Ulrich Berger and Helmut Schwichtenberg. Program extraction from classical proofs. In D. Leivant, editor, Logic and Computational Complexity, International Workshop LCC '94, Indiapolis, IN, USA, October 1994, pages 77-97, 1995.  Home/Search   Document Not in Database   Summary   Related Articles   Check

.... typed calculi, see, e.g. 27, 28, 8] and applied to the compilation and optimization of typed languages, see, e.g. 19, 44] Grin s discovery initiated a series of studies on the computational content of classical proofs where CPS translations are a frequently employed tool, see, e.g. [13, 33, 38, 39, 34, 12, 4, 42, 24, 25, 35, 36, 6]. Inductive and coinductive types, see, e.g. 31, 29, 20, 15, 40] are syntactic representations for initial algebras (such as natural numbers and lists) resp. nal coalgebras (such as conatural numbers and streams) in typed calculi. Despite being pervasive in the type theoretical literature ....

U. Berger and H. Schwichtenberg. Program extraction from classical proofs. In D. Leivant, ed., Selected Papers from Int. Workshop on Logical and Computational Complexity, LCC'94, vol. 960 of Lect. Notes in Comp. Sci., pp. 77-97. Springer-Verlag, 1995.

....interpretation. For a general survey on various forms of realizability interpretations see [124] For a treatment of modified realizability in the context of bounded arithmetic see [29] For applications of modified realizability to the extraction of a program from a specific proof see [9] [10]. For applications of other forms of realizability for the extraction of programs from proofs see [54] Applications of (a new monotone variant of) modified realizability to semiconstructive systems (i.e. system based on E HA # but with various highly nonconstructive classical comprehension ....

.... = 0) Since G # G # G holds by intuitionistic logic, we get HA # #y(t A0 (x, y) 0) and hence HA # #x#yA 0 (x, y) # For more information on the A translation see [100] Applications of (refined) combinations of negative translation, A translation and realizability can be found e.g. in [10], 11] 12] 105] For the interesting refinement of the A translation mentioned above see [28] and for applications of that refinement [2] Chapter 10 Applications to analysis In this chapter we indicate the applicability of the proof theoretic techniques discussed in chapters 6 8 to ....

Berger,U., Schwichtenberg, H., Program extraction from classical proofs. In: Leivant, D. (ed.), Logic and Computational Complexity Workshop LCC'94. Springer LNCS 960, pp. 77-97 (1995).

.... typed calculi, see, e.g. 27, 28, 8] and applied to the compilation and optimization of typed languages, see, e.g. 19, 44] Grin s discovery initiated a series of studies on the computational content of classical proofs where CPS translations are a frequently employed tool, see, e.g. [13, 33, 38, 39, 34, 12, 4, 42, 24, 25, 35, 36, 6]. Inductive and coinductive types, see, e.g. 31, 29, 20, 15, 40] are syntactic representations for initial algebras (such as natural numbers and lists) resp. nal coalgebras (such as conatural numbers and streams) in typed calculi. Despite being pervasive in the type theoretical literature ....

U. Berger and H. Schwichtenberg. Program extraction from classical proofs. In D. Leivant, ed., Selected Papers from Int. Workshop on Logical and Computational Complexity, LCC'94, vol. 960 of Lect. Notes in Comp. Sci., pp. 77-97. Springer-Verlag, 1995.

.... 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 recursively enumerable) The material of the following section on Proof Mining is substantially based on [AF98] [BS95], BSBar] Bus95] Koh98a] Koh93a] Tro73] and [Tv88] 4 2 Proof Mining The general purpose of Proof Mining is to extract from a given proof of a formula A in a system A some constructive content. By constructive content we normally mean a realizing term for the existential quantifiers ....

....A and generates as output realizing terms for A through an iterative process. Due to the high complexity of completely formalizing proofs, however, just simple examples have been treated so far. Further information on this approach to Proof Mining can be found in [Mur90] and most recently in [BS95] and [BSBar] where a refined combination of negative translation and A translation is described in order to reduce the complexity of the extracted term. In [BSS01] some proofs in arithmetic have been analyzed. Another application of realizability for automated extraction of programs from ....

