Monadic Logic Research Articles

Succinctness Gap between Monadic Logic and Duration Calculus

In [8, 11] the expressive completeness of the Propositional fragment of Duration Calculus relative to monadic first-order logic of order was established. In this paper we show that there is at least an exponential blow-up in every meaning preserving translation from monadic logic to PDC. Hence, there exists an exponential gap between the succinctness of monadic logic and that of duration calculus.

Expressive Completeness of Duration Calculus

This paper compares the expressive power of first-order monadic logic of order, a fundamental formalism in mathematical logic and the theory of computation, with that of the propositional version of duration calculus (PDC), a formalism for the specification of real-time systems. Our results show that the propositional duration calculus is expressively complete for first-order monadic logic of order. Our semantics for PDC conservatively extends the standard semantics to all positive (including infinite) length intervals. Hence, in view of the expressive completeness, liveness properties can be specified in PDC. This observation refutes a widely believed misconception that the duration calculus cannot specify liveness properties.

Varieties of Monadic Heyting Algebras Part II: Duality Theory

In this paper we continue the investigation of monadic Heyting algebras which we started in [2]. Here we present the representation theorem for monadic Heyting algebras and develop the duality theory for them. As a result we obtain an adequate topological semantics for intuitionistic modal logics over MIPC along with a Kripke-type semantics for them. It is also shown the importance and the effectiveness of the duality theory for further investigation of monadic Heyting algebras and logics over MIPC.

On the decidability of continuous time specification formalisms

We consider an interpretation of monadic second-order logic of order in the continuous time structure of finitely variable signals and show the decidability of monadic logic in this structure. The expressive power of monadic logic is illustrated by providing a straightforward meaning preserving translation into monadic logic of three typical continuous time specification formalism: temporal logic of reals, restricted duration calculus and the propositional fragment of mean value calculus. As a by-product of the decidability of monadic logic we obtain that the above formalisms are decidable even when extended by quantifiers.

Names, Verbs and Sentences

Metaphysicians often declare that there are large ontological differences (properties versus individuals, universals versus particulars) correlated with the linguistic distinction between names and verbs. Gaskin argues against all such declarations on the grounds that we may quantify with equal ease over the referents of both types of expression. However, his argument must be wrong, given the massive differences between first- and second-order qualification. Its only grain of truth is that these differences show up only in the logic of relations, and not also in monadic logic.

Resolution-based approach to compatibility analysis of interacting automata

The problem of compatibility analysis of two interacting automata arises in the development of algorithms for reactive systems if partial nondeterministic automata serve as models for both the reactive system and its environment. Intuitively, two interacting automata are compatible if a signal from one automaton always induces a defined transition in the other. We present the method for solving this problem in the automated design of reactive system algorithms specified by formulas of a first-order monadic logic. Compatibility analysis is automatically performed both for initial specifications of the algorithm and its environment and for any specification of the same kind obtained as a result of changes made by the designer. So, there is a need in the development of methods for solving this problem at the level of logical specification. The methods based on the resolution principle proved to be very helpful for this purpose.

Monadic logic programs and functional complexity

Problems related to the complexity and to the decidability of several languages weaker than Prolog are studied in this paper. In particular, monadic logic programs, that is, programs containing only monadic functions and monadic predicates, are considered in detail. The functional complexity of a monadic logic program is the language of all words ƒ 1… ƒ k such that the literal p(ƒ 1(…(ƒ k(a))…)) is a logical consequence of the program. The relationship between several subclasses of monadic programs, their functional complexities and the corresponding automata is studied. It is proved that the class of monadic programs corresponds exactly to the class of regular languages. As a consequence, the “SUCCESS” problem is decidable for that class. It is also proved that the success set of a specific subclass of monadic programs (“simple” programs) corresponds exactly to regular languages with star-height not exceeding 1.

On Expressive Completeness of Duration and Mean Value Calculi (Extended Abstract)

This paper compares the expressive power of first-order monadic logic of order, a fundamental formalism in mathematical logic and the theory of computation, with that of two formalisms for the specification of real-time systems, the propositional versions of duration and mean value calculi.Our results show that the propositional mean value calculus is expressively complete for monadic first-order logic of order. A new semantics for the chop operator used in these real-time formalisms is also proposed, and the expressive completeness results achieved in the paper indicate that the new definition might be more natural than the original one. We provide a characterization of the expressive power of the propositional duration calculus and investigate the connections between the propositional duration calculus and star-free regular expressions. Finally, we show that there exists at least an exponential gap between the succinctness of the propositional duration (mean value) calculus and that of monadic first-order logic of order.

Zero-one laws with variable probability

One of this author's favorite theorems has long been the Zero-One law discovered independently by Glebskii et al. [12] and Ron Fagin [10]. Let A be any first-order property of graphs and let µn(A) be the proportion of labelled graphs on n vertices for which A holds. ThenThis result has inspired much work by logicians, generally in the direction of showing (1) for some powerful languages. Thus it is known [5] that (1) holds when A is a sentence in fixed point logic and it is known [13] that (1) does not always hold when A is a sentence in second-order monadic logic. Here, however, we explore recent work in a totally different direction. Let G(n, p) denote the random graph on n vertices with edge probability p. (In §2 we define the random structures we will deal with.) A property A is an event in the probability space and Pr[G(n, p) ╞ A] is well defined. When p = 1/2, each labelled graph on n vertices has equal weight so that (1) may be rewrittenFagin's proof actually gives that (2) holds for any constant 0 < p < 1.

