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Branch of logic
com/browse/equational+logic Gries, D. (2010). Introduction to equational logic . Retrieved from https://www.cs.cornell.edu/home/gries/Logic/Equational.html
Equational_logic
Theorem in equational logic
In logic, Birkhoff's theorem in equational logic states that an equality t = u is a semantic consequence of a set of equalities E, if and only if t =
Birkhoff's theorem (equational logic)
Birkhoff's_theorem_(equational_logic)
Mathematical problem
with his HSP theorem that the equational theory of R ≥ 0 {\displaystyle \mathbb {R} _{\geq 0}} is equal to the equational theory of all commutative semirings
Tarski's high school algebra problem
Tarski's_high_school_algebra_problem
Implementation of rewriting logic
rewriting logic. It is similar in its general approach to Joseph Goguen's OBJ3 implementation of equational logic, but based on rewriting logic rather than
Maude_system
Automated theorem proofer
Prover9 is an automated theorem prover for first-order and equational logic developed by William McCune. Prover9 is the successor of the Otter theorem
Prover9
prover for full first-order logic with equality. It is based on the equational superposition calculus and uses a purely equational paradigm. It has been integrated
E_(theorem_prover)
Branch of logic using category theory to study mathematical structures
category. A classic example is the correspondence between theories of βη-equational logic over simply typed lambda calculus and Cartesian closed categories.
Categorical_logic
Theory of algebraic structures in general
form of identities, or equational laws. An example is the associative axiom for a binary operation, which is given by the equation x ∗ (y ∗ z) = (x ∗ y) ∗ z
Universal_algebra
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 quantified
First-order_logic
Topics referred to by the same term
Look up equational in Wiktionary, the free dictionary. Equational may refer to: Equative (disambiguation), a construction in linguistics something pertaining
Equational
Mathematical term; concerning axioms used to derive theorems
In mathematics and logic, an axiomatic system or axiom system is a standard type of deductive logical structure, used also in theoretical computer science
Axiomatic_system
Branch of logic
function Categorical logic Combinational logic Combinatory logic Conceptual graph Disjunctive syllogism Entitative graph Equational logic Existential graph
Propositional_logic
The superposition calculus is a calculus for reasoning in equational logic. It was developed in the early 1990s and combines concepts from first-order
Superposition_calculus
Reasoning about equations with free variables
logic, algebraic logic is the reasoning obtained by manipulating equations with free variables. What is now usually called classical algebraic logic focuses
Algebraic_logic
Topics referred to by the same term
representation theorem for distributive lattices Birkhoff's theorem (equational logic), stating that syntactic and semantic consequence coincide This disambiguation
Birkhoff's_theorem
Interactive theorem prover software
algorithms Prover9 – is an automated theorem prover for first-order and equational logic QED manifesto – Proposal for a computer-based database of all mathematical
Proof_assistant
for analyzing programs through the use of algebraic structures and equational logic. Algebraic semantics represents programs and data types as algebras—mathematical
Algebraic semantics (computer science)
Algebraic_semantics_(computer_science)
Mathematical assumptions
(1968). "Equational logic". Notre Dame J. Formal Logic. 9 (3): 212–226. doi:10.1305/ndjfl/1093893457. MR 0246753. Meredith, C. A. (1969). "Equational postulates
Minimal axioms for Boolean algebra
Minimal_axioms_for_Boolean_algebra
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
System of formal deduction in logic
axiomatisation and which describes classical equational logic. We deal with a minimal language for this logic, where formulas use only the connectives ¬
Hilbert_system
Algebraic manipulation of "true" and "false"
propositional logic and equational theorems of Boolean algebra. Every tautology Φ of propositional logic can be expressed as the Boolean equation Φ = 1, which
Boolean_algebra
Polish mathematician
students alike. Kalicki published 13 papers on logical matrices and equational logic in the five years before his death. "Jan Kalicki biography". Archived
Jan_Kalicki
being a•(a → a)* ≤ a. Unlike models of the equational theory of Kleene algebras (the regular expression equations), the star operation of action algebras
Action_algebra
Decidable theory of equality
first-order logic, all valid formulas are provable using axioms of first-order logic and the equality axioms (see also equational logic). Decidability
Theory_of_pure_equality
Book by George Boole
Aristotle's logic to formulas in the form of equations—by itself a revolutionary idea. Second, in the realm of logic's problems, Boole's addition of equation solving
The_Laws_of_Thought