[Article contains additional citation context not shown here]

U. Berger and H. Schwichtenberg. Program extraction from classical proofs. In D. Leivant, editor, Logic and Computational Complexity Workshop (LCC'94), Lecture Notes in Computer Science, volume 960, pages 77-- 97. Springer Verlag, 1995.

....from classical proofs are based on the A translation. This can even be carried out for full set theory, as shown by Friedman (a good presentation can be found in [Be85] section VIII.3) A lot of research is carried out for extracting practical programs using the A translation, see for instance [BS95] or [Sch92] However, since Martin Lof s Type Theory is already a programming language, we believe, that our approach allows to switch more easier between classical proofs and direct programming. Further, in KP I U one has constructions corresponding precisely to the di#erent type ....

Berger, U. and Schwichtenberg, H.: Program Extraction from Classical Proofs. To appear in Proceedings of the LCC conference,1995

....least k such that b i k 6b i k 1 , then i j ;i j 1 , is a pair as in the statement. This proof is classical, but may be transformed into a constructive proof by a variant of a translation devised by H.M. Friedman, and to the resulting proof one then applies the so called modi#ed realizability (cf. [2]) in order to extract an algorithm. The variant of Friedman s translation is chosen so as to keep the complexity of formulas down as much as possible, since this results in simpler extracted algorithms when applying the realizability interpretation. For our example the result is the following ....

U. Berger, H. Schwichtenberg, Program extraction from classical proofs, in: Leivant, D., (Ed.), Logic and Computational Complexity, Internat. Workshop LCC '94, Indianapolis, Lecture Notes in Computer Science, vol. 960, Springer, Berlin, 1995.

....to encompass concurrent and non deterministic computation. Why do this We shall mention just one, fairly concrete motivation. Consider the wellestablished paradigm of extracting functional programs from (Intuitionistic or Classical) proofs, using the Curry Howard isomorphism or realizability [GLT89, BS94]. Can we analogously find a suitable combination of a logic and a realizability universe such that we can extract interesting concurrent programs communication protocols, distributed algorithms, security protocols from proofs of their specifications Two important caveats should be registered ....

U. Berger and H. Schwichtenberg. Program extraction from classical proofs. In D. Leivant, ed. Logic and Computational Complexity, Springer Lecture Notes in Computer Science vol. 960, 77--97, 1995.

....classical proofs, there has been a great deal of interest in classical proofs. The following is a tentative (and incomplete) classification: ffl Algorithm extraction, control operators: Griffin [14] Murthy [22] Krivine [21] de Groote [9] Nakano [23] Hirokawa [16] Schwichtenberg and Berger [4], Coquand [6] etc. ffl Formal systems and calculi: Girard [11, 12] Parigot [24] Berardi and Barbanera [2] Danos, Joinet and Schellinx [8] etc. ffl Proofs and semantics of cut elimination: Girard [11] Hofmann [17] Coquand [5] Pfenning [25] Herbelin [15] etc. Of these Parigot s ideas ....

U. Berger and H. Schwichtenberg. Program extraction from classical proofs. In D. Leivant, editor, Proceedings LCC 94. 1995.

....proofs of finite and or discrete mathematics. These problems belong to the class of problems called 5 0 2 , which theories of classical proof execution normally treat. For the latter case (combinatorics) even some positive results of program extraction from classical proofs have been reported in [9,20]. They reported algorithms computing solutions of Higman s lemma are extracted from classical proofs. These works are based on the line of Griffin s work. Berardi and his students [2] are developing techniques giving computational contents to classical proofs in a very different way by game ....

Berger, U. and Schwichtenberg, H.: Program extraction from classical proofs, Logic and Computational Complexity, Lecture Notes in Computer Science 960, pp.77-97, 1995.

