Inductive definitions over finite structures

Monotone inductive definitions occur frequently throughout mathematical logic. The set of formulas in a given language and the set of consequences of a given axiom system are examples of (monotone) inductively defined sets. The class of Borel subsets of the continuum can be given by a monotone inductive definition. Kleene's inductive definition of recursion in a higher type functional (see [6]) is fundamental to modern recursion theory; we make use of it in §2.Inductive definitions over the natural numbers have been studied extensively, beginning with Spector [11]. We list some of the results of that study in §1 for comparison with our new results on inductive definitions over the continuum. Note that for our purposes the continuum is identified with the Baire space ωω.It is possible to obtain simple inductive definitions over the continuum by introducing real parameters into inductive definitions over N—as in the definition of recursion in [5]. This is itself an interesting concept and is discussed further in [4]. These parametric inductive definitions, however, are in general weaker than the unrestricted set of inductive definitions, as is indicated below.In this paper we outline, for several classes of monotone inductive definitions over the continuum, solutions to the following characterization problems:(1) What is the class of sets which may be given by such inductive definitions ?(2) What is the class of ordinals which are the lengths of such inductive definitions ?These questions are made more precise below. Most of the results of this paper were announced in [2].

Recursive Definitions and Fixed-Points

Ordinal Recursion and Inductive Definitions

Recursively invariant β-recursion theory

Algorithmic Procedures, Generalized Turing Algorithms, and Elementary Recursion Theory

We study the complexity of computable and Σ$_{\rm 1}^{\rm 0}$ inductive definitions of sets of natural numbers. For we example, we show how to assign natural indices to monotone Σ$_{\rm 1}^{\rm 0}$-definitions and we use these to calculate the complexity of the set of all indices of monotone Σ$_{\rm 1}^{\rm 0}$-definitions which are computable. We also examine the complexity of new type of inductive definition which we call weakly finitary monotone inductive definitions. Applications are given in proof theory and in logic programming.

Continuous-time computation with restricted integration capabilities

With his Introduction to Operative Logic and Mathematics1, which first appeared in 1955, Paul Lorenzen became an exponent of an approach to the foundations of logic and mathematics, which is both formalistic and intuitionistic in spirit. Formalistic because its basis is the purely syntactical handling of symbols—or ‘figures’, as Lorenzen preferred to say —, and intuitionistic because the insight into the validity of admissibility statements justifies the laws of logic. It is also intuitionistic with respect to its result, as Heyting’s formalism of intuitionistic logic is legitimatised this way. Along with taking formal calculi as its basis, the notion of an inductive definition becomes fundamental. Together with a theory of abstraction and the idea of transfinitely interating inductive definitions, Lorenzen devised a novel foundation for mathematics, many aspects of which still deserve attention. When he wrote his Operative Logic, neither a full-fledged theory of inductive definitions nor a proof-theoretic semantics for logical constants was available. A decade later, Lorenzen’s inversion principle was used and extended by Prawitz (Prawitz 1965) in his theory of natural deduction, and in the 1970s, the idea of inversion was used for a logical semantics in terms of proofs by Dummett, Martin-Lof, Prawitz and others. Prawitz and others. Another aspect which makes Lorenzen’ theory interesting from a modern point of view, is that in his protologic he anticipated certain views of rule-based reasoning and free equality which much later became central to the theory of resolution and logic programming.

Our goal is to define a type of partial recursive functions in constructive type theory. In a series of previous articles, we studied two different formulations of partial functions and general recursion. We could obtain a type only by extending the theory with either an impredicative universe or with coinductive definitions. Here we present a new type constructor that eludes such entities of dubious constructive credentials. We start by showing how to break down a recursive function definition into three components: the first component generates the arguments of the recursive calls, the second evaluates them, and the last computes the output from the results of the recursive calls. We use this dissection as the basis for the introduction rule of the new type constructor. Every partial recursive function is associated with an inductive domain predicate; evaluation of the function requires a proof that the input values satisfy the predicate. We give a constructive justification for the new construct by interpreting it into the base type theory. This shows that the extended theory is consistent and constructive.

We show that deciding the winner of the r-moves Ehrenfeucht-Fraisse game on two finite structures A and B, over any fixed signature Σ that contains at least one binary and one ternary relation, is PSPACE complete. We consider two natural modifications of the EF game, the one-sided r-moves EF game, where the spoiler can choose from the first structure A only, and therefore the duplicator wins only if B satisfies all the existential formulas of rank at most r that A satisfies; and the k-alternations r-moves EF game (for each fixed k), where the spoiler can choose from either structure, but he can switch structure at most k times, and therefore the duplicator wins iff A and B satisfy the same first order formulas of rank at most r and quantifier alternation at most k (defined in the paper). We show that deciding the winner in both the one-sided EF game and the k-alternations EF game is also PSPACE complete.

