Siefkes result that a fragment with full comprehension and induction of secondorder peanos arithmetic is complete w. The complexity of firstorder and monadic secondorder logic. Hence new techniques for optimizing this continuous process is needed for developing efficient reasoners on streaming data. Monadic secondorder logic with arbitrary monadic predicates. As is well known, monadic first order logic is not postcompleteit has consistent proper extensions.
The monadic second order theory of under successor s1s. For example, there have been proposals for xml query languages whose expressivenesses are provably mso. Higher order logic daniel leiv an t con ten ts 1 in tro duction. Monadic secondorder logic with arbitrary monadic predicates 39. Just as in first order logic, second order logic may include nonlogical symbols in a particular second order language. In logic, the monadic predicate calculus also called monadic firstorder logic is the fragment of firstorder logic in which all relation symbols in the signature are monadic that is, they take only one argument, and there are no function symbols. Dustin wehr december 18, 2007 buc his theorem establishes the equivalence of the satis ability relation for monadic.
Rabinovich department of computer science, raymond and beverly sackier faculty of exact sciences, tel aviv university, tel aviv 69978, israel received october 1995. But this is a legitimate sentence of secondorder logic. Individually the properties of graphs could be studied in a logical language referred to as monadic secondorder logic. Firstorder logic, secondorder logic, and completeness. Xml transformation language based on monadic second. Monadic second order logic formulas are also not allowed to contain second order functional variables. The logic based foundation allows to express the schema satisfaction of transformations as the validity of mso formulas over graph structures. In mathematical logic, monadic secondorder logic mso is the fragment of secondorder logic where the secondorder quantification is limited to quantification over sets. Where firstorder and monadic secondorder logic coincide. In this paper, we give superexponential lower bounds for fixedparameter tractable modelchecking problems for first order and monadic second order logic.
Abstract argumentation via monadic second order logic. Here, v 1 0 is a secondorder 1place predicate variable. We discuss the completeness of an axiomatization of monadic secondorder logic mso on in nite words. Monadic secondorder logic in practice 91 even with this extended set of operators, it is often more convenient to ex press regular languages in terms of positions and corresponding letters. We express a wide range of semantics within the proposed framework, including the standard semantics due to dung, semistable, stage, cf2, and resolutionbased semantics. We present the standard material on determinization and minimization, as well as an account of the equivalence of finite. Functions in monadic third order logic and related topics. I was wondered, if there was a paper where anyone has proved some. Section 6 discusses the role of higher order logic in. Monadic secondorder logic formulas are also not allowed to contain secondorder functional variables. In model theory, a model companion of a theory is a. Graph structure and monadic secondorder logic request pdf.
We study on which classes of graphs first order logic fo and monadic second order logic mso have the same expressive power. Similar to wellknown results for monadic secondorder logic over trees, we provide a translation of this logic into a class of automata, relative to the class of coalgebras that admit a treelike supporting kripke frame. A functional monadic second order theory of infinite trees. We study the question of whether, for a given class of finite graphs, one can define, for each graph of the class, a linear ordering in monadic secondorder logic, possibly with the help of.
We extend the weak monadic secondorder logic of one successor for finite strings m2lstr to symbolic alphabets by allowing character predicates to range over decidable quantifier free theories instead of finite alphabets. Monadic second order logic monadic secondorder logic with successor msols is a small fragment of secondorder logic, and an extension of rstorder logic with successor fols. We conclude with an introduction to the syntactic monoid, and as an application give a proof of the equivalence of first order definability and aperiodicity. There are many ways to further extend secondorder logic. Secondorder logic monadic version and henkin semantics. The semantics of the logic determines whether a dosed m2l formula r holds on w. Note that for binary numbers represented by subsets, addition is definable even in ws1s. Second order logic sol is an extension of first order logic with quantifiers and variables that range over subsets of the universe of discourse, and hence is a higher order logic of stage 2. Jan 22, 2016 in logic and mathematics second order logic is an extension of first order logic, which itself is an extension of propositional logic.