....about a typical use of nonlocal control, using a classical typing. Since the original discovery that Felleisen s control operator C could be given a type corresponding to the law of double negation elimination, a great deal of work has been done on the computational meaning of classical proof [2, 3, 6, 24, 33, 36]. However, these ideas have not been exploited in the context of program development or verification. To this end, we have shown how a limited use of classical reasoning in a proof can produce a program extraction which includes a nonlocal control operator. Furthermore, the control operator is ....

U. Berger and H. Schwichtenberg. Program extraction from classical proofs. In D. Leivant, editor, Logic and Computational Complexity, pages 77--97, Berlin, 1994. Springer.

....provided one augments functions by appropriate control constructs. In particular he proposed the tautology : A ) A as the type for Felleisen s C operator. A spate of research into the semantics and computational contents of classical proofs ensued (some of which quite independently of Griffin s) [6, 13, 25, 28, 2, 21, 4, 7, 42], etc. Church s calculus is by now widely accepted as the logical basis of functional programming. A goal of our research is to find the calculus of functional computation with first class access to the flow of control, or functional computation with control, for short. In Sec tion 2 of ....

U. Berger and H. Schwichtenberg. Program extraction from classical proofs. In D. Leivant, editor, Proceedings LCC 94. 1995.

....from classical proofs are based on the A translation. This can even be carried out for full set theory, as shown by Friedman (a good presentation can be found in [Be85] Sect. 1VIII.3) A lot of research is carried out for extracting practical programs using the A translation, see for instance [BS95] or [Sch92] However, since Martin Lof s Type Theory is already a programming language, we believe that our approach allows to switch more easily between classical proofs and direct programming. Further, in KPI U one has constructions corresponding precisely to the different type constructors ....

Berger, U. and Schwichtenberg, H.: Program Extraction from Classical Proofs. In: D. Leivant (Ed.): Logic and Computational Complexity, LCC '94, Indianapolis, October 1994, Springer Lecture Notes in Computer Science 960, pp. 77--97.

....nat nat and well founded trees with branching over the natural numbers (Kleene s O) have this property. We can therefore extract constructive proofs and programs from classical proofs of Pi 2 sentences with this restriction on the types. 1 Introduction In a series of articles [Be93] Be95] [BS95a], BS95b] BS96] Sch91] Sch92] Sch93a] Sch93b] Sch94] Sch95] Schwichtenberg and Berger have studied the extraction of programs from classical proofs. This method relies on the fact, that in Pi 1 assumptions, only decidable atomic formulas are used. When working in HA we have no ....

Berger, U., Schwichtenberg, H.: Program extraction from classical proofs. In: Leivant, D. (Ed.): Logic and Computational Complexity, Springer Lecture Notes in Computer Science 960, 1995, pp. 77 -- 97.

....about a typical use of nonlocal control, using a classical typing. Since the original discovery that Felleisen s control operator C could be given a type corresponding to the law of double negation elimination, a great deal of work has been done on the computational meaning of classical proof [2, 3, 6, 22, 30, 33]. However, these ideas have not been exploited in the context of program development or verification. To this end, we have shown how a limited use of classical reasoning in a proof can produce a program extraction which includes a nonlocal control operator. Furthermore, the control operator is ....

U. Berger and H. Schwichtenberg. Program extraction from classical proofs. In International Workshop on Logic and Computational Complexity, 1994.

.... losing grip on computational aspects, b) extracted programs will contain less junk, as demonstrated in [6] where a short and ecient normalisation program for the simply typed lambda calculus is extracted, c) classical logic can be treated more smoothly [7] avoiding expensive proof translations [27, 12, 13, 14], and (d) crucial axioms of analysis like the classical quanti er free axiom of choice will be realisable (this is inspired by [4] In this connection it will be interesting to compare our work with the approach in [35] to program development in analysis via G odel s Dialectica Interpretation, ....

U. Berger, H. Schwichtenberg. Program Extraction from Classical Proofs. Logic and Computational Complexity (LCC '94), LNCS 960, 77-97, 1995.