: The question of whether branching-time temporal logic or linear-time temporal logic is best for reasoning about concurrent programs is one of the most controversial issues in logics of programs. Concurrent programs are usually modelled by labelled state-transition graphs in which some state is designated as the initial state. For historical reasons such graphs are called Kripke structures. In linear temporal logic operators are provided for describing events along a single time path (i.e., along a single path in a Kripke structure). In a branching-time logic the temporal operators quantify over the futures that are possible from a given state (i.e., over the possible paths that lead from a state). It is well known that the two types of temporal logic have different expressive powers. Linear temporal logic, for example, can express certain fairness properties that cannot be expressed in branching-time temporal logic. On the other hand, certain practical decision problems like model checking are easier for branching-time temporal logic than for linear temporal logic. This paper provides further insight on which type of logic is best. It is shown that if two finite Kripke structures can be distinguished by some formula that contains both branching-time and linear-time operators, then the structures can be distinguished by a formula that contains only branching time operators. Specifically, we show that if two finite Kripke structures can be distinguished by some formula of the logic CTL (i.e., if there is some CTL formula that is true in one but not in the other), then they can be distinguished by some formula of the logic CTL.

This chapter introduces inductively defined sets and simple inductive data types such as natural numbers, words, lists and trees with emphasis on definition patterns. For these sets and data types, recursive equations in Backus-Naur form are introduced and it is shown how, e.g., propositional formulae and simple while programs can be specified in this format. Next it is explained how functions can be defined inductively on such sets and data types. Examples include the factorial function, Harmonic numbers, Fibonacci numbers, addition and multiplication on natural numbers and functions that compute the length of a word or list and the height of a tree. In contrast, recursive functions are defined in a way that does not necessarily lead to terminating evaluations. Recursive procedures are also described, with examples presented of various sorting algorithms and the Towers of Hanoi puzzle.

We have developed a logic grammar for circuit analysis. Circuits are viewed as sentences, and their elements as words. Circuit structures are defined by logic grammar called DCSG. A set of grammar rules, when converted into Prolog clauses, forms a logic program which execute top-down parsing. When we apply our methods to fundamental circuit analysis, problems of recursive definition occur. Recursive definitions often cause looping problems in logic programming. We will discusses several problems of recursive definition in circuit analysis.KeywordsLogic GrammarCircuit AnalysisParsing CircuitsDCGDCSG

Answer Set Programming (ASP) is a rule-based language rooted in traditional Logic Programming, Databases, Knowledge Representation, and Nonmonotonic Reasoning. It offers a flexible language for declarative problem solving, with support of efficient general-purpose solvers and reasoners. The larger part of this article provides an introduction to ASP, with a historical perspective, a definition of the core language, a guideline to knowledge representation, and an overview of existing ASP solvers. One part focuses on one commonly used feature: aggregates and generalized atoms. The inclusion of aggregates in ASP (and Logic Programming at large) has long been motivated, however there are some issues with semantics to be addressed, in particular when aggregates occur in recursive definitions. Very similar considerations are needed when coupling ASP with other formalisms, which we collectively refer to as “generalized atoms”. An overview of these semantic challenges and proposals for addressing them is provided, along with an overview of complexity results and system support.

This paper focuses on computing general first-order parallel and prioritized circumscription with varying constants. We propose linear translations from general first-order circumscription to first-order theories under stable model semantics over arbitrary structures, including Tr_v for parallel circumscription and Tr^s_v for conjunction of parallel circumscriptions (further for prioritized circumscription). To improve the efficiency, we give an optimization \Gamma_{\exists} to reduce logic programs in size when eliminating existential quantifiers during the translations. Based on these results, a general first-order circumscription solver, named cfo2lp, is developed by calling answer set programming (ASP) solvers. Using circuit diagnosis problem and extended stable marriage problem as benchmarks, we compare cfo2lp with a propositional circumscription solver circ2dlp and an ASP solver with complex optimization metasp on efficiency. Experimental results demonstrate that for problems represented by first-order circumscription naturally and intuitively, cfo2lp can compute all solutions over finite structures. We also apply our approach to description logics with circumscription and repairs in inconsistent databases, which can be handled effectively.