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Invariants used to prove properties of loops
In computer science, a loop invariant is a property of a program loop that is true before (and after) each iteration. It is a logical assertion, sometimes
Loop_invariant
Type of compiler optimization
programming, loop-invariant code consists of statements or expressions (in an imperative programming language) that can be moved outside the body of a loop without
Loop-invariant_code_motion
Property that is not changed by mathematical transformations
In mathematics, an invariant is a property of a mathematical object (or a class of mathematical objects) which remains unchanged after operations or transformations
Invariant_(mathematics)
Programming language
supports formal specification through preconditions, postconditions, loop invariants, loop variants, termination specifications and read/write framing specifications
Dafny
Generic term for compiler optimization
computer science, code motion, which includes code hoisting, code sinking, loop-invariant code motion, and code factoring, is a blanket term for any process that
Code_motion
Topics referred to by the same term
whose value doesn't change during program execution Loop invariant, a property of a program loop that is true before (and after) each iteration A data
Invariant
How software progresses through its implementation
the exit condition and the loop invariant are satisfied. Loop invariants are used to monitor specific properties of a loop during successive iterations
Control_flow
Rules to verify computer program correctness
B\wedge P\}}}} Here P is the loop invariant, which is to be preserved by the loop body S. After the loop is finished, this invariant P still holds, and moreover
Hoare_logic
Increasing execution speed and reducing the overheads associated with loops
once before the loop begins, if the resultant quantity of the calculation will be the same for every loop iteration (i.e., a loop-invariant quantity). This
Loop_optimization
Self-balancing binary search tree data structure
the root; then the reassigned constellation satisfies the respective loop invariant. A possibly numbered triangle with a black circle atop represents a
Red–black_tree
Compiler optimization
each time through the loop. Loop invariants are essentially constants within a loop, but their value may change outside of the loop. Induction variables
Strength_reduction
Problem in computer science
values of current_sum seen so far, cf. line 7 of the algorithm. As a loop invariant, in the j {\displaystyle j} th step, the old value of current_sum holds
Maximum_subarray_problem
Analysis of computer programs without executing them
Concolic testing Fuzzing Invariant inference Program slicing Testing Formal methods Concepts Curry–Howard correspondence Loop invariant Refinement Side effect
Static_program_analysis
relation by the iteration of a while loop under some invariant conditions, thereby ensuring its termination. A loop variant whose range is restricted to
Loop_variant
Of a function, an additional effect besides returning a value
while (b = 3) {} // b = 3 evaluates to 3, which then casts to true so the loop is infinite Action at a distance (computer programming) Don't-care term Effect
Side effect (computer science)
Side_effect_(computer_science)
Compiler that optimizes generated code
multiplication by a loop index with addition. Loop optimization acts on the statements that make up a loop, such as a for loop, for example loop-invariant code motion
Optimizing_compiler
Method of computer program specification
empty, or do-nothing command. There is no need for a loop invariant or least fixed point. Loops with multiple intermediate shallow and deep exits work
Predicative_programming
Low-space search for a majority element
(previously empty) set of copies of m (and set c to 1). In all cases, the loop invariant is maintained. After the entire sequence has been processed, it follows
Boyer–Moore majority vote algorithm
Boyer–Moore_majority_vote_algorithm
Validates computer program operations
Ernst. 2011. Leveraging existing instrumentation to automatically infer invariant-constrained models. In Proceedings of the 19th ACM SIGSOFT symposium and
Typestate_analysis
Graphical representation of a computer program or algorithm
empty, but passes like loop-invariant code motion could populate it. Mpre is called the loop pre-header, and Mloop would be the loop header. A control flow
Control-flow_graph
parallelized.[citation needed] Loop unswitching moves loop invariants out of a loop, reducing the number of conditional branches in each loop. Keith Cooper; Linda
