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Logics for computability are formulations of logic that capture some aspect of computability as a basic notion. This usually involves a mix of special
Logics_for_computability
Framework for studying interactive computational tasks through logic
Interactive computation Logic Logics for computability G. Japaridze, Introduction to computability logic. Annals of Pure and Applied Logic 123 (2003), pages
Computability_logic
Deductive system for computable functions by Dana Scott
Logic of Computable Functions (LCF) is a deductive system for computable functions proposed by Dana Scott in 1969 in a memorandum unpublished until 1993
Logic_of_Computable_Functions
1970s automated theorem prover
Logic for Computable Functions (LCF) is an interactive automated theorem prover developed at Stanford and Edinburgh by Robin Milner and collaborators in
Logic for Computable Functions
Logic_for_Computable_Functions
Study of computable functions and Turing degrees
these areas, computability theory overlaps with proof theory and effective descriptive set theory. Basic questions addressed by computability theory include:
Computability_theory
Subfield of mathematics
(also known as computability theory). Research in mathematical logic commonly addresses the mathematical properties of formal systems of logic such as their
Mathematical_logic
Value indicating the relation of a proposition to truth
Multi-valued logics (such as fuzzy logic and relevance logic) allow for more than two truth values, possibly containing some internal structure. For example
Truth_value
Realizability Theory: Ties constructive logic to computability — proofs correspond to algorithms. Topos Logic: Internal logics of topoi (generalized spaces) are
Constructive_logic
Overview of and topical guide to logic
Classical logic Computability logic Deontic logic Dependence logic Description logic Deviant logic Doxastic logic Epistemic logic First-order logic Formal
Outline_of_logic
Topics referred to by the same term
computation, a scientific area concerned with computing with mathematical formulas Symbolic dynamics, a method for modeling dynamical systems by a discrete
Symbolic
Ability to solve a problem by an effective procedure
Computability is the ability to solve a problem by an effective procedure. It is a key topic of the field of computability theory within mathematical logic
Computability
Problem-solving procedures with certain characteristics
mathematical logic, and computability theory, an effective method or effective procedure is a finite-time, deterministic procedure for solving a problem
Effective_method
Formal systems of logic that significantly differ from standard logical systems
Non-classical logics (sometimes alternative logics) are formal systems that differ in a significant way from standard logical systems such as propositional
Non-classical_logic
Whether a decision problem has an effective method to derive the answer
Modal logic, Oxford Logic Guides, vol. 35, Oxford University Press, ISBN 978-0-19-853779-3, MR 1464942 Davis, Martin (2013) [1958], Computability and Unsolvability
Decidability_(logic)
Academic journal
The Journal of Logic and Computation is a peer-reviewed academic journal focused on logic and computing. It was established in 1990 and is published by
Journal of Logic and Computation
Journal_of_Logic_and_Computation
well-formed formulae of some formal language, is a natural-valued function. Computability theory is essentially based on natural numbers and natural (or integer)
Integer-valued_function
Propositional calculus in which there are more than two truth values
(1993). Many-valued logics. Clarendon Press. ISBN 978-0-19-853787-8. S. Gottwald, A Treatise on Many-Valued Logics. Studies in Logic and Computation, vol
Many-valued_logic
mathematical logic topics. For traditional syllogistic logic, see the list of topics in logic. See also the list of computability and complexity topics for more
List of mathematical logic topics
List_of_mathematical_logic_topics
presents "Turing's Thesis", asserting the identity of computability in general with computability by Turing machines, as an equivalent form of Church's
Timeline of mathematical logic
Timeline_of_mathematical_logic
Set with algorithmic membership test
In computability theory, a set of natural numbers is computable (or decidable or recursive) if there is an algorithm that computes the membership of every
Computable_set
Study of correct reasoning
higher-order logics are logics in the strict sense. When understood in a wide sense, logic encompasses both formal and informal logic. Informal logic uses non-formal
Logic
Thesis on the nature of computability
In computability theory, the Church–Turing thesis is a thesis about the nature of computable functions. It states that a function on the natural numbers
Church–Turing_thesis
Tool for performing logic operations
Mathematics portal Logics for computability Stanhope Demonstrator Jevons, William Stanley. "xxiii". Elementary Lessons in Logic. Barrett, Lindsay; Connell
Logical_machine
Logic circuitry that requires low temperatures to achieve superconductivity
Superconducting quantum computing is the application of superconducting logics in quantum computing. Superconducting digital logic circuits use single flux
Superconducting_computing
Type of formal logic
formula is not a tautology in deontic modal logic, since what ought to be true can be false. Modal logics are formal systems that include unary operators
Modal_logic
Higher-order logic (HOL) automated theorem prover