Characteristic of some logical systems
include: SLD resolution on Horn clauses, superposition on equational clausal first-order logic, and Robinson's resolution on clause sets. The latter is
Completeness_(logic)
1969 non-fiction book by G. Spencer-Brown
conventional logic. However, conventional logic relies mainly on the rule modus ponens; thus conventional logic is ponential. The equational-ponential dichotomy
Laws_of_Form
Higher-dimensional generalized digraph
A. Burroni. Higher-dimensional word problems with applications to equational logic. TCS, 115(1):43--62, 1993. R. Street. Limits indexed by category-valued
Polygraph_(mathematics)
ISBN 0444863885. Lawvere, F. William (1969). Eckmann, B. (ed.). "Ordinal sums and equational doctrines". Seminar on Triples and Categorical Homology Theory. Lecture
Doctrine_(mathematics)
Branch of mathematics
2020 Grätzer 2008, pp. 7–8 Bahturin 2013, p. 346 Pratt 2022, § 3.2 Equational Logic Mal’cev 1973, pp. 210–211 Mal’cev 1973, pp. 210–211 Cohn 2012, p. 162
Algebra
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)
Type of formal logic
Paraconsistent logic is a type of non-classical logic that allows for the coexistence of contradictory statements without leading to a logical explosion
Paraconsistent_logic
Symbol representing a mathematical concept
to an initial algebra). Theories with a non-empty set of equations are known as equational theories. The satisfiability problem for free theories is
Function_symbol
American mathematician (1911–1996)
representation theorem Birkhoff's HSP theorem Birkhoff's theorem (equational logic) Birkhoff–von Neumann theorem Birkhoff–Kakutani theorem Pierce–Birkhoff
Garrett_Birkhoff
Subfield of mathematics
Mathematical logic is the study of formal logic within mathematics. Major subareas include model theory, proof theory, set theory, and recursion theory
Mathematical_logic
Theory of logic to account for observations from quantum theory
In the mathematical study of logic and the physical analysis of quantum foundations, quantum logic is a set of rules for manipulation of propositions
Quantum_logic
ISSN 0002-5240. MR 0743465. S2CID 121598599. Pöschel, R. (1989). "The equational logic for graph algebras". Z. Math. Logik Grundlag. Math. 35 (3): 273–282
Graph_algebra
Formal system of logic
In mathematics and logic, a higher-order logic (HOL) is a form of logic that is distinguished from first-order logic by additional quantifiers and, sometimes
Higher-order_logic
Computer programming paradigm
unstructured programs. The value-free style of FP is closely related to the equational logic of a cartesian-closed category. The canonical function-level programming
Function-level_programming
Algorithmic process of solving equations
In logic and computer science, specifically automated reasoning, unification is an algorithmic process of solving equations between symbolic expressions
Unification (computer science)
Unification_(computer_science)
Burroni (1993). Higher dimensional word problems with applications to equational logic (PDF). Theoretical Computer Science. n-monoid at the nLab v t e
N-monoid
Technical treatment of Boolean algebras
Boolean algebras are models of the equational theory of two values; this definition is equivalent to the lattice and ring definitions. Boolean algebra
Boolean algebras canonically defined
Boolean_algebras_canonically_defined
Mathematical use of "for all" and "there exists"
In mathematical logic, quantifiers are formal counterparts of natural-language adjectives like all, some, most, few, etc. which indicate the number of
Quantifier_(logic)
The history of logic deals with the study of the development of the science of valid inference (logic). Formal logics developed in ancient times in India
History_of_logic
Programming paradigm based on formal logic
reconcile the logic-based declarative approach to knowledge representation with Planner's procedural approach. Hayes (1973) developed an equational language
Logic_programming
Existence of values making formula true
determining whether a sentence of first-order logic is satisfiable is not decidable. In universal algebra, equational theory, and automated theorem proving,
Satisfiability
Reconfigurable digital circuit element
programmable logic device (PLD) is an electronic component used to build reconfigurable digital circuits. Unlike digital logic constructed using discrete logic gates
Programmable_logic_device
(Sep 1994). "Set Constraints in Some Equational Theories". Proc. 1st Int. Conf. on Constraints in Computational Logics (CCL). LNCS. Vol. 845. Springer. pp
Set_constraint
Solving symbolic inequations
(help) Comon, Hubert (1990). "Equational Formulas in Order-Sorted Algebras". Proc. ICALP. Comon shows that the first-order logic theory of equality and sort
Dis-unification
which consists of predicates and Horn clauses for logic programming, and functions and equations for functional programming. ALF was designed to be genuine
Algebraic Logic Functional programming language
Algebraic_Logic_Functional_programming_language
Logic gate type
