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Form of second-order logic
In mathematical logic, monadic second-order logic (MSO) is the fragment of second-order logic where the second-order quantification is limited to quantification
Monadic_second-order_logic
Form of logic that allows quantification over predicates
In logic and mathematics, second-order logic is an extension of first-order logic, which itself is an extension of propositional logic. Second-order logic
Second-order_logic
Fragment of first-order logic
In logic, the monadic predicate calculus (also called monadic first-order logic) is the fragment of first-order logic (also called predicate calculus)
Monadic_predicate_calculus
On linear-time algorithms for graph logic
theorem is the statement that every graph property definable in the monadic second-order logic of graphs can be decided in linear time on graphs of bounded treewidth
Courcelle's_theorem
Formal language theorem
language is regular if and only if it can be defined by a formula in monadic second-order logic (MSO). The theorem is due to Julius Richard Büchi, Calvin Elgot
Büchi–Elgot–Trakhtenbrot theorem
Büchi–Elgot–Trakhtenbrot_theorem
Number denoting a graph's closeness to a tree
logic of graphs using monadic second order logic, then it can be solved in linear time on graphs with bounded treewidth. Monadic second order logic is
Treewidth
Generalization of depth-first search trees
is a planar graph. A characterization of Trémaux trees in the monadic second-order logic of graphs allows graph properties involving orientations to be
Trémaux_tree
Formal system of logic
In mathematics and logic, a higher-order logic (abbreviated HOL) is a form of logic that is distinguished from first-order logic by additional quantifiers
Higher-order_logic
Class of languages studied in formal language theory in computer science
ω-regular languages are precisely the ones definable in a particular monadic second-order logic called S1S. Wolfgang Thomas, "Automata on infinite objects." In
Omega-regular_language
Extension of propositional modal logic
ST_{y}(\phi )\end{aligned}}} Recall that monadic second order logic (MSO) extends first-order logic (FO) with second order quantifications over subsets. The
Modal_μ-calculus
Proof technique in model theory
logics, such as fixpoint logics and pebble games for finite variable logics; extensions are powerful enough to characterise definability in monadic second-order
Ehrenfeucht–Fraïssé_game
Computational problem with high complexity
Satisfiability of the Weak Monadic Second-Order Logic of One Successor (WS1S) Satisfiability of W. V. O. Quine's fluted fragment of first-order logic β-convertibility
Nonelementary_problem
Mathematical theory
building They admire only one another also cannot be interpreted in monadic second-order logic. This is because predicates such as "are shipmates", "are meeting
Plural_quantification
Logical formulation of graph properties
first-order logic of graphs concerns sentences in which the variables and predicates concern individual vertices and edges of a graph, while monadic second-order
Logic_of_graphs
It does not hold for monadic second order logic. As pointed out by Yuri Gurevich, zero-one law was proven for first-order logic by Yu. V. Glebskii, D
Zero–one_law_(logic)
Term in mathematical logic
spectra of first-order logic with the successor relation. The set of ultimately periodic sets is the set of spectra of monadic second-order logic with a unary
Spectrum_of_a_sentence
Field of computer science
in monadic second-order logic and state machines in the form of digital circuits. Program synthesis Model checking Church, Alonzo (1962). "Logic, arithmetic
Reactive_synthesis
Topics referred to by the same term
for cellular cultivation Mixed-signal oscilloscope Monadic second-order logic, in mathematical logic Michigan Southern Railroad (1989), reporting markm
MSO
Type of logical system
first-order logic (FOL), also called predicate logic, predicate calculus, or quantificational logic, is a type of formal system. First-order logic uses
First-order_logic
System including an indeterminate value
three-valued logic (also trinary logic, trivalent, ternary, or trilean, sometimes abbreviated 3VL) is any of several many-valued logic systems in which
Three-valued_logic
Formal language that can be expressed using a regular expression
be accepted by a read-only Turing machine it can be defined in monadic second-order logic (Büchi–Elgot–Trakhtenbrot theorem) it is recognized by some finite
Regular_language
Decidable theory of equality
fragment of more expressive decidable theories, including monadic class of first-order logic (which also admits unary predicates and is, via Skolem normal
Theory_of_pure_equality
Topics referred to by the same term
Leadership, a business degree Microsoft Online Services Monadic second-order logic, a form of logic in which one can quantify over sets Msol or solar mass
MSOL
Alternative mathematical ordering
Courcelle, Bruno; Engelfriet, Joost (April 2011), Graph Structure and Monadic Second-Order Logic, a Language Theoretic Approach (PDF), Cambridge University Press
Cyclic_order
straightforward to express the monochromatic triangle problem in the monadic second-order logic of graphs (MSO2), by a logical formula that asserts the existence
Monochromatic_triangle
