L. T. McCarty. Clausal Intuitionistic Logic. II. Tableau Proof Procedure. Journal of Logic Programming, 5(2):93--132, 1988.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....rather than the forms of representation, will prevent me from delving here into the details of representational technicalities. Timothy Williamson [110] discusses intuitionistic modal epistemic logic. Africk [1] draws a comparison between classical logic and intuitionistic logic. McCarty s [70] clausal intuitionist logic is the logic foundation of Gordon s [29] Pleadings Game, a normative formalization and computational model of civil pleading , where claims have to be proven by two players, the plainti and the defendant. One criterion current in logic approaches to AI distinguishes ....

McCarty, L. T.: Clausal Intuitionistic Logic, II: Tableau Proof Procedures. The Journal of Logic Programming, Vol. 5, 1988, pp. 93-132.

....constructions described in [9] since it makes it possible to provide scope to individual, function, and predicate constants. The logic described in this paper is related to logics considered by many researchers in logic programming and, most recently, in theorem proving and type theory. See [3, 5, 7, 8, 14] for the description of closely logics applied to logic programming. Similar logics, especially higher order versions, have been used as meta languages in specifying and implementing theorem provers [2, 18, 19] The logic presented here is most closely related to the first order hereditary Harrop ....

L. McCarty, Clausal Intuitionistic Logic I. Fixed Point Semantics, Journal of Logic Programming 5(1), March 1988, 1 -- 31.

....of interest of x. SLD based resolution Pret s search for an inconsistency between the observations and a particular candidate model is based on SLD resolution. The language in which observations and the ODE theory are expressed is that of generalized Horn clause intuitionistic logic (GHCIL) (McCarty 1988). Roughly, GHCIL clauses are Horn clauses that also allow embedded implications in their bodies. The special atomic formula falsum may only appear as the head of a clause. Such clauses express fundamental reasons for inconsistencies, e.g. that a system cannot be oscillating and non oscillating at ....

McCarty, L. T. 1988. Clausal intuitionistic logic I. Fixed-point semantics. J. Logic Progr. 5:1-31.

....we present the background for the problem addressed by us. 3 Background In this section we summarize some work that, to our knowledge, are the principal antecedents of ours. Firstly, works which aim for extensions of definite logic programming with both, negation and intuitionistic implication [27, 7], are presented. Then, extensions oriented towards providing definite logic programming with structuring mechanisms [30, 16] are considered. 3.1 Negation in Intuitionistic Logic Programming Definite logic programming extended with intuitionistic implication is called intuitionistic logic ....

....in the body of clauses, it is called embedded implication also. This section concretely refers to two works which, in some way, extend intuitionistic logic programming with negation. The Theory of Clausal Intuitionistic Logic The theory of clausal intuitionistic logic was developed by McCarty in [27, 28]. In this work, an extension of Horn clause logic is considered. This extension consist of adding the next types of clauses 3 : Negation Rules Embedded Rules A ( B A ( B 1 ) B 2 ) 1) A ) B A ) B 1 ) B 2 ) 2) 3 We use a notation close to the original one. 24 Since the rules in (2) ....

L.T. McCarty. Clausal intuitionistic logic I. fixed point semantics. Journal of Logic Programming, 5:1--31, 1988.

....Pret s knowledge representation and reasoning formalism is declarative: both object level and meta level knowledge are represented as rst order logical formulae. The language in which observations and the ODE theory are expressed is that of generalized Horn clause intuitionistic logic (GHCIL)[62]. Roughly, GHCIL clauses are Horn clauses that also allow embedded implications in their bodies. The special atomic formula falsum which means inconsistency may only appear as the head of a clause. Such clauses are often called integrity constraints; they express fundamental reasons for ....

L. McCarty. Clausal intuitionistic logic I. Fixed-point semantics. The Journal of Logic Programming, 5:1-31, 1988.

.... logical fragment was proved to preserve the proof theoretic properties at the basis of the logic paradigm, namely to be a uniform proof system [14] Higher order Hereditary Harrop formulas underlie the languages Prolog[15] and Isabelle [18] first order Hereditary Harrop formulas are explored in [9, 10]. In addition to embedded implications, Hereditary Harrop formulas allow universal quantifications in the bodies of clauses. The combination of these two extensions provides a rich notion of dynamic scoping, which adresses important issues in as various fields as theorem proving [6, 18, 5] ....