Enabling_transformation
Function with unusual fractal properties
in the loop that can possibly affect the invariants are in the last two lines, and these can be shown to preserve the truth of both invariants as long
Minkowski's question-mark function
Minkowski's_question-mark_function
Substitution box used in the Rijndael cipher
void initialize_aes_sbox(uint8_t sbox[256]) { uint8_t p = 1, q = 1; /* loop invariant: p * q == 1 in the Galois field */ do { /* multiply p by 3 */ p = p
Rijndael_S-box
Specification language for Java programs
or throw an exception. invariant Defines an invariant property of the class. loop_invariant Defines a loop invariant for a loop. also Combines specification
Java_Modeling_Language
Graph data structure
to several data structure invariants. Two e-nodes are equivalent if they are in the same e-class. The congruence invariant states that an e-graph must
E-graph
formal specification through function preconditions, postconditions and loop invariants. The language uses flow-sensitive typing also termed flow typing. The
Whiley_(programming_language)
Computer program for the Boolean satisfiability problem
Concolic testing Fuzzing Invariant inference Program slicing Testing Formal methods Concepts Curry–Howard correspondence Loop invariant Refinement Side effect
SAT_solver
Topological quantum field theory
gauge-invariant operators. The most often studied class of gauge invariant operators are Wilson loops. A Wilson loop is the holonomy around a loop in M
Chern–Simons_theory
Computer science field
Concolic testing Fuzzing Invariant inference Program slicing Testing Formal methods Concepts Curry–Howard correspondence Loop invariant Refinement Side effect
Model_checking
Gauge field loop operator
field theory, Wilson loops are gauge invariant operators arising from the parallel transport of gauge variables around closed loops. They encode all gauge
Wilson_loop
Theory of quantum gravity merging quantum mechanics and general relativity
around a closed loop is written W γ [ A ] {\displaystyle W_{\gamma }[A]} and is called a Wilson loop. Thus Wilson loops are gauge invariant. The explicit
Loop_quantum_gravity
Concept in computer science
category. It required the user to input pre/post specs, loop invariants, and resource invariants for locks. It introduced a method of symbolic execution
Separation_logic
is that it performs (a form of) common subexpression elimination and loop-invariant code motion at the same time. In addition, PRE can be extended to eliminate
Partial-redundancy elimination
Partial-redundancy_elimination
Methodology of programming
control flow blocks such as for loops, while loops and if statements. Such languages may not be ideal for invariants-first programming, since they force
Invariant-based_programming
Process of analyzing computer program behavior
the edges represent the flow of control. By identifying code blocks and loops a CFG becomes a starting point for compiler-made optimizations. Data-flow
Program_analysis
Optimization replacing a function call with that function's source code
matching parameter, or part of the function may be "hoisted out" of a loop (via loop-invariant code motion). Register allocation can be done across the larger
Inline_expansion
C++ framework for compiler development
apply existing optimizations (e.g., common subexpression elimination, loop-invariant code motion) on custom dialects by means of traits and interfaces. These
MLIR_(software)
Finite, symmetry-reduced model of loop quantum gravity
couplings, then the resulting loop quantum geometry has no definitive discrete gaps or a minimum size. Consequently, in scale-invariant LQC, the Big Bang is shown
Loop_quantum_cosmology
Approach to static program analysis
and floating-point operations". WING'12 - 4th International Workshop on Invariant Generation. Manchester, United Kingdom: 16. Regehr, John; Duongsaa, Usit
Abstract_interpretation
Concepts in theoretical computer science
course on distributed systems in Munich. It assumed that properties are invariant under stuttering. The formal definition of safety given above appears
Safety and liveness properties
Safety_and_liveness_properties
Description of gauge theories using loop operators
represented singular objects. As we shall see the loop formalism goes far beyond a simple gauge invariant description, in fact it is the natural geometrical
Loop representation in gauge theories and quantum gravity
Loop_representation_in_gauge_theories_and_quantum_gravity
Algorithm used in modular arithmetic
