R. Harrop. Concerning formulas of the types A!BC, A!(Ex)B(x) in intuitionistic formal systems. J. Symbolic Logic, 25(1):27--23, 1960.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....in fl or in any uncancelled premiss. Definition 1 A formula is a Harrop formula if it is 1. an atomic formula, 2. of the form ff fi where ff, fi are Harrop, 3. oe ff) where ff (but not necessarily ) is Harrop or 4. 8xff where ff is Harrop. Harrop formulae are named for Ronald Harrop (see his [4]) Intuitively they contain no essential use of 9 or . We first associate formal terms with Harrop formulae. If ff is atomic, then its associate is A ff where A is a term constant. If ff is (fi fl) then its associate is (F fi ; G fl ) fi fl where F fi , G fl are associates of fi and ....

R. Harrop, Concerning formulas of the types A ! B C, A ! (Ex)B(x) in Intuitionistic Formal Systems, J. Symb. Logic 25 27--32 (1960).

....of 30 smaller than syntactic representations of proofs. 3 A Nondeterministic Higher Order Logic Interpreter There are several choices in the design of a nondeterministic logic interpreter. The simplest one is an interpreter for rst order Horn logic or maybe rst order hereditary Harrop formulas [9]. In such an interpreter the existential choices consist of what witnesses to chose for proving existentials. Such choices are postponed and solved by rst order uni cation. The disjunctive choices (sometimes called backchaining choices) consist of what clause to use at each step. These choices ....

R. Harrop. Concerning formulas of the types A ! B_C; A ! (Ex)B(x) in intuitionistic formal systems. Journal of Symbolic Logic, pages 27-32, 1960.

....B that is classically provable but not intuitionistically provable. a) Show that B must have order at least 3. b) Show that the smallest such formula (counting logical connectives) is Pierce s formula, namely ( p oe q) oe p) oe p. 21 4 Hereditary Harrop Formulas 4. 1 Harrop formulas In [Har60], Harrop studied a class of formulas that can be defined as follows. Let B be a syntactic variable for arbitrary first order formulas and let H be defined by H : A j B oe H j 8 x H j H 1 H 2 : An H formula is often called a Harrop formula. The main theorem regarding these formulas is that ....

....formulas is that the six reduction rules mentioned in Subsection 3.2 are intuitionistically satisfied when they are applied to sequents of the form Sigma : H Gamma B where H is a finite collection of Harrop formulas and B is an arbitrary Harrop formula. Actually, what is mentioned explicitly in [Har60] correspond to the OR and INSTAN reductions: the other four reductions are simple to show. If a set of formulas H satisfy the OR and INSTAN reductions, those formulas are often said to satisfy, respectively, the disjunctive and existential property. Harrop formulas do not, however, constitute an ....

R. Harrop. Concerning formulas of the types A ! B C; A ! (Ex)B(x) in intuitionistic formal systems. Journal of Symbolic Logic, pages 27--32, 1960.

.... Sp h1 ; gi(A B) 1 I) Gamma 1 [ Gamma Sp 1 and Sp case(x : A:d : C; y : B:e : C; h1 ; gi : A B) C ( E) 20 In fact these can readily be transformed into programs in the usual programming languages (such as C , ML, etc. 21 Harrop formulae are named for Ronald Harrop (see his [10]) 22 The substitution that we need is as in [18] supplemented by substitution of the extra terms we have because of the additional logical rules. However, if we use the implementation of the reductions given in [7] for case and select, then we do not need any extra clauses. 23 We have ....

Harrop, R. Concerning formulas of the types A ! B C, A ! (Ex)B(x) in Intuitionistic Formal Systems, J. Symb. Logic, 25, (1960), 27-32.

....of course, the original system was consistent because, for each term t without free variables, we have A(t; g f(t) where g f(t) denotes the appropriate (variable free) term of the language. 6 Harrop formulae and code optimization Harrop formulae are named for Ronald Harrop (see his [10]) Intuitively they contain no essential use of 9 or . More importantly the (sub )terms that we associate with Harrop formulae have no computational content . That is to say, the (sub )term itself makes no contribution to the computation, i.e. normalization, process when we are simplifying a ....

Harrop, R. Concerning formulas of the types A ! B C, A ! (Ex)B(x) in Intuitionistic Formal Systems, J. Symb. Logic, 25, (1960), 27-32.

....the theory is closed. It is well known that, in contrast to classical propositional logic, intuitionistic propositional logic IPC, has admissible rules which are not derivable. Probably the first nonderivable admissible rule known for this logic is the rule :A (B C) A B) A C) stated by Harrop (1960). Extensions of this rule which are as well admissible but not derivable followed [Mints 76] Citkin 77] but the question whether there were other admissible rules for IPC than the ones known remained open. In 1975 Friedman posed the problem whether it is decidable if a rule is an admissible rule ....