Although, i understand the automata and people in papers have tried to explain the relation to mso to me, they always assume a strong background in logic and an understanding of mso. In previous work we have offered a reconstruction of this argument which locates its source in the conflict between the neutrality of secondorder logic and its alleged entanglement with mathematics. Monadic secondorder logic and the verification of graph. For example, there is no way in fol to say that a and b have some property in common. Download torrent graph structure and monadic secondorder logic encyclopedia of mathematics and its applications, 8 pdf epub free. The language lr denoted by r is the set of strings that make r hold. Monadic second order logic monadic secondorder logic with successor msols is a small fragment of second order logic, and an extension of rstorder logic with successor fols.
Then by exploiting the two established properties of uncal called bisimulationgenericity and compactness, we re. Monadic secondorder logic and lineartime algorithms for graphs of bounded treewidth damon kaller b. On this book, these two options of graph construction are introduced collectively for the primary time in a presentation that unifies and synthesizes analysis during the last 25 years. In this paper, we describe a solution to an incremental reasoning problem on an expressive logic, namely monadic second order logic, and point out further research directions in this area. Monadic second order logic with arbitrary monadic predicates 39. Secondorder logic wikipedia, the free encyclopedia. Incremental reasoning on monadic secondorder logics with.
Recall that the monadic secondorder logic is accepted as a kind of an universal decidable logic for specifying discretetime temporal behavior. Secondorder logic in logic and mathematics secondorder logic is an extension of firstorder logic, which itself is an extension of propositional logic. The modelchecking problem for a logic l on a class c of structures asks whether a given lsentence holds in a given structure in c. Monadic secondorder logic on finite sequences microsoft. How i learned to stop worrying and love the incompleteness theorems 3 logic, in order to then give a slightly more detailed overview of secondorder logic and compare the foundational merit of each. While monadic secondorder logic is a prominent logic for specifying languages of finite words, it lacks the power to compute quantitative properties, e. Notable examples are linear temporal logic ltl 18 and the weak monadic second order logic of one successor ws1s 9. Monadic second order logic and automata on in nite words. On translations of temporal logic of actions into monadic secondorder logic a. Quantitative monadic secondorder logic request pdf. The theory we present initially is the simply typed theory of sets, equiv. Similar results hold for cliquewidth bounded graphs and monadic secondorder logic without edge set quantifications with cubic time because of the parsing step. These logics are typically equipped with operators that can describe the order between events appearing in a given sequence and oper ators for describing the kind of events that can appear. As is well known,monadic first order logic is not postcompleteit has consistent proper extensions.
Since these contain no free varaibles, they are sentencesof secondorder logic. As a result, secondorder logic has much more expressive power than fol does. Monadic second order logic as the model companion of temporal. Similar to wellknown results for monadic second order logic over trees, we provide a translation of this logic into a class of automata, relative to the class of. The overgeneration argument is a prominent objection against the modeltheoretic account of logical consequence for secondorder languages. Notable ex amples are linear temporal logic ltl 18 and the weak monadic secondorder logic of one successor ws1s 9. Monadic secondorder logic is firstorder logic plus quantification over sets. The aim of this talk is to transfer basic ideas from robinsons style modeltheoretic algebra to the realm of algebraic logic. The purpose of this article is to introduce monadic secondorder logic as a practical means of specifying regularity. Download graph structure and monadic secondorder logic. Monadic secondorder logic and lineartime algorithms for. Deterministic finite automata dfa, tree automata and visibly pushdown automata are all related to monadic second order logic mso. The most obvious is third, fourth, and so on order logic.
These are restricted, however, in that all terms that they form must. We show that for all classes c of graphs that are closed under taking subgraphs, fo and mso have the same expressive power on c if, and only if, c has bounded tree depth. We call this logic, which is able to describe sequences over complex and potentially infinite domains, symbolic m2lstr sm2lstr. Functions in monadic third order logic and related topics m. Similarly, secondorder logic recognizes as formally valid certain inferences that are not fovalid. It is particularly important in the logic of graphs, because of courcelles theorem, which provides algorithms for evaluating monadic secondorder formulas over graphs of bounded treewidth.