L. T. McCarty. Clausal intuitionistic logic I. fixed point semantics. Journal of Logic Programming, 5:1--31, 1988.

....languages. For example, first order and higher order versions of hereditary Harrop formulas (formulas with no positive occurrences of disjunctions or existentials) have been used as both specification languages for proof systems [7, 8, 25, 29] and as the basis of logic programming languages [9, 12, 18, 20, 21]. Much of the expressivity of such systems derives from the the intuitionistic proof rule which states that to prove an implicational goal of the form D oe G in the presence of a proof context Gamma, the context is augmented with D and a proof of G is attempted in the new context. That is, the ....

L. T. McCarty. Clausal intuitionistic logic I. fixed point semantics. Journal of Logic Programming, 5:1 -- 31, 1988.

....knowledge. As Horn clauses are not a particularly large fragment of first order logic, it is perhaps not surprising that this class of formulae has such a relatively strong property. There have been various schemes proposed for logic programming languages which 1 are extensions of Horn clauses [2, 8, 9, 22, 23, 28, 29, 31]. Given these various extensions, it seems natural to ask whether there is a maximal class of formulae which may be used as a programming language. Moreover, there does not seem to be universally agreed criterion which may be used to determine what constitutes a logic programming language, without ....

L.T. McCarty, Clausal Intuitionistic Logic I. Fixed Point Semantics, Journal of Logic Programming 5:1:1-32, 1988.

....in extending the basic formalism to include more expressive constructs than is permitted in the Horn clause framework. However, it is not always clear how the NAF rule may be applied in some of these extensions. For example, for systems in which programs may grow and or shrink during execution [7, 13, 14, 15], the notion of failure may change with time due to changes in the program, and so the notion of negation may change. As pointed out in [7] this does not coincide with the usual notion of negation in mathematical logic, essentially because the NAF rule is not monotonic; that is, if P :A, then ....

L.T. McCarty, Clausal Intuitionistic Logic I. Fixed Point Semantics, Journal of Logic Programming 5:1:1-32, 1988.

....under this update semantics is PSPACE complete. A large account to hypothetical reasoning in databases has been given by Bonner [2, 3, 4] who studied extensions of Horn clause logic in which rules can hypothetically add facts to a relational database by means of embedded implications [22]. For example, a rule of the form A B[add : C] intuitively means infer A if inserting C allows the inference of B. Note, however, that the fact C is always consistent with the current database, and that no facts are removed from the database in the course of a hypothetical addition. This is a ....

L.T. McCarty. Clausal Intuitionistic Logic. I. Fixed-Point Semantics. Journal of Logic Programming, 5:1--31, 1993.

.... of C(X) should we delete from the database Must we test all possible instantiations This problem has been partially solved by McCarty, who has developed an SLD style proof procedure for hypothetical insertions, a proof procedure based on top down inference and unification, as in Prolog [48]. For hypothetical deletions, a simple syntactic restriction suffices. The idea is to ensure that all variables are bound to values in the database during inference. This can be done in several ways. Perhaps the simplest way is to require that every variable in a rule appear in an atomic premise ....

L.T. McCarty. Clausal Intuitionistic Logic. II. Tableau Proof Procedures. Journal of Logic Programming (JLP), 5(2):93--132, 1988.

.... for citizenship if your father would be eligible if he were still alive [33] And McCarty, also motivated by legal applications, has developed a wide class of hypothetical rules for computer based consultation systems, especially 121 systems for reasoning about corporate tax law and estate tax law [47, 49, 57]. Theoretical work on hypothetical inference has also been carried out, largely by the logic programming community. Most of this work focuses on the hypothetical insertion of atoms into a database. One reason for this focus is that hypothetical insertion fits neatly into a well known logical ....

.... logic [29] Gabbay was the first to show that intuitionistic logic models hypothetical insertion [32] Working independently, McCarty and Miller extended this result to operations that create new constant symbols during inference, and they developed fixpoint semantics based on intuitionistic logic [47, 50]. Earlier, Statman showed that inference in intuitionistic logic is PSPACE complete in the propositional case [59] In [10, 11] we show that the complexity of answering queries with hypothetical insertions is PSPACE complete in the function free predicate case (i:e: the case of greatest interest ....

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L.T. McCarty. Clausal Intuitionistic Logic. I. Fixed-Point Semantics. Journal of Logic Programming (JLP), 5(1):1--31, 1988.