R. Harrop. Concerning formulas of the types A ! BC, A ! (Ex)B(x) in intuitionistic formal systems. Journal of Symbolic Logic 25, 1960.

....formula if it is 1. an atomic formula, 2. of the form ff fi where ff, fi are Harrop, 3. of the form ( oe ff) where ff (but not necessarily ) is Harrop, 4. of the form 8xff where ff is Harrop, or 5. of the form 8f ff where ff is Harrop. Harrop formulae are named for Ronald Harrop (see his [6]) Intuitively they contain no essential use of 9, 9 1 or . We first associate formal terms with Harrop formulae. If ff is atomic, then its associate is A ff where A is a term constant. If ff is (fi fl) then its associate is (F fi ; G fl ) fi fl where F fi , G fl are associates of ....

R. Harrop, Concerning formulas of the types A ! B C, A ! (Ex)B(x) in Intuitionistic Formal Systems, J. Symb. Logic 25 27--32 (1960).

....themselves, that is, to formulas satisfying C j C. The existence of values of open objects is of interest, for instance, in partial evaluation [7] and pattern matching [2, 4] When optimizing programs extracted from proofs, an important role is played by sets corresponding to Harrop formulas [6] since they are without computational content [12, 16] We will define what it means for a set to be without computational content and then show that a set is without computational content if and only if it slashes itself; the sets satisfying this condition strictly contain the Harrop sets ....

R. Harrop. Concerning formulas of the types A ! B C, A ! (9x)B(x) in intuitionistic formal systems. Journal of Symbolic Logic, 25:27--32, 1960.

....in fact the class of D formulas defined by the productions: D : A j D D j 8xD j B oe A where B denotes an arbitrary first order formula and A, as before, stands for an atomic formula. The formulas in this class are known as Harrop Formulas for they satisfy the condition, introduced by Harrop [44], that they contain no 36 strictly 6 positive occurrence of disjunctions or existential quantifiers. Harrop formulas enjoy an important computational property proved by Harrop: if a formula B, arbitrary, follows (intuitionistically) from a set of Harrop formulas P, then there exists a sequent ....

R. Harrop. Concerning Formulas of the types A ! B C, A ! (Ex)(B(x)) in Intuitionistic Formal Systems. Journal of Symbolic Logic, 25(1):27--32, 1960.

.... e 2 Eq( Pix 2 C)D(x) x:b(x) x:d(x) and by Eq introduction and introduction we get that e:eq is an object of (6) 2 The details of the full proof can be found in Salvesen [8] 4 Harrop formulas We will define a class of stable formulas corresponding to Harropformulas in predicate logic [2]. The definition is made in two steps where the first is a straightforward translation of the definition of Harrop formulas given for instance in [12] to type theory. The second step is made by reflection of the first, using the universe U . So first we define the class of Harrop formulas by the ....

R. Harrop. Concerning formulas of the types A ! B C , A ! (Ex)B(x) in intuitionistic formal systems. Journal of Symbolic Logic Vol. 25, No. 1, March 1960, pp. 27-32.

....the extracted programs are well typed and the extracted types can be inferred for them by the type inference system of Figure 2.10. In order to preserve well typedness, we distinguish between positive and negative uninformative formulas. The positive uninformative formulas are the Harrop[Har60] formulas: Harrop formulas H : j t 1 = t 2 j H H j 8x : H j A oe H where A is any formula. For these formulas we extract the unit element ( Programs can be further simplified by noting that an expression extracted from A oe B, where A is Harrop, will have type unit ) where is the ....

Ronald Harrop. Concerning formulas of the types A ! B C; A ! (Ex)B(x) in intuitionistic formal systems. Journal of Symbolic Logic, 25(1):27--32, March 1960.

....a formula B that is classically provable but not intuitionistically provable. a) Show that B must have order at least 3. b) Show that the smallest such formula (counting logical connectives) is Pierce s formula, namely ( p oe q) oe p) oe p. 5 Hereditary Harrop Formulas 5. 1 Harrop Formulas In [Har60], Harrop studied a class of formulas that can be defined as follows. Let B be a syntactic variables for arbitrary first order formulas and let H be defined by H : A j B oe H j 8 x H j H 1 H 2 : An H formula is often called a Harrop formula. The main theorem regarding these formulas is that ....