.... problem of eliminating extensionality see [15] In order to extend it to PA plus extra axioms (e.g. DC) one has to nd realizers for the negative translation, N , of , where is replaced by a 0 1 formula (regarding negation, C, as de ned by C ) For technical reasons we follow [4] and combine the Dragalin Friedman Leivant trick and modi ed realizability: instead of replacing by a 0 1 formula we slightly change the de nition of modi ed realizability by regarding y mr as an (uninterpreted) atomic formula. More formally we de ne y mr : P (y) where P is a new unary ....

.... 8z 9y o B(z; y) via negative translation HA N m 8y (B(z; y) where m denotes derivability in minimal logic, i.e. ex falso quodlibet is not used. Now, soundness of modi ed realizability (which holds for our abstract version of modi ed realizability and minimal logic [4]) together with the assumption on allows us to extract from this proof a closed term M such that HA Mzmr (8y (B(z; y) i.e. HA 8f o o (8y (B(z; y) P (fy) P (Mzf) Replacing P by y:B(z; y) remember the closure property of ) we conclude HA 8z ....

[Article contains additional citation context not shown here]

U. Berger and H. Schwichtenberg, Program extraction from classical proofs, Logic and computational complexity workshop (lcc'94), lecture notes in computer science (D. Leivant, editor), vol. 960, Springer Verlag, 1995, pp. 77-97.

....implemented a refinement of the A translation which does not replace all atomic formulas R by R A A. In our example this is only necessary for formulas Q ; in general, it can be decided easily whether an atomic formula has to be replaced or not. For more information on this refinement we refer to (Berger and Schwichtenberg, 1995). Now, the MINLOG system transforms our classical proof into a constructive one. We show this automatically generated proof in tree form below. Due to lack of space we graphically contracted consecutive applications of elimination rules to one rule. Similarly for consecutive introduction rules. A ....

Berger, U. and H. Schwichtenberg: 1995, `Program Extraction from Classical Proofs'. In: D. Leivant (ed.): Logic and Computational Complexity, LCC '94, Vol. 960 of Lecture Notes in Computer Science. pp. 77--97.

....a refinement of the A translation which does not replace all atomic formulas R by (R A) A. In our example this is only necessary for formulas Q[ Delta] in general, it can be decided easily whether an atomic formula has to be replaced or not. For more information on this refinement we refer to (Berger and Schwichtenberg, 1995). Now, the MINLOG system transforms our classical proof into a constructive one. We show this automatically generated proof in tree form below. Due to lack of space we graphically contracted consecutive applications of elimination rules to one rule. Similarly for consecutive introduction rules. A ....

Berger, U. and H. Schwichtenberg: 1995, `Program Extraction from Classical Proofs'. In: D. Leivant (ed.): Logic and Computational Complexity, LCC '94, Vol.

No context found.

Ulrich Berger and Helmut Schwichtenberg. Program extraction from classical proofs. In D. Leivant, editor, Logic and Computational Complexity, International Workshop LCC '94, Indiapolis, IN, USA, October 1994, pages 77-97, 1995.

No context found.

Ulrich Berger and Helmut Schwichtenberg. Program extraction from classical proofs. In D. Leivant, editor, Logic and Computational Complexity, International Workshop LCC '94, Indiapolis, IN, USA, October 1994, pages 77--97, 1995.

No context found.

Ulrich Berger and Helmut Schwichtenberg. Program extraction from classical proofs. In D. Leivant, editor, Logic and Computational Complexity, International Workshop LCC '94, Indiapolis, IN, USA, October 1994, pages 77--97, 1995.

No context found.

U. Berger and H. Schwichtenberg. Program extraction from classical proofs. In D. Leivant, editor, Logic and Computational Complexity Workshop (LCC'94), volume 960 of Lecture Notes in Computer Science, pages 77--97. Springer, Berlin, 1995.

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U. Berger and H. Schwichtenberg. Program Extraction from Classical Proofs. In Logic and Computational Complexity, volume 960 of LNCS, pages 77--97, 1995.

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U. Berger and H. Schwichtenberg. Program extraction from classical proofs. In Daniel Leivant, editor, Logic and Computational Complexity, pages 77--97. Springer, Berlin, 1994.

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