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The generalized completeness of Horn predicate logic as a programming language
... The import of that result is that in general, it is impossible to effectively query the unique stable model of such programs. Marek et al. [33] constructed finite predicate logic programs whose stable models could code up the paths through any infinitely Vol. 30 branching recursive tree so that the problem of deciding whether a finite predicate logic program has a stable model is Σ 1 1 -complete. For such reasons, researchers have focused on finite predicate logic programs without function symbols. ...
... , a n , ¬b 1 , . . . , ¬b m (1) where c, a 1 , . . . , a n , b 1 , . . . ...
... Let ω = {0, 1, 2, . . .} denote the set of natural numbers and let [x, y] denote the standard pairing function 1 2 (x 2 + 2xy + y 2 + 3x + y) and, for n ≥ 2, we let [x 0 , . . . , ...
We study the recognition problem in the metaprogramming of finite normal predicate logic programs. That is, let be a computable first-order predicate language with infinitely many constant symbols and infinitely many n-ary predicate symbols and n-ary functions symbols for all . Then we can effectively list all the finite normal predicate logic programs over . Given some property of finite normal predicate logic programs over , we define the index set to be the set of indices e such that has property . We classify the complexity of the index set within the arithmetic hierarchy for various natural properties of finite predicate logic programs. For example, we determine the complexity of the index sets relative to all finite predicate logic programs and relative to certain special classes of finite predicate logic programs of properties such as (i) having no stable models, (ii) having no recursive stable models, (iii) having at least one stable model, (iv) having at least one recursive stable model, (v) having exactly c stable models for any given positive integer c, (vi) having exactly c recursive stable models for any given positive integer c, (vii) having only finitely many stable models, (viii) having only finitely many recursive stable models, (ix) having infinitely many stable models and (x) having infinitely many recursive stable models.
... Breaking up a big axiom system to many small ones and indicating their interconnections with interpretations between them is one natural way of structuring a complex theory. 42 Using small and well-understood theories is important in foundational thinking, too, see [Friedman, H., On foundational thinking 1. FOM (Foundations of Mathematics) Posting, Archives www.cs.nyu.edu, January 20, 2004]. ...
... The same idea was discovered about the same time independently by other researchers, too, and relativistic computation became one branch of unconventional computation. 42 [Burstall, R., Goguen, J.: Putting theories together to make specifications. In: Proc. ...
This is a two-in-one autobiography. If one omits the word “joint” from the title, omits István Németi from the authors, and changes “we” everywhere to “we, with István”, one gets a separate autobiography for Hajnal. Likewise, if one omits the word “joint”
from the title, omits Hajnal Andréka from the authors, and changes “we” everywhere to “we, with Hajnal”, one gets a separate autobiography for István.
In what follows, numbered references such as [4] refer to our joint publication list in this volume. Other publications are also referred to, these are given explicitly in the text.
Q>_e is the effective list of all finite predicate logic programs. is the list of recursive trees. We modify constructions of Marek, Nerode, and Remmel [25] to construct recursive functions f and g such that for all indices e, (i) there is a one-to-one degree preserving correspondence between the set of stable models of Q_e and the set of infinite paths through T_{f(e)} and (ii) there is a one-to-one degree preserving correspondence between the set of infinite paths through T_e and the set of stable models of Q_{g(e)}. We use these two recursive functions to reduce the problem of finding the complexity of the index set I_P for various properties P of normal finite predicate logic programs to the problem of computing index sets for primitive recursive trees for which there is a large variety of results [6], [8], [16], [17], [18], [19]. We use our correspondences to determine the complexity of the index sets of all programs and of certain special classes of finite predicate logic programs of properties such as (i) having no stable models, (ii) having at least one stable model, (iii) having exactly c stable models for any given positive integer c, (iv) having only finitely many stable models, or (vi) having infinitely many stable models.