In this paper, we describe a solution to an incremental reasoning problem on an expressive logic, namely monadic secondorder logic, and point out further research directions in. This paper studies the expressive power that an extra. Jerome keisler and wafik boulos lotfallah abstract. Free download graph structure and monadic secondorder logic encyclopedia of mathematics and its applications, 8 pdf. We propose the formalism of monadic second order logic mso as a unifying framework for representing and reasoning with various semantics of abstract argumentation. Generalizing standard monadic secondorder logic for kripke models, we introduce monadic secondorder logic interpreted over coalgebras for an arbitrary set functor. Monadic second order logic as the model companion of. Monadic second order logic first order logic extended with set variables r. The main focus of this paper is on bisimulationinvariant mso, and more particularly on giving a novel modeltheoretic approach to it. The complexity of firstorder and monadic secondorder. Notable examples are linear temporal logic ltl 18 and the weak monadic secondorder logic of one successor ws1s 9.
Boolos has suggested a plural interpretation of secondorder logic for two purposes. On translations of temporal logic of actions into monadic. If there is only one base type x, these are rel x n and op. In standard semantics, a structure consists of a domain and interpretations for nonlogical symbols as in first order logic. As an analogy to rstorder logic being a basis for relational queries, monadic secondorder logic mso has gradually stabilizing its position as a foundation of xml processing. Firstorder logic in its broadest sense, we take logic to mean the study of correct reasoning. The subjectpredicate form of atomic statements recall the distinction in sentential logic between the following sentences. In logic and mathematics secondorder logic is an extension of firstorder logic, which itself is an extension of propositional logic. Second order logic is in turn extended by higher order logic.
In a monadic sequential presentation scheme, each assessor will be exposed to at least two objects but only one at a time, i. Monadic secondorder logic describes theories with only one type x called the base type, and its powerset, rel x 1 full secondorder logic admits a list of base types, and the full type of structures of every kind over them. In this paper, we give superexponential lower bounds for fixedparameter tractable modelchecking problems for firstorder and monadic secondorder logic. A key point is that full second order logic lets you quantify into existence a linear order of the domain. By using modeltheoretic tools, we give an alternative proof of d.
We study on which classes of graphs firstorder logic fo and monadic secondorder logic mso have the same expressive power. Monadic secondorder logic how is monadic secondorder logic abbreviated. Monadic second order logic as the model companion of temporal logic. A model theoretic proof of completeness of an axiomatization of monadic second order logic on infinite words. Similar results hold for cliquewidth bounded graphs and monadic second order logic without edge set quantifications with cubic time because of the parsing step. In standard semantics, a structure consists of a domain and interpretations for nonlogical symbols as in. All atomic formulas are thus of the form, where is a relation symbol and is a variable monadic predicate calculus can be contrasted with. The general principle, already recognized by tarski 1933 1956, is that in higher order logic one can formalize the semanticsdefine truthof lower order logic. Secondorder logicsol is an extension of firstorder logic with quantifiers and variables that range over subsets of the universe of discourse, and hence is a higherorder logic of stage 2 an important fragment is monadic secondorder logic msol, where secondorder quantification is restricted to secondorder unary relations between subsets i. Why is it important for a second order logic to be monadic to be decidable or why is this the wrong question. In logic, the monadic predicate calculus also called monadic first order logic is the fragment of first order logic in which all relation symbols in the signature are monadic that is, they take only one argument, and there are no function symbols. Monadic secondorder logic how is monadic secondorder. Secondorder and higherorder logic stanford encyclopedia of. While monadic second order logic is a prominent logic for specifying languages of finite words, it lacks the power to compute quantitative properties, e.
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