 7 E n ]P means the set of all propositions [V 1 7 E 1 ; V 2 7 E 2 ; V n 7 E n ]P for every proposition P 2 P. 13 See books on mathematical logic by Kleene[Kle67], Hamilton[Ham88] or Ebbinghaus[EFT94] 5 GTS MODELS 86 Note that we are using sets of variables and propositions because in this section, where we are dealing with modeling and validity, the order of variables and propositions in their various lists does not matter. De nition 81 (Logical ....

Stephen Cole Kleene. Mathematical Logic. Wiley, 1967.

....it is implicitly assumed that #, # # T . Moreover, x, y will stand for tuples of variables and t for tuples of terms. We use A, B and C for arbitrary formulas or sentences, A 0 for a quantifier free formulas and A 1 for purely existential formulas. The predicate T (x, y, z) defined by Kleene [Kle67] (see also [Sho67] is true whenever the Turing machine M , encoded by x, on input y halts with computation z (i.e. z is the Gdel number of that computation) The predicate T (x, y, z) is decidable (even primitive recursive) and the predicate T # (x) #yT (x, x, y) is undecidable (but ....

S. C. Kleene. Mathematical Logic. John Wiley and Sons, 1967.

....the behavior of the New York stock market, or the pattern of airline flights over Latin America can. 1992: 200201; see also 1997: 87. Searle s statement of Church s thesis is mistaken. Church s thesis (also known as Turing s thesis and the Church Turing thesis (Church 1936, Turing 1936, Kleene 1967) is a proposition concerning the extent of what can be achieved by a human mathematician who is unaided by any machinery save paper and pencil, and who is working in accordance with mechanical methods, which is to say, methods set out in the form of a finite number of exact instructions that ....

Kleene, S.C. 1967. Mathematical Logic. New York: Wiley.

....introduced by Kleene (with a small flourish of bias in favour of Church) So Turing s and Church s theses are equivalent. We shall usually refer to them both as Church s thesis, or in connection with that one of its . versions which deals with Turing machines as the Church Turing thesis. (Kleene 1967: 232; see also Kleene 1952. In mathematical logic this is the standard use of the term Church Turing thesis . However, it is nowadays not uncommon for a very different proposition, that whatever can be calculated by any machine can be computed by a universal Turing machine, to be called the ....

Kleene, S.C. 1967. Mathematical Logic. New York: Wiley.

....Programs, Bastad, June 1994: Selected papers. Springer Verlag LNCS 996. 3 This example should be compared with the notion of definition as internalized in Automath [dB85] then a derivation of b a from G with b removed, Gnb b a , can be constructed. The former derivation can be much shorter [Kle67] and might be accepted as an indirect notation for the latter. As in the case of the cut rule, this indirectness can be removed by internalizing the deduction theorem as the implication introduction rule of a natural deduction presentation of the logic. In this case, however, the internalization ....

Stephen C. Kleene. Mathematical Logic. Wiley, New York, 1967.

....for representing rules We, first, consider an example of rule used in RUBRIC as follows: baseball championship event. This rule says that baseball championship is an event. One possible interpretation for symbol is the material implication of the Boolean logic having the property: a b a b [11]. For theoretical completeness, we use the material implication to represent the rules, although it is not suitable to the retrieval domain 1 . Using the material implication, we can represent it as x baseball championship(x) event(x) Intuitively, this formula says that if a document is ....

Kleene, S. C. Mathematical Logic. John Wiely & Sons, 1967.

....frameworks (section 4.2) 4.1 Truth based frameworks These frameworks are are extensions of classical logic and deal with specific needs such as modals, partiality or non monotonic reasoning. They consider the notion of truth as primordial. I studied the following: three valued logic [Kle67], modal logic [HC68, Che80] default theory [Rei80] belief revision [Gar88] epistemic logic [Moo80] and cumulative logic [KLM90] None of the these frameworks were successful in modelling appropriately and expressively the flow of information. Three valued logic and modal logic frameworks ....

S. C. Kleene. Mathematical Logic. New York, Wiley, 1967.

....the so called system LLH, is equivalent to standard linear sequent calculus LL. 1 Introduction A result in classical logic which has been widely exploited in logic programming is Herbrand theorem. Several versions of this result are present in the literature; we recall here one of them (see [13]) Herbrand Theorem Let F be a prenex formula of the form 9w8x9y8zA[w; x; y; z] with A quantifier free. F is provable in predicate calculus if and only if a disjunction of the form A[t 11 ; f(t 11 ) t 12 ; g(t 11 ; t 12 ) Delta Delta Delta A[t n1 ; f(t n1 ) t n2 ; g(t n1 ; t n2 ) ....