In mathematics, S2S is the monadic second-order theory with two successors. Its first-order objects are finite binary strings. It is one of the most expressive
S2S_(mathematics)
Graph polynomial generating numbers of matchings
parameter complexity of graph enumeration problems definable in monadic second-order logic" (PDF), Discrete Applied Mathematics, 108 (1–2): 23–52, doi:10
Matching_polynomial
Branch of logic
zeroth-order logic. Sometimes, it is called first-order propositional logic to contrast it with System F, but it is distinct from first-order logic. It deals
Propositional_logic
Whether a decision problem has an effective method to derive the answer
systems extending first-order logic, such as second-order logic and type theory, are also undecidable. The validities of monadic predicate calculus with
Decidability_(logic)
Topics referred to by the same term
research infrastructure Existential monadic second-order logic, a fragment of second-order logic in which all second-order quantifiers must be existential
EMSO
Design pattern in functional programming to build generic types
which lifts a value into the monadic context, and bind : <A,B>(m_a : M(A), f : A -> M(B)) -> M(B) which chains monadic computations. In simpler terms
Monad (functional programming)
Monad_(functional_programming)
Measure of graph complexity
every graph property that can be expressed in MSO1 monadic second-order logic (a form of logic allowing quantification over sets of vertices) has a
Clique-width
Finite-state machine
Deterministic acyclic finite state automaton DFA minimization Monadic second-order logic Powerset construction Quantum finite automaton Separating words
Deterministic finite automaton
Deterministic_finite_automaton
Concept in logic
soundness of the deduction rule described in the previous section. In first-order logic, a substitution is a total mapping σ: V → T from variables to terms;
Substitution_(logic)
Method of graph decomposition
brambles, grid-like minors, and parameterized intractability of monadic second-order logic", Proceedings of the Twenty-First Annual ACM-SIAM Symposium on
Bramble_(graph_theory)
Approach to logic
In logic and formal semantics, term logic, also known as traditional logic, syllogistic logic or Aristotelian logic, is a loose name for an approach to
Term_logic
Type of formal logic
intuitionistic logic to create new intuitionistic connectives and to simulate the monadic elements of intuitionistic first order logic. In the most common
Modal_logic
Limitative results in mathematical logic
Gödel's incompleteness theorems are two theorems of mathematical logic that are concerned with the limits of provability in formal axiomatic theories
Gödel's incompleteness theorems
Gödel's_incompleteness_theorems
Method of deriving conclusions
discourse. An important difference between first-order and second-order logic is that second-order logic is incomplete, meaning that it is not possible
Rule_of_inference
Subfield of mathematics
classical logics such as second-order logic or infinitary logic are also studied, along with Non-classical logics such as intuitionistic logic. First-order logic
Mathematical_logic
Computer science field
more generally implies the tractability of model checking for monadic second-order logic), bounding the degree of every domain element, and more general
Model_checking
Overview of and topical guide to logic
(predicate logic) First-order logic First-order predicate Formation rule Free variables and bound variables Generalization (logic) Monadic predicate calculus
Outline_of_logic
American scientist (1839–1914)
contributions to logic, such as theories of relations and quantification. C. I. Lewis wrote, "The contributions of C. S. Peirce to symbolic logic are more numerous
Charles_Sanders_Peirce
Set of sentences in a formal language
first-order logic, the most important case, it follows from the completeness theorem that the two meanings coincide. In other logics, such as second-order logic
Theory_(mathematical_logic)
Mathematical use of "for all" and "there exists"
\exists } . Other quantifiers are only definable within second-order logic or higher-order logics. Quantifiers have been generalized beginning with the
Quantifier_(logic)
Set that intersects every one of a family of sets
Bruno Courcelle; Joost Engelfriet (2012). Graph Structure and Monadic Second-Order Logic: A Language-Theoretic Approach. Cambridge University Press. p
Transversal_(combinatorics)
Determining the answers to a query on a database
path queries, etc., up to logical formalisms like first-order logic or monadic second-order logic. For instance, for Boolean conjunctive queries, the complexity
Query_evaluation
Graph formed by complementation and disjoint union
Courcelle's theorem may be used to test any property in the monadic second-order logic of graphs (MSO1) on cographs in linear time. The problem of testing
Cograph
Template that specifies one or more axioms
semantics for second-order logic. Analogously, some first-order set-theoretic schemata can be represented by quantifying over classes or higher-order objects
Axiom_schema
Components of a mathematical or logical formula
In mathematical logic, a term is an arrangement of dependent/bound symbols that denotes a mathematical object within an expression/formula. In particular
Term_(logic)
Set of rules defining correctly structured programs