Inverse monoids, trees and context-free languages

This paper is concerned with a study of inverse monoids presented by a set X X subject to relations of the form e i = f i {e_i} = {f_i} , i ∈ I i \in I , where e i {e_i} and f i {f_i} are Dyck words, i.e. idempotents of the free inverse monoid on X X . Some general results of Stephen are used to reduce the word problem for such a presentation to the membership problem for a certain subtree of the Cayley graph of the free group on X X . In the finitely presented case the word problem is solved by using Rabin’s theorem on the second order monadic logic of the infinite binary tree. Some connections with the theory of rational subsets of the free group and the theory of context-free languages are explored.

Graph Types

Recursive data structures are abstractions of simple records and pointers. They impose a <em> shape</em> invariant, which is verified at compile-time and exploited to automatically generate code for building, copying, comparing, and traversing values without loss of efficiency. However, such values are always tree shaped, which is a major obstacle to practical use. We propose a notion of <em> graph types </em>, which allow common shapes, such as doubly-linked lists or threaded trees, to be expressed concisely and efficiently. We define regular languages of <em> routing expressions</em> to specify relative addresses of extra pointers in a canonical spanning tree. An efficient algorithm for computing such addresses is developed. We employ a second-order monadic logic to decide well-formedness of graph type specifications. This logic can also be used for automated reasoning about pointer structures.

On automata on infinite trees

In the paper first of all we analyze new conditions for acceptance of sets of infinite trees recognized by Muller automata and for which equivalent Büchi automata can be given. We define a new model of infinite tree automata, called limited Muller automata, and investigate their characterization in weak monadic logic. In the second part we extend the notion of control automaton to the case where more than one language is used to control the set of infinite words labeling each path of a successful run on a tree. We prove that using boolean operators in defining acceptance conditions for control automata, a characterization in terms of control automata for M-automata, introduced by Nivat and Saoudi (1988), and other classes of automata can be given. We introduce the And-control automaton and use this notion to prove properties of boolean closure of deterministic classes of automata.

More on monadic logic. part C: Monadically interpreting in stable unsuperstable ℱ and the monadic theory of $$^\omega \lambda $$

We continue investigating the strength of monadic logic in elementary classes. Mainly we show that all stable unsuperstable theories with finite vocabulary are either among the (easily definable class of) hopelessly complicated or are essentially as complicated as a variant of the tree $$^{\omega \geqslant } \lambda $$ .

Inductive inference of logic programs based on algebraic semantics

In this paper we present a new inductive inference algorithm for a class of logic programs, calledlinear monadic logic programs. It has several unique features not found in Shapiro’s Model Inference System. It has been proved that a set of trees isrational if and only if it is computed by a linear monadic logic program, and that the rational set of trees is recognized by a tree automaton. Based on these facts, we can reduce the problem of inductive inference of linear monadic logic programs to the problem of inductive inference of tree automata. Further several efficient inference algorithms for finite automata have been developed. We extend them to an inference algorithm for tree automata and use it to get an efficient inductive inference algorithm for linear monadic logic programs. The correctness, time complexity and several comparisons of our algorithm with Model Inference System are shown.

Notes on monadic logic. Part B: Complexity of linear orders in ZFC

In those notes we prove in ZFC: (a) that the monadic theory of linear order (syntactically) interprets and has the same Lowenheim number as second order logic (the interpretation is semantical but not in the “classical” way), (b) a parallel (weaker) result for the monadic logic for completely metrizable spaces. The main results are in §§5, 6.

More on monadic logic part D: A note on addition of theories

We improve somewhat some of the results from Baldwin and Shelah [BSh 156], closing a small gap therein (see [BSh 156], pages 248, lines 19 ff.; 255, second and third paragraphs; 256, following 3.2.10; and 257 following 3.2.12).

Notes on monadic logic. part A. Monadic theory of the real line

The second-order theory of the continuum in the Cohen extension of a set-theoretic universe is interpreted in the monadic theory of the real line and may be interpreted in the monadic topology of Cantor’s discontinuum as well.

On random models of finite power and monadic logic

For any property θ of a model (or graph), let μ n( θ) be the fraction of models of power n which satisfy θ, and let μ( θ) = lim n →∞ μ n( θ) if this limit exists. For first-order properties θ, it is known that μ(θ) must be 0 or 1. We answer a question of K. Compton by proving in a strong way that this 0–1 law can fail if we allow monadic quantification (that is, quantification over sets) in defining the sentence θ. In fact, by producing a monadic sentence which codes arithmetic on n with probability μ = 1, we show that every recursive real is μ(θ) for some monadic θ.

Monadic logic and löwenheim numbers

We investigate the monadic logic of trees with ω + 1 levels, the monadic topology of the product space ωλ and a strengthening of monadic logic for trees with ω levels.

The characterization of monadic logic

The first section of this paper is concerned with the intrinsic properties of elementary monadic logic (EM), and characterizations in the spirit of Lindström [2] are given. His proofs do not apply to monadic logic since relations are used, and intrinsic properties of EM turn out to differ in certain ways from those of the elementary logic of relations (i.e., the predicate calculus), which we shall call EL. In the second section we investigate connections between higher-order monadic and polyadic logics.EM is the subsystem of EL which results by the restriction to one-place predicate letters. We omit constants (for simplicity) but take EM to contain identity. Let a type be any finite sequence (possibly empty) of one-place predicate letters. A model M of type has a nonempty universe ∣M∣ and assigns to each predicate letter P of a subset PM of ∣M∣.Let us take a monadic logic L to be any collection of classes of models, called L-classes, satisfying the following:1. All models in a given L-class are of the same type (called the type of the class).2. Isomorphic models lie in the same L-classes.3. If and are L-classes of the same type, then and are L-classes.