condition t = 1 and halt, the last loop invariant implies that R2 = n. We can alternately express the loop invariants using the order of the elements: ord
Tonelli–Shanks_algorithm
Mutual exclusion algorithm
never accessed in the loop. It can then remove the writes to those variables from the loop, using a process called loop-invariant code motion. It would
Dekker's_algorithm
Compiler optimization technique
Traditional optimizations such as common subexpression elimination and loop invariant hoisting often focus on eliminating redundant computation. Since computation
Rematerialization
Feedback controller
A closed-loop controller or feedback controller is a control loop which incorporates feedback, in contrast to an open-loop controller or non-feedback
Closed-loop_controller
O(n) sorting algorithm
a (movable) index into bucket b, so Kb−1 ≤ Lb ≤ Kb. We maintain the loop invariant that each bucket is divided by Lb into an unclassified prefix (Ai for
Flashsort
Concept in computer science
successors(n) Changed = Changed U { s }; } } Dead-code elimination Loop-invariant code motion Reachable uses Static single assignment form Aho, Alfred
Reaching_definition
dead code elimination alias analysis loop unrolling induction variable elimination loop-invariant code motion loop reversal Sunitha, K.V.N. (2013). Compiler
Vbcc
Task to construct a program meeting a formal specification
synthesis also has applications to superoptimization and inference of loop invariants. During the Summer Institute of Symbolic Logic at Cornell University
Program_synthesis
Concolic testing Fuzzing Invariant inference Program slicing Testing Formal methods Concepts Curry–Howard correspondence Loop invariant Refinement Side effect
Hyperproperty
Compiler optimization
recomputing an arithmetic result when it is needed. Global value numbering Loop-invariant code motion Steven Muchnick; Muchnick and Associates (15 August 1997)
Common subexpression elimination
Common_subexpression_elimination
Keyword used in some programming languages to tag variables
sample with the read in the loop to the second sample without the read in the loop as part of the common loop-invariant code-motion optimization, and
Volatile (computer programming)
Volatile_(computer_programming)
Case in parallel computing
respected during execution to prevent incorrect results. In general, loop invariant dependencies and lexically forward dependencies can be easily vectorized
Automatic_vectorization
Improving the efficiency of software
understanding of compilers and machine code can improve performance. Loop-invariant code motion and return value optimization are examples of optimizations
Program_optimization
Reformulation of Floyd-Hoare logic
predicate INV, called the Loop INVariant, typically supplied by a programmer: To show total correctness, we also have to show that the loop terminates. For this
Predicate transformer semantics
Predicate_transformer_semantics
Set of software engineering methods
make the slice an executable slice we merely add the end brace for the for loop and the declaration of i. The resulting static executable slice is shown
Program_slicing
Fundamental problem in computer science
program size and can even be infinite in the case of programs with unbounded loop iterations. Therefore, any program analysis that attempts to explore control-flow
Path_explosion
Compiler optimization
detriment to performance. [citation needed] Additionally, loop inversion allows safe loop-invariant code motion.[citation needed][clarification needed] [clarification
Loop_inversion
JavaScript and WebAssembly engine
inlining, linear-scan register allocation, dead code elimination, and loop-invariant code motion. The compiler can emit fast native code translations of
SpiderMonkey
problem of computing dependencies within loops, which is a significant and nontrivial problem, is tackled by loop dependence analysis, which extends the
Dependence_analysis
execution. For example, a loop invariant is a condition that is true at the beginning and the end of every execution of a loop. iteration Is the repetition
Glossary_of_computer_science
Overview of and topical guide to algorithms
(computer science) — property that an algorithm satisfies its specification Loop invariant — condition used to prove correctness of iterative algorithms Recurrence
Outline_of_algorithms
Topics referred to by the same term
people Outliner, filter for viewing In computing, hoisting may refer to: Loop-invariant code motion, a compiler optimization Variable hoisting, scope rule in
Hoist
Form of differential geometry