by non-textual symbols. Most symbols denote functions or operators. A monadic function takes as its argument the result of evaluating everything to its
APL_syntax_and_symbols
performing formal logic such as the Stanhope Demonstrator or Jevon's logic piano. logic of attributes See monadic first-order logic. logic of conditionals
Glossary_of_logic
Mathematical logic concept
In logic and mathematics, contraposition, or transposition, refers to the inference of going from a conditional statement into its logically equivalent
Contraposition
Study of the properties of logical systems
propositional logic (Emil Post 1920), Consistency of first-order monadic predicate logic (Leopold Löwenheim 1915) Consistency of first-order predicate logic (David
Metalogic
Mapping of mathematical formulas to a particular meaning
view, structures are the objects used to define the semantics of first-order logic, cf. also Tarski's theory of truth or Tarskian semantics. For a given
Structure (mathematical logic)
Structure_(mathematical_logic)
Symbol representing a property or relation in logic
In the semantics of logic, predicates are interpreted as relations. For instance, in a standard semantics for first-order logic, the formula R ( a ,
Predicate_(logic)
Impossible task in computing
Church and Alan Turing in 1936. By the completeness theorem of first-order logic, a statement is universally valid if and only if it can be deduced using
Entscheidungsproblem
School of thought in philosophy of mathematics
is an extension of logic, some or all of mathematics is reducible to logic, or some or all of mathematics may be modelled in logic. Bertrand Russell and
Logicism
Assignment of meaning to the symbols of a formal language
explicitly included in first-order theories when equality is treated this way. This second approach is sometimes called first order logic with equality, but many
Interpretation_(logic)
Order whose elements are all comparable
statements hold for all total orders. Using interpretability in S2S, the monadic second-order theory of countable total orders is also decidable. There are several
Total_order
Depth of nesting of quantifiers in a formula
In mathematical logic, the quantifier rank of a formula is the depth of nesting of its quantifiers. It plays an essential role in model theory. The quantifier
Quantifier_rank
Mathematical structure
first used by Michael Rabin for proving decidability of S2S, the monadic second-order theory with two successors. It has been further observed that tree
Infinite-tree_automaton
Theorem in mathematical logic
In mathematical logic, the compactness theorem states that a set of first-order sentences has a model if and only if every finite subset of it has a model
Compactness_theorem
Existence of values making formula true
respect to a fixed logic defining the syntax of allowed symbols, such as first-order logic, second-order logic or propositional logic. Rather than being
Satisfiability
Number of arguments required by a function
Abraham Robinson follows Quine's usage. In philosophy, the adjective monadic is sometimes used to describe a one-place relation such as 'is square-shaped'
Arity
aimed to express all of arithmetic in terms of logic. Frege's work laid the groundwork for much of modern logic and was highly influential, though it encountered
Mathematical_object
characterizations based on an equivalent form of automata and monadic second-order logic. Aho, Sethi & Ullman 1988, p. 203 Aho, Sethi & Ullman 1988, pp
Operator-precedence_grammar
Statement that is taken to be true
requires the use of second-order logic. The Löwenheim–Skolem theorems tell us that if we restrict ourselves to first-order logic, any axiom system for
Axiom
Logical connective AND
In logic, mathematics and linguistics, and ( ∧ {\displaystyle \wedge } ) is the truth-functional operator of conjunction or logical conjunction. The logical
Logical_conjunction
Logical statement with variables, predicates, and quantifiers over objects
predicate. First-order predicate calculus Monadic predicate calculus Flew, Antony (1984), A Dictionary of Philosophy: Revised Second Edition, Macmillan
First-order_predicate
Algebraic manipulation of "true" and "false"
first-order logic. Although the development of mathematical logic did not follow Boole's program, the connection between his algebra and logic was later
Boolean_algebra
Algebraization of first-order logic
In mathematical logic, predicate functor logic (PFL) is one of several ways to express first-order logic (also known as predicate logic) by purely algebraic
Predicate_functor_logic
Number of matchings in a graph
parameter complexity of graph enumeration problems definable in monadic second-order logic" (PDF), Discrete Applied Mathematics, 108 (1–2): 23–52, doi:10
Hosoya_index
System of logic in mathematics and philosophy
Malinowski (eds.): Trends in Logic: 50 Years of Studia Logica, Trends in Logic 20: 177–212. Wang, J. T.; Xin, X. L. (2022-06-01). "Monadic algebras of an involutive
Łukasiewicz_logic
Axiom of set theory
{\displaystyle X} contains exactly one element. This can be formalized in first-order logic as: ∀ x ( ∃ e ( e ∈ x ∧ ¬ ∃ y ( y ∈ e ) ) ∨ ∃ a ∃ b ∃ c ( a ∈ x ∧ b ∈
Axiom_of_choice
Operation in algebra and mathematics
be monadic if it has a left adjoint F forming a monadic adjunction. For example, the free–forgetful adjunction between groups and sets is monadic, since
