Monadic Second-order Properties Research Articles

Monadic second-order incorrectness logic for GP 2

Program logics typically reason about an over-approximation of program behaviour to prove the absence of bugs. Recently, program logics have been proposed that instead prove the presence of bugs by means of under-approximate reasoning, which has the promise of better scalability. In this paper, we present an under-approximate program logic for GP 2, a rule-based programming language for manipulating graphs. We define the proof rules of this program logic extensionally, i.e. independently of fixed assertion languages, then instantiate them with a morphism-based assertion language able to specify monadic second-order properties on graphs (e.g. path properties). We show how these proof rules can be used to reason deductively about the presence of forbidden graph structure or failing executions. Finally, we prove our ‘incorrectness logic’ to be sound, and our extensional proof rules to be relatively complete.

Short Monadic Second Order Sentences about Sparse Random Graphs

In this paper, we study zero-one laws for the Erdös--Rényi random graph model $G(n,p)$ in the case when $p = n^{-\alpha}$ for $\alpha>0$. For a given class $\mathcal{K}$ of logical sentences about graphs and a given function $p=p(n)$, we say that $G(n,p)$ obeys the zero-one law (w.r.t. the class $\mathcal{K}$) if each sentence $\varphi\in\mathcal{K}$ is either asymptotically almost surely (a.a.s.) true or a.a.s. false for $G(n,p)$. In this paper, we consider first order properties and monadic second order properties of bounded quantifier depth $k$, that is, the length of the longest chain of nested quantifiers in the formula expressing the property. We call zero-one laws for properties of quantifier depth $k$ the zero-one $k$-laws. The main results of this paper concern the zero-one $k$-laws for monadic second order (MSO) properties. We determine all values $\alpha>0$, for which the zero-one $3$-law for MSO properties does not hold. We also show that, in contrast to the case of the $3$-law, there are infinitely many values of $\alpha$ for which the zero-one $4$-law for MSO properties does not hold. To this end, we analyze the evolution of certain properties of $G(n,p)$ that may be of independent interest.

Monadic second-order properties of very sparse random graphs

We study asymptotical probabilities of first order and monadic second order properties of Bernoulli random graph G(n,n−a). The random graph obeys FO (MSO) zero-one k-law (k is a positive integer) if, for any first order (monadic second order) formulae with quantifier depth at most k, it is true for G(n,n−a) with probability tending to 0 or to 1. Zero-one k-laws are well studied only for the first order language and a<1. We obtain new zero-one k-laws (both for first order and monadic second order languages) when a>1. Proofs of these results are based on the earlier studies of first order equivalence classes and our study of monadic second order equivalence classes. The respective results are of interest by themselves.

Fly-automata for checking [formula omitted] graph properties

A more descriptive but too long title would be: Constructing fly-automata to check properties of graphs of bounded tree-width expressed by monadic second-order formulas written with edge quantifications. Such properties are called MSO2 in short. Fly-automata (FA) run bottom-up on terms denoting graphs and compute “on the fly” the necessary states and transitions instead of looking into huge, actually unimplementable tables. In previous works, we have constructed FA that process terms denoting graphs of bounded clique-width, in order to check their monadic second-order (MSO) properties (expressed by formulas without edge quantifications). Here, we adapt these FA to incidence graphs, so that they can check MSO2 properties of graphs of bounded tree-width. This is possible because: (1) an MSO2 property of a graph is nothing but an MSO property of its incidence graph and (2) the clique-width of the incidence graph of a graph is linearly bounded in terms of its tree-width. Our constructions are actually implementable and usable. We detail concrete constructions of automata in this perspective.

A Gentle Introduction to Applications of Algorithmic Metatheorems for Space and Circuit Classes

Algorithmic metatheorems state that if a problem can be described in a certain logic and the inputs are structured in a certain way, then the problem can be solved with a certain amount of resources. As an example, by Courcelle’s Theorem, all monadic second-order (“in a certain logic”) properties of graphs of bounded tree width (“structured in a certain way”) can be solved in linear time (“with a certain amount of resources”). Such theorems have become valuable tools in algorithmics: if a problem happens to have the right structure and can be described in the right logic, they immediately yield a (typically tight) upper bound on the time complexity of the problem. Perhaps even more importantly, several complex algorithms rely on algorithmic metatheorems internally to solve subproblems, which considerably broadens the range of applications of these theorems. This paper is intended as a gentle introduction to the ideas behind algorithmic metatheorems, especially behind some recent results concerning space and circuit classes, and tries to give a flavor of the range of their applications.