convert logic equations from Karnaugh and Quine–McCluskey logic reductions. Most logic optimization result in a sum-of-products or product-of-sums logic expression
AND-OR-invert
Field-programmable semiconductor devices
Programmable Array Logic (PAL) is a family of programmable logic device semiconductors used to implement logic functions in digital circuits that was
Programmable_Array_Logic
Subfield of automated reasoning and mathematical logic
below. E is a high-performance prover for full first-order logic, but built on a purely equational calculus, originally developed in the automated reasoning
Automated_theorem_proving
Relationship between programs and proofs
proofs of intuitionistic propositional logic and the combinators of typed combinatory logic share a common equational theory, the theory of cartesian closed
Curry–Howard_correspondence
Spanish computer scientist (born 1950)
Computer Science using equational logic, rewriting logic, and the theory of general logics. He is the inventor of rewriting logic and the main developer
Jose_Meseguer
Study of discrete mathematical structures
studied in discrete mathematics include integers, graphs, and statements in logic. By contrast, discrete mathematics excludes topics in "continuous mathematics"
Discrete_mathematics
Logical formalism using combinators instead of variables
Combinatory logic is a notation to eliminate the need for quantified variables in mathematical logic. It was introduced by Moses Schönfinkel and Haskell
Combinatory_logic
Type of logical formula
mathematical logic and logic programming, a Horn clause is a logical formula of a particular rule-like form that gives it useful properties for use in logic programming
Horn_clause
Paradoxical assertion
In philosophy and logic, the classical liar paradox or liar's paradox or antinomy of the liar is the statement of a liar that they are lying: for instance
Liar_paradox
Index of articles associated with the same name
self-reference", as in first-order logic and other logic uses, where it is contrasted with "allowing some self-reference" (higher-order logic) In detail, it may refer
First-order
This is a list of mathematical logic topics. For traditional syllogistic logic, see the list of topics in logic. See also the list of computability and
List of mathematical logic topics
List_of_mathematical_logic_topics
Formal system for transcribing expressions into equivalent terms
Abstract rewriting from the practical perspective of solving problems in equational logic. Gérard Huet, Confluent Reductions: Abstract Properties and Applications
Abstract_rewriting_system
Basic circuit in quantum computing
computation, a quantum logic gate (or simply quantum gate) is a basic quantum circuit operating on a small number of qubits. Quantum logic gates are the building
Quantum_logic_gate
Soviet-born Israeli mathematician (1944–2024)
Sci. v. 9, 2(2007), 3-10 A. Tarski. Equational logic and equational theories of algebras. Contrib. to math. Logic. Hannover, 1966, (Amst. 1968), 275-288
Avraham_Trahtman
of algebraic structures generalizing the notion of variety by allowing equational conditions on the axioms defining the class. A trivial algebra contains
Quasivariety
of functional programming. The machine code can be optimized using the equational form of a theory of computation. Using CAM, the various mechanisms of
Categorical_abstract_machine
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
Type of functional equation (mathematics)
In mathematics, a differential equation is an equation that relates one or more unknown functions and their derivatives. In applications, the functions
Differential_equation
2005-01-25 D.M. Gabbay and O.Rodrigues, "Probabilistic Argumentation: An Equational Approach", Logica Universalis, 2015. doi:10.1007/s11787-015-0120-1
Probabilistic_argumentation
Description of non-logical symbols
In mathematical logic, a signature is a description of the non-logical symbols of a formal language. In universal algebra, a signature lists the operations
Signature_(logic)
Irish mathematician (1904–1976)
Formal Logic. 4 (3): 171–187. doi:10.1305/ndjfl/1093957574. C.A. Meredith and A.N. Prior (1968). "Equational logic". Notre Dame Journal of Formal Logic. 9
Carew_Arthur_Meredith
Mathematical formula expressing equality
equations List of scientific equations named after people Term (logic) Theory of equations Cancelling out As such an equation can be rewritten P – Q = 0
Equation
English mathematician and philosopher (1815–1864)
differential equations and algebraic logic, and is best known as the author of The Laws of Thought (1854), which contains Boolean algebra. Boolean logic, essential
George_Boole
Description of a quantum-mechanical system
The Schrödinger equation is a partial differential equation that governs the wave function of a non-relativistic quantum-mechanical system. Its discovery
Schrödinger_equation
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
Extension of modal logic
In logic, philosophy, and theoretical computer science, dynamic logic is an extension of modal logic capable of encoding properties of computer programs