compact metric space X is a metric invariant of X, defined to be the least length of a noncontractible loop in X (i.e. a loop that cannot be contracted to a
Systolic_geometry
Proof method in mathematical logic
whenever S has fewer nodes than T. Coinduction Initial algebra Loop invariant, analog for loops Hopcroft, John E.; Rajeev Motwani; Jeffrey D. Ullman (2001)
Structural_induction
Topics referred to by the same term
LICM may refer to: Long Island Children's Museum Loop-invariant code motion This disambiguation page lists articles associated with the title LICM. If
LICM
Study of mathematical knots
century, invariants such as "quantum" knot polynomials, Vassiliev invariants and hyperbolic invariants were discovered. These aforementioned invariants are
Knot_theory
Inverse of a finite difference
computations with direct formulas, enabling strength reduction, loop-invariant code motion, and dependence analysis without iterative re-evaluation of
Indefinite_sum
Model for a random simple path
conformally invariant. These arguments allowed to show that certain measurables of loop-erased random walk are (in the limit) conformally invariant, and that
Loop-erased_random_walk
Concolic testing Fuzzing Invariant inference Program slicing Testing Formal methods Concepts Curry–Howard correspondence Loop invariant Refinement Side effect
Polyvariance
Type of graph invariant
In mathematics, a Tutte–Grothendieck (TG) invariant is a type of graph invariant that satisfies a generalized deletion–contraction formula. Any evaluation
Tutte–Grothendieck_invariant
Invariance of operations under geometric translation
Analogously, an operator A on functions is said to be translationally invariant with respect to a translation operator T δ {\displaystyle T_{\delta }}
Translational_symmetry
Object-oriented programming language
supports a "check instruction" (a kind of "assert"), loop invariants, and loop variants (which guarantee loop termination). Void-safe ability, like static typing
Eiffel_(programming_language)
element R i , j {\displaystyle \mathrm {R} _{i,j}} gives a scale invariant (unit-invariant) measure of the dependence of output j {\displaystyle j} on input
Relative_gain_array
Isomorphism of commutative rings constructed in the theory of Lie algebras
{h}})} of a Cartan subalgebra h {\displaystyle {\mathfrak {h}}} that are invariant under the Weyl group W {\displaystyle W} . Let g {\displaystyle {\mathfrak
Harish-Chandra_isomorphism
Tools for optimizing binary code
copy propagation, constant propagation, strength reduction, loop invariant code motion and loop unrolling. Target binaries had to be relinked to its custom
Binary_optimizer
Loop seen as a trivial knot
is the least knotted of all knots. Intuitively, the unknot is a closed loop of rope without a knot tied into it, unknotted. To a knot theorist, an unknot
Unknot
by the programmer or the compiler such as pre/post-conditions and loop invariants (a form of proof-carrying code). VC generators are often coupled with
Verification condition generator
Verification_condition_generator
Mathematical invariant of a knot or link
polynomial discovered by Vaughan Jones in 1984. Specifically, it is an invariant of an oriented knot or link which assigns to each oriented knot or link
Jones_polynomial
Process to determine relationships among statements
detect not only invariant commands in loops, but also larger code fragments such as an inner loop. The analysis also detects quasi-invariants of arbitrary
Loop_dependence_analysis
Simplest non-trivial closed knot with three crossings
requires the construction of a knot invariant that distinguishes the trefoil from the unknot. The simplest such invariant is tricolorability: the trefoil
Trefoil_knot
Programming language
t n = -(n 1) if >(n 2) then fib else exit exit: return x2 Where the loop invariant of fib is that x1 is the (i+2-1)th and x2 is the (i+2)th Fibonacci number
Flow_chart_language
Topological structure in loop quantum gravity
These structures are employed in loop quantum gravity as a version of quantum foam. The covariant formulation of loop quantum gravity provides the best
Spin_foam
free. Another use for fresh variables involves the development of loop invariants in formal program verification, where it is sometimes useful to replace
Fresh_variable
Knot invariant is an invariant defined on knots which is invariant under ambient isotopies of the knot. Finite type invariant is a knot invariant that
List_of_knot_theory_topics
Branch of engineering and mathematics
