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Joined 2026-09-21 Message
Incoming links: Search algorithm
Action schema 2026-10-24
A parameterized action definition with typed parameters, preconditions, add effects and delete effects. Replacing its parameters by type-correct objects yields a ground action. Type restrictions and static inequalities can remove invalid instantiations before the action is considered by a search algorithm or SATPlan.
AND-OR search 2026-10-24
A search algorithm whose OR nodes represent controllable choices and whose AND nodes require solutions for every possible outcome of a chosen action. In nondeterministic planning, a successful OR choice needs successful continuations for all its AND children. In contingent planning, those children can represent different observations and updated belief states. Cycle handling must match the required solution concept: ignoring a repeated state is not a proof that the goal will inevitably be reached.
Artificial intelligence Created 2026-10-18 Updated 2026-11-05
Artificial intelligence studies computational systems that perform tasks involving inference, learning, planning, perception or action under uncertainty. A classifier can learn to map measured features to labels; a planning system can search for a sequence of actions satisfying a goal. The capabilities depend on the model, training information and evaluation setting. Success on one benchmark does not by itself establish general competence or human-like understanding. Machine learning, symbolic reasoning and search algorithms offer different, sometimes combined, ways of building such systems.
Complete search algorithm 2026-10-24
A search algorithm is complete for a specified class of problems if it finds a solution whenever a solution exists in that class. This definition need not require termination on unsolvable instances; an additional termination guarantee should be stated separately. A finite branching factor makes breadth-first tree search complete for finite-depth solutions, while depth-first tree search additionally needs a reason it cannot be trapped on an infinite branch.
Goal state 2026-10-25
A state satisfying a problem's goal test. Whether its generation or its later selection terminates a search algorithm affects optimality for algorithms such as A-star search. A goal state's remaining optimal cost is zero, whereas its accumulated path cost need not be zero.
Optimal search algorithm 2026-10-24
A search algorithm is optimal for a specified objective if its returned solution has the best objective value among all solutions. The objective is part of the claim: minimizing the number of actions does not generally maximize a reward assigned to terminal states. Breadth-first tree search optimizes total cost when all actions have the same positive cost.
In terms of minimum plan length, allowing more stacks can only help:Every legal plan using at most stacks remains legal with available: simply ignore the new stack. The inequality need not be strict. For , one move suffices for every .
With , put each of the smaller coins on its own spare stack, move the largest coin to the destination, and then place the smaller coins on it in decreasing size order. This uses moves. Every smaller coin must leave the initial stack before the largest coin's first move. It cannot move just once directly to the final stack and stay there, since that would block the largest coin's eventual arrival. Thus each smaller coin requires at least two moves, while the largest coin requires at least one. HenceCompared with the three-stack value , the physical task can therefore become substantially easier.
The difficulty of finding an optimal plan is a different issue. Each legal state is determined by assigning every coin to one of the labelled stacks, whose size order is then forced. There are such assignments, so the state space and possible branching factor grow with . An uninformed search algorithm can therefore face more alternatives even when the optimal plan becomes shorter. Likewise, a simple recursive strategy for three stacks need not immediately supply the best strategy with more stacks. The defensible answer is that the required number of moves never increases, while computational search effort need not decrease.
Use a linear search over every pair of array indices.For a worst-case lower bound, consider an input in which every inspected element differs from . If an element remains uninspected, changing just that element to gives an indistinguishable execution requiring the opposite answer. Therefore a correct exact search algorithm must inspect all elements in the worst case.
PROCEDURE SearchAll(x: INTEGER; VAR i, j: INTEGER): BOOLEAN;
VAR r, c: INTEGER;
BEGIN
FOR r := 0 TO M-1 DO
FOR c := 0 TO N-1 DO
IF f[r,c] = x THEN i := r; j := c; RETURN TRUE END
END
END;
RETURN FALSE
END SearchAll;The printed phrase “at least ” uses the upper-bound notation for a lower-bound claim. Its precise interpretation isThis is an adversary argument for access to arbitrary independent array entries.