Monad_(category_theory)
In logic, a statement which is always true
In mathematical logic, a tautology (from Ancient Greek: ταυτολογία) is a formula that is true regardless of the interpretation of its component terms
Tautology_(logic)
Standard system of axiomatic set theory
constructed in first-order logic. Some formulations of first-order logic include identity; others do not. If the variety of first-order logic in which one is
Zermelo–Fraenkel_set_theory
Mathematical logic concept
given model. The well-formed terms and propositions of ordinary first-order logic have the following syntax: Terms: t ≡ c ∣ x ∣ f ( t 1 , … , t n ) {\displaystyle
Atomic_formula
Formal language concept
over nested words are exactly the set of languages described by monadic second-order logic with two unary predicates call and return, linear successor and
Nested_word
Logical formulation of recursion
In mathematical logic, fixed-point logics are extensions of classical predicate logic that have been introduced to express recursion. Their development
Fixed-point_logic
parsers, defied rewriting in a monadic form. The formal concept of arrows was developed to explain these exceptions to monadic code, and in the process, monads
Arrow_(computer_science)
Mathematical model for deduction or proof systems
which, in order to avoid confusion, are usually called metatheorems. A logical system is a deductive system (most commonly first order logic) together
Formal_system
Type of planar graph
bounded treewidth by finite state tree automata is definable in the monadic second-order logic of graphs, has been proven for the k {\displaystyle k} -outerplanar
K-outerplanar_graph
Relationship between programs and proofs
{\displaystyle \Box } in modal logic and staged computation possibility ◊ {\displaystyle \Diamond } in modal logic and monadic types for effects The λI calculus
Curry–Howard_correspondence
Sequence of words formed by specific rules
In logic, mathematics, computer science, and linguistics, a formal language is a set of strings whose symbols are taken from a set called "alphabet".
Formal_language
Class of formal logics
Classical logic (or standard logic) or Frege–Russell logic is the intensively studied and most widely used class of deductive logic. Classical logic has had
Classical_logic
Basic notion of sameness in mathematics
of symbolic logic. There are generally two ways that equality is formalized in mathematics: through logic or through set theory. In logic, equality is
Equality_(mathematics)
Many-valued logic in which truth values comprise a continuous range
In logic, an infinite-valued logic (or real-valued logic or infinitely-many-valued logic) is a many-valued logic in which truth values comprise a continuous
Infinite-valued_logic
In mathematical logic, a well-formed formula with no free variables
In mathematical logic, a sentence (or closed formula) of a predicate logic is a Boolean-valued well-formed formula with no free variables. A sentence can
Sentence_(mathematical_logic)
Characteristic of some logical systems
propositional logic and first-order predicate logic are semantically complete, but not syntactically complete (for example, the propositional logic statement
Completeness_(logic)
Theorem of mathematical logic
Robinson's joint consistency theorem is an important theorem of mathematical logic. It is related to Craig interpolation and Beth definability. The classical
Robinson's joint consistency theorem
Robinson's_joint_consistency_theorem
American philosopher and logician (1940–1996)
Boolos argued that if one reads the second-order variables in monadic second-order logic plurally, then second-order logic can be interpreted as having no
George_Boolos
Term in logic and deductive reasoning
In logic, soundness can refer to either a property of arguments or a property of formal deductive systems. An argument is sound if (and only if) it is
Soundness
Theorem in mathematical logic
mathematical logic, Lindström's theorem (named after Swedish logician Per Lindström, who published it in 1969) states that first-order logic is the strongest
Lindström's_theorem
Problem in computer science
consistent) and complete effective axiomatization of all true first-order logic statements about natural numbers. Then we can build an algorithm that
Halting_problem
Logical principle
In logic, the law of excluded middle or the principle of excluded middle states that for every proposition, either this proposition or its negation is
Law_of_excluded_middle
Computation model defining an abstract machine
rational integers. The Entscheidungsproblem [decision problem for first-order logic] is solved when we know a procedure that allows for any given logical
Turing_machine
Kind of proof calculus
work on natural deduction, and included applications for modal and second-order logic. In natural deduction, a proposition is deduced from a collection
Natural_deduction
Subfield of automated reasoning and mathematical logic
revised second edition in 1927. Russell and Whitehead thought they could derive all mathematical truth using axioms and inference rules of formal logic, in
Automated_theorem_proving
MONADIC SECOND-ORDER-LOGIC
MONADIC SECOND-ORDER-LOGIC
MONADIC SECOND-ORDER-LOGIC
MONADIC SECOND-ORDER-LOGIC
MONADIC SECOND-ORDER-LOGIC
MONADIC SECOND-ORDER-LOGIC
MONADIC SECOND-ORDER-LOGIC
MONADIC SECOND-ORDER-LOGIC
MONADIC SECOND-ORDER-LOGIC