.... approaches to higher type logic programming essentially amount to allowing implications to appear within program bodies, thus permitting a kind of modular structure [28] Such embedded implications have behavioral similarities with intuitionistic implication, and as such have been investigated in [26,27]. Either an intuitionistic version of negation, or negation as failure can be added. If negation as failure is added, a three valued approach seems most natural, 17] Example 37 Suppose we have a directed graph, and we wish to write a program that can determine whether there is a path from one ....

McCarty, L. T. Clausal intuitionistic logic I. fixed point semantics. Journal of Logic Programming 5 (1988), 1--31.

....M , the logic containing the set of formulas occurring in all stable sets. M is called the autoepistemic logic of . 1 We study the intuitionistic M and provide it with a semantics. Intuitionism has been little explored in nonmonotonic logics, mainly by Gabbay in [12, 13] and by McCarty in [19]. In [12] Mp is interpreted in an intuitionistic model, as a possible intuitionistic extension of the current world in which the sentence p is realized ( models) In [13] a fixed point equation is introduced for models, in the McDermott Doyle s style. Fischer Servi extends Gabbay s work by ....

T. McCarty, Clausal intuitionistic logic I. Fixed-point semantics, Journal of Logic Programming, 5(1):1-31, 1988.

....previous, unsuccessful paths are lost altogether. In our system, previously proved formulae are maintained explicitly and therefore can be passed to other reasoning modules. The language of the logic system presented in this paper is that of Generalized Horn Clause Intuitionistic Logic (GHCIL) McC88a, McC88b] The inference engine can be briefly characterized as a GHCIL reasoner with declarative meta level control and explicit representation of previously derived knowledge. The next three sections describe the GHCIL language, the meta level control, and the explicit representation of ....

....is: For all people P : P is considered a dedicated person if P is working under the assumption that there is some unfinished work W that is assigned to P . Thus, embedded implications can be seen as hypothetical statements. For a more detailed discussion of clausal intuitionistic logic, see [McC88a, McC88b] In our system, there are several distinguished predicates that may occur in GHCIL clauses. One of them is falsum: GHCIL clauses having falsum as their head indicate contradictory situations. Negation as failure is not suitable for our purposes because it destroys the equivalence of ....

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L. T. McCarty. Clausal intuitionistic logic I. Fixed-point semantics. The Journal of Logic Programming, 5:1--31, 1988.

....these. There are some logical systems that can be used as a basis of logic programming and that contain natural notions of scope for program clauses and constants. For example, the logic of hereditary Harrop formulas, parts of which were developed independently by Gabbay and Reyle [GR84] McCarty [McC88a, McC88b], and Miller [Mil86, Mil89b, Mil90] allows for a simple stack based structuring of the runtime program and set of constants. The modal logic of Giordano, Martelli, and Rossi [GMR88] provides an interesting variation on hereditary Harrop formulas that has a different runtime structuring of ....

L. T. McCarty. Clausal intuitionistic logic I. fixed point semantics. Journal of Logic Programming, 5:1--31, 1988.

.... all of this work seems to be: How to conjecture an appropriate induction schema In this extended abstract, we will show how to formulate induction schemata in secondorder intuitionistic logic [27] and how to search for these schemata in a logic programming language based on embedded implications [17, 18]. This is a report on work in progress, and it relies heavily on two concrete examples. One example ( Red and Green Blocks ) is a variant of a familiar problem in common sense reasoning; the other example ( Naive Reverse ) is a standard problem from the logic programming literature. We use these ....

....of our work, abstracted from [20] The two examples are presented in Sections 3 and 4. Section 5 then outlines our current and future investigations into inductive reasoning. 2 Theoretical Background The framework for our work is the language of intuitionistic embedded implications presented in [17, 18]. A similar language is studied in [23] and forms the basis of the PROLOG program 2 Proceedings of the 1992 Prolog Workshop ming language [25] Prior work on essentially the same language appears in [8, 7] The class of intuitionistic embedded implications is given by: Definition 2.1: ffl An ....

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L.T. McCarty. Clausal intuitionistic logic. II. Tableau proof procedures. Journal of Logic Programming, 5(2):93--132, 1988.

.... all of this work seems to be: How to conjecture an appropriate induction schema In this extended abstract, we will show how to formulate induction schemata in secondorder intuitionistic logic [27] and how to search for these schemata in a logic programming language based on embedded implications [17, 18]. This is a report on work in progress, and it relies heavily on two concrete examples. One example ( Red and Green Blocks ) is a variant of a familiar problem in common sense reasoning; the other example ( Naive Reverse ) is a standard problem from the logic programming literature. We use these ....