....be put in prenex form, so that it does not make much sense trying to get exactly this kind of theorem in linear logic. Indeed, the same kind of phenomenon occurs also in other non classical logics, for instance intuitionistic logic. However, the idea underlying the proof of the classical theorem [13] may be exploited also in the linear context : the universally quantified variables can be expressed as functions of the existential ones. Let us call this technique Herbrandization. Notice that when formulae are Skolemized the existential variables are expressed as functions of the universal ....

S.C.Kleene 1967. Mathematical Logic, J.Wiley and Sons.

....when the rules are read upwards, a different version of (8 ) has to be adopted: 8 ) Gamma; ff(t) 8xff(x) Delta Gamma; 8xff(x) Delta The main problem in automatizing proof search in this calculus is the choice of the term t in such a rule. Sometimes the rule has been stated as follows [11, 7]: 8 ) Gamma; ff(t 1 ) ff(t k ) 8xff(x) Delta Gamma; 8xff(x) Delta where t 1 ; t k are terms from the language up to a given depth. This allows the mechanization of the calculus to perform validity checking. In [10, 2] it is proposed to delay the choice of the terms ....

S.C. Kleene. Mathematical Logic. John Wiley & Sons, New York, 1966.

.... [ If one had an algorithm which would deliver the code for such an enumeration, one could apply it to the Turing machine grammar, then execute the code with a particular starting tape and examine the resulting list to see if the Turing machine would have halted, thus solving the Halting Problem [16]. That is impossible, so there is no way of reliably getting the code for such an enumeration. Note that the Turing machine attribute grammar (17) is very simple. As a corollary to this proposition, one cannot expect that the compilation method favoured here will generally result in an ....

S.C. Kleene, Mathematical Logic, New York, 1967.

....F 2 (succ(0) S 0 ) Delta Delta Delta. Thus M 0 j= 8x)F 2 (x; S 0 ) a contradiction. Therefore there is no model M 0 of D such that M 2 S ff M 0 . We now show that there is a model M 1 of D such that for any sentence in L SA , M 1 j= iff M 2 j= By Skolem s theorem (cf. Kleene [7], page 326) there is a firstorder structure M such that for any sentence in L SA , M 2 j= iff M j= and (M 2 ; 0; succ) and (M ; 0; succ) are not isomorphic, i.e. M 2 and M are not isomorphic on sort object. In particular, M j= F 1 (do(A; S 0 ) 8x)F 2 (x; do(A; S 0 ) ....

S. C. Kleene. Mathematical Logic. John Wiley & Sons, Inc., 1967.

.... (Pure variable derivation) A pure variable derivation is a derivation such that no type variable occurs both free and bound in the same sequent (in the unlabelled and labelled systems Co and Co (cut) This notion of pure variable derivations is comparable with that of Kleene s (see [Kle67]) Lemma 6 Every derivation in Co or in Co (cut) is = to any derivation obtained by safely renaming bound type variables in types and terms, without capturing free type variables. Proof: By induction on the size of the derivation. In the case of (8 left) use the identity [ae=X]oe = ....

S.C. Kleene. Mathematical logic, Wiley, 1967.

....models, than (eq appl2) 3.5 Pure variable derivations Definition (Pure variable derivation) A pure variable derivation is a derivation in which no type variable occurs both free and bound in the same sequent. This notion of a pure variable derivation is comparable with that used by Kleene (see [Kle67]) Lemma 6 Every derivation in Co is = to any derivation obtained by safely renaming bound type variables in types and terms, without capturing free type variables. Proof: By induction on the size of the derivation. In the case of (8 left) use the identity [ae=X ]oe = ae=X 0 ] X 0 =X ....

S.C. Kleene. Mathematical logic, Wiley, 1967.

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Kleene, S.: Mathematical Logic, Wiley, 1967.

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S.C. Kleene, Mathematical Logic, John Wiley and Sons, New York, 413 pp. (1967).

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S. Kleene, Mathematical Logic, Wiley (1967)

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S. C. Kleene. Mathematical logic. John Wiley & Sons, 1967.

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Kleene, S. C. Mathematical Logic. Wiley, New York, 1968, Ch. V, pp. 223--282.

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S.C. Kleene, Mathematical logic, John Wiley & Sons, Inc., New York-London-Sydney, 1967.

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S.C. Kleene, Mathematical Logic. Wiley Interscience, New York, 1967.

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Kleene, S. C. Mathematical Logic. Wiley, New York, 1968, Ch. V, pp. 223--282.

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