Dynamic_logic_(modal_logic)
Programming language
the symbol “=:=” is used for equational constraints in order to provide a syntactic distinction from defining equations. Similarly, extra variables (i
Curry_(programming_language)
Derivation of the laws of probability theory
nature of the conjunction in propositional logic, the consistency with logic gives a functional equation saying that the function g {\displaystyle g}
Cox's_theorem
Concept in logic
original expression. Where ψ and φ represent formulas of propositional logic, ψ is a substitution instance of φ if and only if ψ may be obtained from
Substitution_(logic)
Function that outputs either true or false
Bit Boolean data type Boolean algebra (logic) Boolean domain Boolean logic Propositional calculus Truth table Logic minimization Indicator function Predicate
Boolean-valued_function
Relativistic quantum mechanical wave equation
In particle physics, the Dirac equation is a relativistic wave equation derived by British physicist Paul Dirac in 1928. In its free form, or including
Dirac_equation
Mapping of mathematical formulas to a particular meaning
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 theory
Structure (mathematical logic)
Structure_(mathematical_logic)
American mathematician (1920-2005)
Rosenbloom's research includes analysis, special functions, differential equations, logic, and the teaching of mathematics. In the academic year 1959–1960 he
Paul_C._Rosenbloom
Formal logic whose entailment relation is not monotonic
A non-monotonic logic is a formal logic whose entailment relation is not monotonic. In other words, non-monotonic logics are devised to capture and represent
Non-monotonic_logic
Functional equation characterizing associative binary operations
Triangular norm Aggregation function Abel equation Cox's theorem Jaynes, Edwin T. (2003). "2". Probability Theory: The Logic of Science. Cambridge University Press
Associativity_equation
Process in digital electronics and integrated circuit design
Logic optimization is a process of finding an equivalent representation of the specified logic circuit under one or more specified constraints. This process
Logic_optimization
Replacing subterm in a formula with another term
Handbook of Logic in Artificial Intelligence and Logic Programming, Volume 1. Jürgen Avenhaus and Klaus Madlener. "Term rewriting and equational reasoning"
Rewriting
(help) Kirchner, C.; Kirchner, H. (2014). "Equational Logic and Rewriting" (PDF). Handbook of the History of Logic. 9: 255–282. doi:10.1016/B978-0-444-51624-4
Abstract_rewriting_machine
McCune proved the conjecture using the automated theorem prover EQP (equational prover). EQP was developed by McCune while working at the Mathematics
Robbins_algebra
Relativistic wave equation in quantum mechanics
In particle physics, the Klein–Gordon equation is a relativistic wave equation for spinless particles. It was discovered 1926 as the relativistic generalization
Klein–Gordon_equation
Computer science and logic conference
influential. Leo Bachmair, Nachum Dershowitz, Jieh Hsiang, "Orderings for Equational Proofs" E. Allen Emerson, Chin-Laung Lei, "Efficient Model Checking in
Symposium on Logic in Computer Science
Symposium_on_Logic_in_Computer_Science
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
Topics referred to by the same term
superposition of univariate functions Superposition calculus, used in logic for equational first-order reasoning Dalton's law Law of superposition in geology
Superposition (disambiguation)
Superposition_(disambiguation)
of mathematical logic; see also history of logic. 1847 – George Boole proposes symbolic logic in The Mathematical Analysis of Logic, defining what is
Timeline of mathematical logic
Timeline_of_mathematical_logic
Necessary condition for optimality associated with dynamic programming
variable as of the next-to-last period decision.[clarification needed] This logic continues recursively back in time, until the first period decision rule
Bellman_equation
combinatorial logic (such as AND OR gates) and a flip-flop. In other words, a small Boolean logic equation can be built within each macrocell. This equation will
Simple programmable logic device
Simple_programmable_logic_device
Research association in computer science
Interest Group on Logic and Computation. It publishes a news magazine (SIGLOG News), and has the annual ACM–IEEE Symposium on Logic in Computer Science
ACM_SIGLOG
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
Concurrent constraint logic programming is a version of constraint logic programming aimed primarily at programming concurrent processes rather than (or
Concurrent constraint logic programming
Concurrent_constraint_logic_programming
Polynomial equation whose integer solutions are sought
Diophantine equation is a polynomial equation with integer coefficients, for which only integer solutions are of interest. A linear Diophantine equation equates
Diophantine_equation
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