....formulas is that the six reduction rules mentioned in Subsection 4.2 are intuitionistically satisfied when they are applied to sequents of the form Sigma : H Gamma B where H is a finite collection of Harrop formulas and B is an arbitrary Harrop formula. Actually, what is mentioned explicitly in [Har60] correspond to the OR and INSTAN reductions: the other four reductions are simple to show. If a set of formulas H satisfy the OR and INSTAN reductions, those formulas are often said to satisfy, respectively, the disjunctive and existential property. Harrop formulas do not, however, constitute an ....

R. Harrop. Concerning formulas of the types A ! B C; A ! (Ex)B(x) in intuitionistic formal systems. Journal of Symbolic Logic, pages 27--32, 1960.

....mutually recursive syntax rules: G : j A j G G j G G j 8xG j 9xG j D oe G; D : A j G oe A j 8x D j D D: We shall use G 2 and D 2 to refer to the classes of G and D formulas so defined. There is a correspondence between these D formulas and those described by the logician Harrop [Har60, Tro73]. Assuming that the symbol B represents arbitrary formulas, the so called Harrop formulas are equivalent in intuitionistic and minimal logic to the H formulas defined by the rule H : A j B oe A j 8xH j H H: An interesting property of Harrop formulas, proved in [Har60] is the following: if P ....

.... logician Harrop [Har60, Tro73] Assuming that the symbol B represents arbitrary formulas, the so called Harrop formulas are equivalent in intuitionistic and minimal logic to the H formulas defined by the rule H : A j B oe A j 8xH j H H: An interesting property of Harrop formulas, proved in [Har60], is the following: if P is a finite set of Harrop formulas and C is a non atomic formula, then P I C only if there is an I proof of P Gamma C in which the last inference rule introduces the logical connective of C. Thus, an I proof of a sequent whose antecedent is a set of Harrop formulas ....

R. Harrop. Concerning formulas of the types A ! BC; A ! (Ex)B(x) in intuitionistic formal systems. Journal of Symbolic Logic, 25:27--32, 1960.

....derivable. On the other hand, there are natural theories which have admissible rules which are not derivable. An example is intuitionistic propositional logic IPC. Probably the first nonderivable admissible rule known for this logic is the rule :A (B C) A B) A C) stated in 1960 in [Harrop 60] Extensions of this rule which are as well admissible but not derivable followed [Mints 76] Citkin 77] but the question whether there were other admissible rules for IPC than the ones known remained open. In 1975 Friedman posed the problem whether it is decidable if a rule is an admissible rule ....

R. Harrop. Concerning formulas of the types A ! BC, A ! (Ex)B(x) in intuitionistic formal systems. Journal of Symbolic Logic 25, 1960.

....with a constructor. 1 Introduction The disjunction and existence properties, that is, AB implies A or B and 9xA(x) implies A(t) for some term t , respectively, were first proved for intuitionistic arithmetic by Kleene [9] using a modification of recursive realizability. Harrop [8] extended Kleene s result by also considering derivations depending on assumptions. Harrop proved C A B implies C A or C B (ED) C 9xA(x) implies C A(t) for some term t (EE) where C is a closed formula not containing any strictly positive occurrences of and 9 ; such a formula is ....

R. Harrop. Concerning formulas of the types A ! BC , A ! (9x)B(x) in intuitionistic formal systems. Journal of Symbolic Logic, 25:27--32, 1960.

.... : and ; like in Prolog. Since program clauses contain no free variable 4 The name Harrop formula refers to R. Harrop s works on conditions such that a formula A ) BC (resp. A ) 9xB(x) is provable if and only if either A ) B or A ) C is provable (resp. A ) B(t) is provable for some t) [22]. Harrop formulas have this property at the top level, and hereditary Harrop formulas have it at any level. This property is important to give a computational meaning to logical formulas. This work had been developed before logic programming as was A. Horn s work [24] founding the notion of Horn ....

R. Harrop. Concerning formulas of the types A!BC, A!(Ex)B(x) in intuitionistic formal systems. J. Symbolic Logic, 25(1):27--23, 1960.

....4 and 5 we introduce the semantics domains and denotational semantics respectively; and finally in sections 6 and 7 we present our operational model and semantics properties of the denotation. 2 First Order Hereditary Harrop Formulas The first order hereditary Harrop formulas (fohh formulas) [6, 8] are divided in two groups: the G formulas (goals) and the D formulas (definite clauses) They are defined by the following syntax rules, where A is an atom: G : A j G G j G G j 9x:G j D oe G j 8x:G D : A j G oe A j D D j 8x:D We assume that the formulas are defined over a first order ....

R. Harrop. Concerning formulas of the types A ! B C, A ! (Ex)B(x) in intuitionistic formal systems. Journal of Symbolic Logic, pages 27--32, 1960. 12 APPIA-GULP-PRODE'98

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