it provides a meta-logic (a weak type theory), which is used to encode object logics like first-order logic (FOL), higher-order logic (HOL) or Zermelo–Fraenkel
Isabelle_(proof_assistant)
Application of logical methods to philosophical problems
extended logics and deviant logics. Logic itself can be defined as the study of valid inference. Classical logic is the dominant form of logic and articulates
Philosophical_logic
Erasmus Mundus co-operative MSc program
artificial intelligence, formal specification and verification, logic and computability. This basic knowledge is then applied to areas like natural language
European Master Program in Computational Logic
European_Master_Program_in_Computational_Logic
Mathematical function that can be computed by a program
Computable functions are the basic objects of study in computability theory. Informally, a function is computable if there is an algorithm that computes
Computable_function
In computability theory, a maximal set is a coinfinite computably enumerable subset A of the natural numbers such that for every further computably enumerable
Maximal set (computability theory)
Maximal_set_(computability_theory)
Molecule that performs a logical operation
fields such as molecular electronics, biosensing, DNA computing, nanorobotics, and cell imaging. For logic gates with a single input, there are four possible
Molecular_logic_gate
Existence of values making formula true
satisfiability for an input formula in a given logic may differ from that of deciding finite satisfiability; in fact, for some logics, only one of them
Satisfiability
System for reasoning about vagueness
etc. In mathematical logic, there are several formal systems of "fuzzy logic", most of which are in the family of t-norm fuzzy logics. The most important
Fuzzy_logic
Study of mathematical analysis seen through computability theory
mathematics and computer science, computable analysis is the study of mathematical analysis from the perspective of computability theory. It is concerned with
Computable_analysis
In computability theory, a Turing degree [X] is high if it is computable in 0′, and the Turing jump [X′] is 0′′, which is the greatest possible degree
High_(computability)
Computation model defining an abstract machine
data from given input data. Computability theory, which studies computability of functions from inputs to outputs, and for which Turing machines were invented
Turing_machine
System including an indeterminate value
with the more commonly known bivalent logics (such as classical sentential or Boolean logic) which provide only for true and false. Emil Leon Post is credited
Three-valued_logic
Various systems of symbolic logic
Yurii Medvedev’s logic of finite problems, or Giorgi Japaridze’s computability logic. Yet such semantics persistently induce logics properly stronger
Intuitionistic_logic
Mathematical logic concept
In computability theory, a set S of natural numbers is called computably enumerable (c.e.), recursively enumerable (r.e.), semidecidable, partially decidable
Computably_enumerable_set
Concept in computer science
this property that is referred to as charge recovery logic, adiabatic circuits, or adiabatic computing (see adiabatic process). Although in practice no nonstationary
Reversible_computing
Portuguese logician (born 1951)
logics for information systems, and the use of category theory in the combination ("fibring") of multiple types of logic. She is Professor for Logic and
Cristina_Sernadas
Family of formal knowledge representation
Description logics (DL) are a family of formal knowledge representation languages. Many DLs are more expressive than propositional logic but less expressive
Description_logic
Mathematical theory of data types
between logics and programming languages. The implication in logic, "A → {\displaystyle \to } B" resembles a function from type "A" to type "B". For a variety
Type_theory
Type of formal logic
paraconsistent logic is that it rejects the principle of explosion. As a result, paraconsistent logics, unlike classical and other logics, can be used to
Paraconsistent_logic
Approach to formal semantics
independence-friendly logic and certain extensions of linear and intuitionistic logics turn out to be special fragments of computability logic, obtained merely
Game_semantics
Interpretation of intuitionistic logic
) {\displaystyle P\vee (\neg P)} . Curry–Howard correspondence Logics for computability Van Atten 2022. Troelstra, A. (1991). "History of Constructivism
Brouwer–Heyting–Kolmogorov interpretation
Brouwer–Heyting–Kolmogorov_interpretation
Brazilian logician
theory and semantics of many-valued logics and paraconsistent logics. His tableau method for many-valued logics generalized all previous treatments of
Walter_Carnielli
Book by Raymond Smullyan
abstract models of computing. With this analogy in hand, one can explore advanced topics in the mathematical theory of computability, such as Church–Turing
To_Mock_a_Mockingbird
Problem in computer science
In computability theory, the halting problem is the decision problem of, given an arbitrary computer program and an input, determining whether said program
Halting_problem
Symbol representing a property or relation in logic
interpretation given to them. While first-order logic only includes predicates that apply to individual objects, other logics may allow predicates that apply to collections
Predicate_(logic)
Formal system of logic
logics with their standard semantics are more expressive, but their model-theoretic properties are less well-behaved than those of first-order logic.