In this paper, we discuss the copy complexity of Horn formulas with respect to unit resolution. A Horn formula is a boolean formula in conjunctive normal form (CNF) with at most one positive literal per clause. Horn formulas find applications in a number of domains, such as program verification (abstract interpretation) and logic programming (answer set programming). Quantified Horn clauses are used extensively in temporal verification of universal properties. Resolution is one of the oldest proof systems (refutation systems) for the boolean satisfiability problem (SAT), when the input is presented in conjunctive normal form (CNF). It is both sound and complete, although inefficient, when compared to other stronger proof systems for boolean formulas. Despite its inefficiency, the simple nature of resolution makes it an integral part of several theorem provers. Unit resolution is a restricted form of resolution in which each resolution step needs to use a clause with only one literal (unit literal clause). While not complete for general CNF formulas, unit resolution is complete for Horn formulas. Read-once resolution is a form of resolution in which each clause (input or derived) may be used in at most one resolution step. As with unit resolution, read-once resolution is incomplete in general and complete for Horn clauses. This paper focuses on a combination of unit resolution and read-once resolution called unit read-once resolution. Unit read-once resolution is incomplete for Horn clauses. In this paper, we study the copy complexity problem in Horn formulas with respect to unit read-once resolution. Briefly, the copy complexity of a formula with respect to unit read-once resolution, is the smallest number k, such that replicating each clause k times guarantees the existence of a unit read-once resolution refutation (UROR). This paper focuses on two problems related to the copy complexity of Horn formulas with respect to unit read-once resolution. We first relate the copy complexity of Horn formulas with respect to unit read-once resolution to the copy complexity of the corresponding Horn constraint system with respect to the addition rule. We also examine a form of copy complexity in which we permit replication of derived clauses, in addition to the input clauses. Finally, we provide a polynomial time algorithm for the problem of checking if a 2-CNF formula has a UROR.
Unification complexity of Horn clause programs is introduced, and its complexity is investigated for various classes of universal Horn formulas. A faithful simulation theorem is proved which associates with every k-tape Turing machine a Horn clause program requiring exactly as many unification steps as the Turing machine. From this it follows that Horn clause programs are computationally complete even in the case of bi-Horn (=Krom) formulas, and that the unification complexity of Horn clause programs is not recursively bounded. The faithful simulation theorem is also used to give a new interpretation to hierarchy theorems in the context of logic programming.
In dieser Arbeit wurde an der über Google Dialogflow gesteuerten Entwicklungsumgebung für logische Programmierung "Speech and Logic IDE" (SLIDE) geforscht. Die Anwendung wurde von Dialogflow zu der Bibliothek Snips NLU überführt, damit ohne Internetanbindung gearbeitet werden kann. Als Hauptteil der Arbeit wurden die logischen Konzepte Variablen, Rekursion und Listen in die Anwendung implementiert. Es wurde eine Benennungsvorschrift eingeführt, die die Anwendung von starren Strukturen löst und es durch rekursive Verarbeitung erlaubt, beliebig komplexe Strukturen zu modellieren. Die Anwendung wurde anschließend im Rahmen der Sekundarstufe I betrachtet. Die behandelten Fragen waren: "Kann SLIDE genutzt werden, um SuS der Sekundarstufe I Wissen zu vermitteln?", "Kann SLIDE genutzt werden, um SuS der Sekundarstufe I die Konzepte Fakten und Regeln zu vermitteln?", "Kann SLIDE genutzt werden, um SuS der Sekundarstufe I die Konzepte Variablen, Rekursion und Listen zu vermitteln?", "Kann SLIDE genutzt werden, um SuS der Sekundarstufe I Wissen außerhalb der mathematischen Domäne zu vermitteln?" Dazu wurden zwei Unterrichtsbeispiele konzipiert, die sich im Deutschunterricht mit Grammatik und Lyrik auseinandersetzen, zwei Themen des niedersächsischen Kerncurriculums aus der Sekundarstufe I. Bei der Unterrichtsgestaltung wurde besonderes Augenmerk auf die neu eingeführten Konzepte gesetzt. Das zweite Unterrichtsbeispiel wurde im Rahmen einer