Fly-automata for checking monadic second-order properties of graphs of bounded tree-width

Every graph property expressible in monadic second-order (MSO) logic, can be checked in linear time on graphs of bounded tree-width by means of finite automata running on terms denoting tree-decompositions. Implementing these automata is difficult because of their huge sizes. Fly-automata (FA) are deterministic automata that compute the necessary states and transitions when running (instead of looking into tables); they overcome this difficulty. Previously, we constructed FA to check MSO properties of graphs of bounded clique-width. An MSO property with edge quantifications (called an MSO2 property) is an MSO property of the incidence graph and, graphs of tree-width k have incidence graphs of clique-width O(k). Our existing constructions can be used for MSO2 properties of graphs of bounded tree-width. We examine concrete aspects of this adaptation.

Automata for the verification of monadic second-order graph properties

The model-checking problem for monadic second-order logic on graphs is fixed-parameter tractable with respect to tree-width and clique-width. The proof constructs finite automata from monadic second-order sentences. These automata recognize the terms over fixed finite signatures that define graphs satisfying the given sentences. However, this construction produces automata of hyper-exponential sizes, and is thus impossible to use in practice in many cases. To overcome this difficulty, we propose to specify the transitions of automata by programs instead of tables. Such automata are called fly-automata. By using them, we can check certain monadic second-order graph properties with limited quantifier alternation depth, that are nevertheless interesting for Graph Theory. We give explicit constructions of automata relative to graphs of bounded clique-width, and we report on experiments.

On the model-checking of monadic second-order formulas with edge set quantifications

We extend clique-width to graphs with multiple edges. We obtain fixed-parameter tractable model-checking algorithms for certain monadic second-order graph properties that depend on the multiplicities of edges, with respect to this “new” clique-width. We define special tree-width, the variant of tree-width relative to tree-decompositions such that the boxes that contain a vertex are on a path originating from some fixed node. We study its main properties. This definition is motivated by the construction of finite automata associated with monadic second-order formulas using edge set quantifications. These automata yield fixed-parameter linear algorithms with respect to tree-width for the model-checking of these formulas. Their construction is much simpler for special tree-width than for tree-width, for reasons that we explain.

Algorithmic correspondence and completeness in modal logic. I. The core algorithm SQEMA

Modal formulae express monadic second-order properties on Kripke frames, but in many important cases these have first-order equivalents. Computing such equivalents is important for both logical and computational reasons. On the other hand, canonicity of modal formulae is important, too, because it implies frame-completeness of logics axiomatized with canonical formulae. Computing a first-order equivalent of a modal formula amounts to elimination of second-order quantifiers. Two algorithms have been developed for second-order quantifier elimination: SCAN, based on constraint resolution, and DLS, based on a logical equivalence established by Ackermann. In this paper we introduce a new algorithm, SQEMA, for computing first-order equivalents (using a modal version of Ackermann's lemma) and, moreover, for proving canonicity of modal formulae. Unlike SCAN and DLS, it works directly on modal formulae, thus avoiding Skolemization and the subsequent problem of unskolemization. We present the core algorithm and illustrate it with some examples. We then prove its correctness and the canonicity of all formulae on which the algorithm succeeds. We show that it succeeds not only on all Sahlqvist formulae, but also on the larger class of inductive formulae, introduced in our earlier papers. Thus, we develop a purely algorithmic approach to proving canonical completeness in modal logic and, in particular, establish one of the most general completeness results in modal logic so far.

Bounded Model Checking for All Regular Properties

The technique of bounded model checking is extended to the linear time μ-calculus, a temporal logic that can express all monadic second-order properties of ω-words, in other words, all ω-regular languages. Experimental evidence is presented showing that the method can be successfully employed for properties that are hard or impossible to express in the weaker logic LTL that is traditionally used in bounded model checking.

Tree-width and the monadic quantifier hierarchy

It is well known that on classes of graphs of bounded tree-width, every monadic second-order property is decidable in polynomial time. The converse is not true without further assumptions. It follows from the work of Robertson and Seymour, that if a class of graphs K has unbounded tree-width and is closed under minors, then K contains all planar graphs. But on planar graphs, three-colorability is NP-complete. Hence, if P≠NP and on K every existential monadic second-order property is in P, then K has bounded tree-width. In other words, for K closed under minors, K is of bounded tree-width iff all monadic second-order properties are decidable in~ P . In this note we prove that in order to characterize classes of graphs of bounded tree-width where the monadic quantifier hierarchy collapses, closure under minors can be replaced by closure under topological minors. Closure under minors of K implies that K is in P, whereas we also note that there is a class of graphs K closed under topological minors which is not even r.e. We also show, that closure under induced subgraphs or even under subgraphs alone does not suffice to show that the collapse of the monadic quantifier hierarchy on K implies that K is of bounded tree-width or clique-width. Other characterizations of classes of bounded tree-width in terms of collapses of the monadic quantifier hierarchy to levels above the existential are discussed.