Higher-order_logic
Interactive theorem proving systems
HOL (Higher Order Logic) denotes a family of interactive theorem proving systems using similar (higher-order) logics and implementation strategies. Systems
HOL_(proof_assistant)
University. Japaridze is best known for his invention of computability logic, cirquent calculus, and Japaridze's polymodal logic. During 1985–1988 Japaridze elaborated
Giorgi_Japaridze
American annual computer science prize
in Computing". Network World. Archived from the original on December 4, 2023. Retrieved June 3, 2015. Homer, Steven and Alan L. (2001). Computability and
Turing_Award
Class of formal logics
classical logic normally only include propositional and first-order logics. In other words, the overwhelming majority of time spent studying classical logic has
Classical_logic
Approach to logic
systems, term logic still plays a significant role in the study of logic. Rather than radically breaking with term logic, modern logics typically expand
Term_logic
American academic and author (1963–2023)
research focused on "cyberlibertarianism, bitcoin, blockchain, and the logic of computing". He had been selected to be a co-editor of Boundary 2 to succeed
David_Golumbia
Study of the semantics, or interpretations, of formal and natural languages
Marcus for modal logics in the early 1960s and later championed by J. Michael Dunn, Nuel Belnap, and Hugues Leblanc for standard first-order logic. James
Semantics_(logic)
School of thought in philosophy of mathematics
in the process theories of classes, sets and mappings, and higher-order logics other than with Henkin semantics have come to be regarded as extralogical
Logicism
System of logic in mathematics and philosophy
called the Łukasiewicz–Tarski logic. It belongs to the classes of t-norm fuzzy logics and substructural logics. Łukasiewicz logic was motivated by Aristotle's
Łukasiewicz_logic
American logician (born 1932)
branch of mathematics that provides a foundation for the theory of programming languages and computability. Dana Stewart Scott was born on October 11, 1932
Dana_Scott
due chiefly to Kleene Turing, A. M. (1937). "Computability and λ-Definability". The Journal of Symbolic Logic. 2 (4): 153–163. doi:10.2307/2268280. ISSN 0022-4812
List of pioneers in computer science
List_of_pioneers_in_computer_science
Type of non-monotonic logic
autoepistemic, default and priority logics, and parallel circumscription. In Proceedings of the Sixth European Workshop on Logics in Artificial Intelligence (JELIA'98)
Default_logic
Type of logical system
symbols. For example, infinitary logics permit formulas of infinite size, and modal logics add symbols for possibility and necessity. First-order logic can
First-order_logic
Method of deriving conclusions
Modal Logics Sider 2010, pp. 133–136, 227–228 Garson 2024, § 3. Deontic Logics Garson 2024, § 1. What is Modal Logic?, § 4. Temporal Logics Sider 2010
Rule_of_inference
Typographic symbol
uses in mathematics, computing, and typography. It has many names, often related to particular meanings: Sheffer stroke (in logic), pipe, bar, or (literally
Vertical_bar
Theory of logic to account for observations from quantum theory
article. For discussion of the similarities and differences between quantum logic and some of these competitors, see § Relationship to other logics. Quantum
Quantum_logic
Computer system simulating intelligence
of the human brain with probabilistic thinking, fuzzy logic and multi-valued logic. Soft computing can process a wealth of data and perform a large number
Computational_intelligence
Type of memory used on processors that require high transfer rate memory
Retrieved 2026-09-14. "Samsung reveals three-phase HBM roadmap that puts logic and compute inside memory — zHBM ultimately stacks DRAM directly on top of the
High_Bandwidth_Memory
Device performing a Boolean function
output. Depending on the context, the term may refer to an ideal logic gate, one that has, for instance, zero rise time and unlimited fan-out, or it may refer
Logic_gate
Impossible task in computing
of Truth" for a history leading to, and a discussion of, his proof. Soare, Robert I., "Computability and recursion", Bull. Symbolic Logic 2 (1996), no
Entscheidungsproblem
Branch of logic
and are only dealt with in nonclassical logics, called erotetic and imperative logics. In propositional logic, a statement can contain one or more other
Propositional_logic
General-purpose functional programming language
Standard ML is a modern dialect of ML, the language used in the Logic for Computable Functions (LCF) theorem-proving project. It is distinctive among
Standard_ML
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
Type of computer applications
Fuzzy logic Granular computing Rough set Type-2 fuzzy sets and systems Vagueness Lotfi Zadeh [7], the father of fuzzy logic, coined the phrase computing with
Perceptual_computing
Affirms the existence of a computable universal function
In computability theory, the UTM theorem, or universal Turing machine theorem, is a basic result about Gödel numberings of the set of computable functions
UTM_theorem
Characteristic of some logical systems