there are two types of control loop: open-loop control (feedforward), and closed-loop control (feedback). In open-loop control, the control action from
Control_theory
Continuous deformation between two continuous functions
is the definition of homotopy groups and cohomotopy groups, important invariants in algebraic topology. In practice, there are technical difficulties in
Homotopy
We say that a loop property P is universal if it is isotopy invariant, that is, P holds for a loop L if and only if P holds for all loop isotopes of L
Isotopy_of_loops
American computer Scientist
theory, integration and combination of decision procedures, lemma and loop invariant generation, as well as his work in computer algebra, which helped to
Deepak_Kapur
Method to control electric motors
time varying coefficients to another with time invariant coefficients resulting in a linear time-invariant system or LTI system. Overview of key competing
Field-oriented_control
loop: for i := 0 to N do A(i) := (A(i) + A(N-i))/2 Approaches that cannot represent symbolic terms (such as the loop-invariant quantity N in the loop
Frameworks supporting the polyhedral model
Frameworks_supporting_the_polyhedral_model
How many times curves wind around each other
In mathematics, the linking number is a numerical invariant that describes the linking of two closed curves in three-dimensional space. Intuitively, the
Linking_number
Breakdown of conformal symmetry at the quantum level
"classically conserved" and the theory is classically scale invariant. However, at loop level the scale current can develop a nonzero divergence. This
Conformal_anomaly
integer-valued invariant in symplectic geometry, microlocal analysis, and semiclassical analysis. It is associated most classically with a loop or path of
Maslov_index
American mathematician
Kervaire invariant one", arXiv:0908.3724 [math.AT] Freed, Daniel S.; Hopkins, Michael J.; Teleman, Constantin (2003), "Twisted K-theory and loop group representations"
Michael_J._Hopkins
Physical theory with fields invariant under the action of local "gauge" Lie groups
differential geometry. A quantity which is gauge-invariant (i.e., invariant under gauge transformations) is the Wilson loop, which is defined over any closed path
Gauge_theory
Theorem in geometric topology
Because the (non)triviality of the fundamental group is known to be invariant under homeomorphism, it follows that the two-dimensional sphere and torus
Poincaré_conjecture
Gauge symmetry cannot be spontaneously broken
invariant operators, whose expectation values need not be zero. The most common ones are Wilson loops and their thermal equivalents, Polyakov loops.
Elitzur's_theorem
Symplectic topology tool
functional on the (universal cover of the) free loop space of a symplectic manifold. SFH is invariant under Hamiltonian isotopy of the symplectomorphism
Floer_homology
Graphical method of determining the stability of a dynamical system
plot of the open loop systems, it can be applied without explicitly computing the poles and zeros of either the closed-loop or open-loop system (although
Nyquist_stability_criterion
LOOP INVARIANT
LOOP INVARIANT
Girl/Female
Hindu
Look, Blessed with beauty, Shape, Beauty
Girl/Female
Arabic, Muslim
Look
Boy/Male
Bengali, Indian
Loop; Autumn
Male
Dutch
, Jehovah's gift (or grace).
Male
French
French form of Latin Lupus, LOUP means "wolf."
Boy/Male
Arabic
The Biblical Lot is the English Language Equivalent
Boy/Male
Hindu, Indian, Rajasthani, Sindhi, Traditional
Look; Beauty; Appearance
Boy/Male
Dutch, German, Hebrew
God will Multiply; God will Add
Boy/Male
British, English
Barrel Maker
Surname or Lastname
English
English : metonymic occupational name for a cooper, from Middle English coupe ‘tub’, ‘container’ (see Cooper). In some cases the surname may have been derived from a pub or house sign.Dutch : from koop ‘purchase’, ‘bargain’, hence a nickname for a haggler or a metonymic occupational name for a merchant.
Boy/Male
Hebrew
God will multiply.
Surname or Lastname
Dutch
Dutch : from a short form of the Germanic personal name Robrecht.Altered spelling of German Rupp.English : variant spelling of Roope.
Girl/Female
Gujarati, Hindu, Indian
Look
Girl/Female
Tamil
Look, Blessed with beauty, Shape, Beauty
Surname or Lastname
English
English : possibly from the Old Norse personal name Tópi, Túpi, a short form of a personal name formed with þórr, name of the Norse god of thunder (see Thor) + a second element with initial b-, for example björn ‘bear’, ‘warrior’. On the other hand, the name is found mainly in Dorset and Devon, which are far from areas of Scandinavian settlement.