Fusion in relational structures and the verification of monadic second-order properties

Relational structures offer a common framework for handling graphs and hypergraphs of various kinds. Operations like disjoint union, the creation of new relations by means of quantifier-free formulas, and relabellings of relations make it possible to denote them using algebraic expressions. It is known that every monadic second-order property of a structure is verifiable in time proportional to the size of such an algebraic expression defining it. We prove here that this result remains true if we also use in these algebraic expressions a fusion operation that fuses all elements of the domain satisfying some unary predicate. The value mapping from these algebraic expressions to the structures they denote is a monadic second-order definable transduction, which means that the structure is definable inside the tree representing the algebraic expression by monadic second-order formulas. It follows (by using results of other articles) that, with this fusion operation, we cannot generate more graph families, but we can generate them with less unary auxiliary predicates. We also obtain clear-cut characterizations of Vertex Replacement and Hyperedge Replacement context-free graph grammars in terms of four types of operations, amongst which is the fusion of vertices satisfying a specified predicate.

The evaluation of first-order substitution is monadic second-order compatible

We denote first-order substitutions of finite and infinite terms by function symbols indexed by the sequences of first-order variables to which substitutions are made. We consider the evaluation mapping from infinite terms to infinite terms that evaluates these substitution operations. This mapping may perform infinitely many nested substitutions, so that a term which has the structure of an infinite string can be transformed into one isomorphic to an infinite binary tree. We prove that this mapping is monadic second-order compatible which means that a monadic second-order formula expressing a property of the output term produced by the evaluation mapping can be translated into a monadic second-order formula expressing this property over the input term. This implies that, deciding the monadic second-order theory of the output term reduces to deciding that of the input term. As an application, we obtain another proof that the monadic second-order properties of the algebraic trees, which represent the behaviours of recursive applicative program schemes, are decidable. This proof extends to hyperalgebraic trees. These infinite trees correspond to certain recursive program schemes with functional parameters of arbitrary high type.

Thue Specifications and Their Monadic Second-order Properties

This paper introduces oriented Thue specifications and their models which are simple directed edge-labeled graphs. The monadic second-order theory of these models is investigated and shown decidable for the class of suffix-bounded rational oriented Thue specifications. In addition, a characterization of prefix-closed context-free languages is given in terms of oriented Thue specifications.

Monadic second-order logic, graph coverings and unfoldings of transition systems

We prove that every monadic second-order property of the unfolding of a transition system is a monadic second-order property of the system itself. An unfolding is an instance of the general notion of graph covering. We consider two more instances of this notion. A similar result is possible for one of them but not for the other.

The monadic second-order logic of graphs IX: Machines and their behaviours

We establish that every monadic second-order property of the behaviour of a machine (transition systems and tree automata are typical examples of machines) is a monadic second-order property of the machine itself. In this way, we clarify the distinction between “dynamic” properties of machines (i.e., properties of their behaviours), and their “static” properties (i.e., properties of the graphs or relational structures representing them). It is important for program verification that the dynamic properties that one wants to verify can be formulated statically, in the simplest possible way. As a corollary of our main result, we also obtain that the monadic theory of an algebraic tree is decidable.

Monadic Second-Order Logic, Graphs and Unfoldings of Transition Systems

We prove that every monadic second-order property of the unfolding&lt;br /&gt;of a transition system is a monadic second-order property of the&lt;br /&gt;system itself. We prove a similar result for certain graph coverings.

The monadic second-order logic of graphs VII: graphs as relational structures

Relational structures form a unique framework in which various types of graphs and hypergraphs can be formalized and studied. We define operations on structures that are compatible with monadic second-order logic, and that are powerful enough to represent context-free graph and hypergraph grammars of various types, namely, the so-called hyperedge replacement, C-edNCE, and separated handle rewriting grammars. Several results concerning monadic second-order properties of the generated sets are obtained in a uniform way. We also give a logical characterization of the equational sets of structures that generalizes the ones obtained by Engelfriet and Courcelle for hyperedge replacement and C-edNCE sets of graphs.

Recursive queries and context-free graph grammars

The semantics of a recursive query in the context of a relational database can be formalized in terms of a context-free graph grammar associated with it. This grammar generates oriented, labeled hypergraphs. The labels of the hyperedges of these hypergraphs, called the computation graphs of the query, are the names of the relations forming the database under consideration. Some properties of recursive queries expressible as monadic second-order properties of their computation graphs are decidable. A few examples are given.

A logical approach to asymptotic combinatorics II: Monadic second-order properties

Etude generale des problemes sur les proprietes exprimables en logique du second ordre monadique. Existence de probabilites asymptotiques etendues