theorem, and neither is its negation). In superintuitionistic and modal logics, a logic is structurally complete if every admissible rule is a derivable implication
Completeness_(logic)
Yes-or-no question that cannot ever be solved by a computer
In computability theory and computational complexity theory, an undecidable problem is a decision problem for which it is proved to be impossible to construct
Undecidable_problem
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)
Form of logic
A free logic is a logic with fewer existential presuppositions than classical logic. Free logics may allow for terms that do not denote any object. Free
Free_logic
In computability theory, a Turing degree [X] is low if its Turing jump [X′] = 0′. A subset S ⊂ N {\displaystyle S\subset \mathbb {N} } is low if its Turing
Low_(computability)
Computer that uses photons or light waves
computing or photonic computing uses light waves produced by lasers or incoherent sources for data processing, data storage or data communication for
Optical_computing
Study of computation
can be computed and what amount of resources are required to perform those computations. In an effort to answer the first question, computability theory
Computer_science
Formal semantics of logic programming languages
{\displaystyle B_{i}} is true, then H {\displaystyle H} is true". Logic programs compute the set of facts that are implied by their rules. Many implementations
Syntax and semantics of logic programming
Syntax_and_semantics_of_logic_programming
Form of logic that allows quantification over predicates
expressiveness of the logics over finite structures; for example, if PH = PSPACE, then adding a transitive closure operator to second-order logic would not make
Second-order_logic
Logic with discrete truth values
finite-valued logic encompasses both finitely many-valued logic and bivalent logic. Fuzzy logics, which allow for degrees of values between "true" and "false", are
Finite-valued_logic
Theorem in mathematical logic
Heinz-Dieter Ebbinghaus Extended logics: the general framework in K. J. Barwise and S. Feferman, editors, Model-theoretic logics, 1985 ISBN 0-387-90936-2 page
Lindström's_theorem
mathematical logic, specifically computability theory, a function f : R → R {\displaystyle f:\mathbb {R} \to \mathbb {R} } is sequentially computable if, for every
Computable_real_function
Statistical interpretation with many tests
Benjamini-Hochberg procedure. For continuous problems, one can employ Bayesian logic to compute m {\displaystyle m} from the prior-to-posterior volume ratio. Continuous
Multiple_comparisons_problem
Reasoning about equations with free variables
models appropriate for the study of various logics (in the form of classes of algebras that constitute the algebraic semantics for these deductive systems)
Algebraic_logic
Many-valued logic in which truth values comprise a continuous range
a system of three-valued logic in 1920. He generalized the system to many-valued logics in 1922 and went on to develop logics with ℵ 0 {\displaystyle \aleph
Infinite-valued_logic
1989 monograph by Marian Pour-El and J. Ian Richards
Computability in Analysis and Physics is a monograph on computable analysis by Marian Pour-El and J. Ian Richards. It was published by Springer-Verlag
Computability in Analysis and Physics
Computability_in_Analysis_and_Physics
Mathematical-logic system
usual for such a proof, computable means computable by any model of computation that is Turing complete. In fact computability can itself be defined via
Lambda_calculus
Academic subfield of computer science
computability theory builds on the halting problem result. Another important step in computability theory was Rice's theorem, which states that for all
Theory_of_computation
Study of the scope and nature of logic
first-order logic, extended logics, and deviant logics. Extended logics accept the basic formalism and the axioms of classical logic but extend them with new
Philosophy_of_logic
Assignment of meaning to the symbols of a formal language
The most commonly studied formal logics are propositional logic, predicate logic and their modal analogs, and for these there are standard ways of presenting
Interpretation_(logic)
Theorem in mathematical logic
characterize first-order logic. Although there are some generalizations of the compactness theorem to non-first-order logics, the compactness theorem
Compactness_theorem
Fundamental theorem in mathematical logic
same is true of all higher-order logics. It is possible to produce sound deductive systems for higher-order logics, but no such system can be complete
Gödel's_completeness_theorem
This is a list of computability and complexity topics, by Wikipedia page. Computability theory is the part of the theory of computation that deals with
List of computability and complexity topics
List_of_computability_and_complexity_topics
American mathematician
theory (particularly the idea of the set-theoretic multiverse), in computability theory, and in group theory. After earning a Bachelor of Science in
Joel_David_Hamkins
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LOGICS FOR-COMPUTABILITY
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LOGICS FOR-COMPUTABILITY
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