Boy/Male
Hindu, Indian
Look
Surname or Lastname
North German
North German : habitational name from any of several places called Loose or Loosey.North German : from a short form of Nikolaus, German form of Nicholas.Dutch : nickname from the adjective loos ‘cunning’, ‘artful’, ‘guileful’.English : variant spelling of Loose.
Surname or Lastname
English (Somerset)
English (Somerset) : habitational name from Look in Puncknowle, Dorset, named in Old English with lūce ‘enclosure’.English : possibly a variant of Luck 3.Northern English and Scottish : from a vernacular pet form of Lucas.Dutch (van Look) : topographic name from look ‘enclosure’ or habitational name from a place named with this word.Thomas Look (b. c. 1622) was in Lynn, MA, by 1646. His son, also called Thomas (b. 1646), moved to Martha’s Vineyard about 1670.
Boy/Male
Hindu
Flower
Boy/Male
Indian, Sanskrit
Natural; Original; Innate; Simply; Loop
LOOP INVARIANT
LOOP INVARIANT
Girl/Female
Tamil
Nadharanjani | நாதாரஂஜநீ
Name of a Raga
Boy/Male
English American Latin
A modern English form of Laurence and the preferred form in America.
Boy/Male
Finnish, German
Army Man
Surname or Lastname
English (chiefly Somerset)
English (chiefly Somerset) : from a pet form of Fulcher.
Girl/Female
Tamil
Ayantika | அயாநà¯à®¤à¯€à®•à®¾Â
Goddess Durga, Goddess Parvati
Girl/Female
Muslim/Islamic
Gentle; name of the daughter of the Prophet (S.A.W)
Surname or Lastname
English
English : patronymic from Sim.
Boy/Male
Hawaiian, Hebrew, Hindu, Indian, Marathi
First
Girl/Female
Gaelic Irish
Gift from God.
Surname or Lastname
English
English : nickname from Middle English cr(o)us(e) ‘bold’, ‘fierce’.English (of Norman origin) : habitational name from a place in France, perhaps Cruys-Staëte in Nord, apparently named with a Gaulish word crodiu ‘hard’.German : northern variant of Krause.Americanized spelling of German Kruse.
LOOP INVARIANT
LOOP INVARIANT
LOOP INVARIANT
LOOP INVARIANT
LOOP INVARIANT
n.
See Loon, the bird.
v. i.
To seem; to appear; to have a particular appearance; as, the patient looks better; the clouds look rainy.
v. t.
To beat in the game of loo by winning every trick.
n.
See 1st Loop.
v. t.
To let hang down; as, to lop the head.
v. i.
To direct the attention (to something); to consider; to examine; as, to look at an action.
n.
The act of looking; a glance; a sight; a view; -- often in certain phrases; as, to have, get, take, throw, or cast, a look.
v. t.
To express or manifest by a look.
v. t.
To bind or fasten with hoops; as, to hoop a barrel or puncheon.
v. t.
To make a loop of or in; to fasten with a loop or loops; -- often with up; as, to loop a string; to loop up a curtain.
v. t.
To influence, overawe, or subdue by looks or presence as, to look down opposition.
v. t.
To break over the poop or stern, as a wave.
n.
Hence; Appearance; aspect; as, the house has a gloomy look; the affair has a bad look.
v. t.
To confine in a coop; hence, to shut up or confine in a narrow compass; to cramp; -- usually followed by up, sometimes by in.
v. t.
To look at; to turn the eyes toward.
n.
A ring; a circular band; anything resembling a hoop, as the cylinder (cheese hoop) in which the curd is pressed in making cheese.
pl.
of Trou-de-loup
n.
Any one of several aquatic, wed-footed, northern birds of the genus Urinator (formerly Colymbus), noted for their expertness in diving and swimming under water. The common loon, or great northern diver (Urinator imber, or Colymbus torquatus), and the red-throated loon or diver (U. septentrionalis), are the best known species. See Diver.
n.
A curve of any kind in the form of a loop.
n.
Expression of the eyes and face; manner; as, a proud or defiant look.