....of our work, abstracted from [20] The two examples are presented in Sections 3 and 4. Section 5 then outlines our current and future investigations into inductive reasoning. 2 Theoretical Background The framework for our work is the language of intuitionistic embedded implications presented in [17, 18]. A similar language is studied in [23] and forms the basis of the PROLOG program 2 Proceedings of the 1992 Prolog Workshop ming language [25] Prior work on essentially the same language appears in [8, 7] The class of intuitionistic embedded implications is given by: Definition 2.1: ffl An ....

[Article contains additional citation context not shown here]

L.T. McCarty. Clausal intuitionistic logic. I. Fixed-point semantics. Journal of Logic Programming, 5(1):1--31, 1988.

.... Several researchers in the logic programming community have pointed out the utility of augmenting Horn clause logic with the ability to hypothetically add facts to a database [11, 10, 20, 24, 16, 4] Others have done as much by investigating the intuitionistic semantics of embedded implications [18, 19]. The complexity and expressibility of these logics has remained an open question however. This paper addresses these issues from a database point of view. In particular, results on data complexity and expressibility are established for the function free, predicate case. This paper also examines ....

.... they have an intuitionistic semantics [10] and Miller has developed fixpoint semantics for the predicate case [20] McCarty has extended this work to a larger class of formulas, and has established interesting semantic results analogous to the unique minimalmodel property of Horn clause logic [18, 19]. It is also known that intuitionistic validity is PSPACE complete in the propositional case [21, 15] To our knowledge, however, we are the first to address the questions of the data complexity and expressibility of this kind of reasoning. In order to establish upper bounds on the data complexity ....

[Article contains additional citation context not shown here]

L.T. McCarty. Clausal Intuitionistic Logic. II. Tableau Proof Procedures. Journal of Logic Programming (JLP), 5(2):93--132, 1988.

.... Several researchers in the logic programming community have pointed out the utility of augmenting Horn clause logic with the ability to hypothetically add facts to a database [11, 10, 20, 24, 16, 4] Others have done as much by investigating the intuitionistic semantics of embedded implications [18, 19]. The complexity and expressibility of these logics has remained an open question however. This paper addresses these issues from a database point of view. In particular, results on data complexity and expressibility are established for the function free, predicate case. This paper also examines ....

....the British Nationality Act. The act contains hypothetical rules such as the following: You are eligible for citizenship if your father would be eligible if he were still alive [11] Such rules are called embedded implications, and they can be implemented as rules of hypothetical addition [18]. The formal properties of hypothetical additions are still being explored. Gabbay has shown that they have an intuitionistic semantics [10] and Miller has developed fixpoint semantics for the predicate case [20] McCarty has extended this work to a larger class of formulas, and has established ....

[Article contains additional citation context not shown here]

L.T. McCarty. Clausal Intuitionistic Logic. I. Fixed-Point Semantics. Journal of Logic Programming (JLP), 5(1):1--31, 1988.

.... The fragment of second order intuitionistic logic that is needed for the application of this induction schema has a complete proof procedure (see [ McCarty, 1992; Miller, 1989 ] The procedure is similar to the first order proof procedure for universally quantified implications discussed in [ McCarty, 1988b ] To prove the second conjunct on the right hand side of Definition 3.2, we replace the predicate variable X with a new predicate constant X , we assert Phi( X) into the rulebase, and we try to prove Phi( Delta X) If this proof succeeds, then we have proven the goal: 8X ) Phi( DeltaX ) ....

L.T. McCarty. Clausal intuitionistic logic. II. Tableau proof procedures. Journal of Logic Programming, 5(2):93--132, 1988.

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L.T. McCarty. Clausal intuitionistic logic. I. Fixed-point semantics. Journal of Logic Programming, 5(1):1--31, 1988.

No context found.

L. T. McCarty. Clausal Intuitionistic Logic. II. Tableau Proof Procedure. Journal of Logic Programming, 5(2):93--132, 1988.

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L. T. McCarty. Clausal Intuitionistic Logic. I. Fixed-Point Semantics. Journal of Logic Programming, 5(1):1--31, 1988.

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L.T. McCarty, Clausal Intuitionistic Logic I. Fixed Point Semantics, Journal of Logic Programming 5:1:1-32, 1988.

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