Zusammenarbeit mit dem Projekthaus Zukunft MINT der Hochschule Hannover zweimalig mit unterschiedlichen 10. Klassen (IGS und Gymnasium) durchgeführt. Die theoretischen Ergebnisse der Arbeit zeigen, dass alle Fragen mit "Ja" beantwortet werden können. In der neuen Version von SLIDE ist es möglich die neuen Konzepte zu modellieren und es ist möglich Unterrichtsbeispiele zu konzipieren, die dieses Wissen vermitteln und sich auf Inhalte des Kerncurriculums beziehen. Die Ergebnisse der Feldexperimente in Form von Fragebögen fallen weniger aussagekräftig aus, da sich die SuS bereits am Ende der Sekundarstufe I befanden und die konzipierten Inhalte somit eine Wiederholung darstellten. Weiter muss anerkannt werden, dass viele Faktoren bei der Befragung nicht berücksichtigt werden konnten. Deswegen können aus den praktischen Versuchen keine umfassenden Schlüsse gezogen werden, eine optimistische Betrachtung zeigt ein generelles Interesse der Anwendung seitens der SuS. Die Erfahrungen legen nahe die Unterrichtsinhalte auf mehrere Unterrichtseinheiten aufzuteilen, damit die Teilnehmer mit Vorwissen an die neuen Konzepte herantreten und sich auf sie konzentrieren können.
This chapter provides a brief introduction to two main semantics of logic programs with negation, the stable-model semantics of Gelfond and Lifschitz, and the well-founded semantics of Van Gelder, Ross, and Schlipf. We present definitions, introduce basic results, and relate the two semantics to each other. We restrict attention to the syntax of normal logic programs and focus on classical results. However, throughout the chapter and in concluding remarks we briefly discuss generalizations of the syntax and extensions of the semantics, and mention several recent developments.
Logic programming has been introduced as programming in the Horn clause subset of first-order logic. This view breaks down for the negation as failure inference rule. To overcome the problem, one line of research has been to view a logic program as a set of iff-definitions. A second approach was to identify a unique canonical, preferred, or intended model among the models of the program and to appeal to common sense to validate the choice of such model. Another line of research developed the view of logic programming as a nonmonotonic reasoning formalism strongly related to Default Logic and Autoepistemic Logic. These competing approaches have resulted in some confusion about the declarative meaning of logic programming. This paper investigates the problem and proposes an alternative epistemological foundation for the canonical model approach, which is not based on common sense but on a solid mathematical information principle. The thesis is developed that logic programming can be understood as a natural and general logic of inductive definitions. In particular, logic programs with negation represent nonmonotone inductive definitions. It is argued that this thesis results in an alternative justification of the well-founded model as the unique intended model of the logic program. In addition, it equips logic programs with an easy-to-comprehend meaning that corresponds very well with the intuitions of programmers.
It has been acknowledged that emerging Web applications require features that are not available in standard rule languages like Datalog or Answer Set Programming (ASP), e.g., they are not powerful enough to deal with anonymous values (objects that are not explicitly mentioned in the data but whose existence is implied by the background knowledge). In this paper, we introduce a new rule language based on ASP extended with function symbols, which can be used to reason about anonymous values. In particular, we define binary frontier-guarded programs (BFG programs) that allow for disjunction, function symbols, and negation under the stable model semantics. In order to ensure decidability, BFG programs are syntactically restricted by allowing at most binary predicates and by requiring rules to be frontier-guarded. BFG programs are expressive enough to simulate ontologies expressed in popular Description Logics (DLs), capture their recent non-monotonic extensions, and can simulate conjunctive query answering over many standard DLs. We provide an elegant automata-based algorithm to reason in BFG programs, which yields a 3ExpTime upper bound for reasoning tasks like deciding consistency or cautious entailment. Due to existing results, these problems are known to be 2ExpTime-hard.
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