Use half-open rectangles . Split the longer dimension at its middle. If rows are the longer dimension, binary search the middle row for its first entry at least . Equality returns immediately; otherwise part (e) leaves only an upper-right and a lower-left rectangle. A middle-column split uses the transposed argument.The lower-bound binary search invariant says that entries before
PROCEDURE SearchRect(x, r0, r1, c0, c1: INTEGER;
VAR i, j: INTEGER): BOOLEAN;
VAR mid, lo, hi, k: INTEGER;
BEGIN
IF (r0 >= r1) OR (c0 >= c1) THEN RETURN FALSE END;
IF r1-r0 >= c1-c0 THEN
mid := r0 + (r1-r0) DIV 2;
lo := c0; hi := c1;
WHILE lo < hi DO
k := lo + (hi-lo) DIV 2;
IF f[mid,k] < x THEN lo := k+1 ELSE hi := k END
END;
IF lo < c1 THEN
IF f[mid,lo] = x THEN i := mid; j := lo; RETURN TRUE END
END;
IF SearchRect(x, r0, mid, lo, c1, i, j) THEN
RETURN TRUE
END;
RETURN SearchRect(x, mid+1, r1, c0, lo, i, j)
ELSE
mid := c0 + (c1-c0) DIV 2;
lo := r0; hi := r1;
WHILE lo < hi DO
k := lo + (hi-lo) DIV 2;
IF f[k,mid] < x THEN lo := k+1 ELSE hi := k END
END;
IF lo < r1 THEN
IF f[lo,mid] = x THEN i := lo; j := mid; RETURN TRUE END
END;
IF SearchRect(x, r0, lo, mid+1, c1, i, j) THEN
RETURN TRUE
END;
RETURN SearchRect(x, lo, r1, c0, mid, i, j)
END
END SearchRect;
PROCEDURE DivideSearch(x: INTEGER; VAR i, j: INTEGER): BOOLEAN;
BEGIN
RETURN SearchRect(x, 0, M, 0, N, i, j)
END DivideSearch;lo are smaller than and entries from hi onward are at least , within the splitting row or column. At its end, either equality is found or the splitting line is strictly separated around . Endpoint cuts lo=c0, lo=c1, lo=r0, or lo=r1 are valid and merely make one candidate rectangle empty.The recursion invariant is that the current rectangle contains every still-possible answer assigned to that call. The discarded rectangles are excluded by monotonicity; both retained rectangles inherit monotonicity. Each nonempty child has a strictly smaller longest dimension after at most two levels, so the recursion terminates, including one-row and one-column cases. The concise construction is
Two complementary rectangles survive a middle-row search
. For a monotone matrix, the middle row's binary search separates entries smaller than from entries greater than when no equality is found. The upper-left and lower-right rectangles are excluded by monotonicity; only the blue and orange rectangles require further search algorithm. Details.
Peter Norvig Created 2026-10-24 Updated 2026-11-05
Peter Norvig is a computer scientist whose work combines artificial intelligence, computer programming and technical education. With Stuart J. Russell, he wrote Artificial Intelligence: A Modern Approach by Stuart Russell and Peter Norvig, which connects search algorithms, logical reasoning, probabilistic reasoning and machine learning through the design of intelligent agents.[ref]
Artificial Intelligence: A Modern Approach by Stuart Russell and Peter Norvig. Author byline and fourth-edition table of contents
His programming explanations also connect compact working examples with general ideas: his spelling-correction tutorial illustrates how candidate generation and a probability model can cooperate in selecting a correction.[ref]
Portrait of Peter Norvig in 2019
. Christopher Michel, photograph; Filetime, existing Commons crop. CC BY-SA 4.0. Source..jpg){kind=link}
Search algorithm Created 2026-10-20 Updated 2026-11-05
A search algorithm looks for an object satisfying a specified condition within a collection or state space. Linear search inspects entries directly; binary search exploits sorted order; breadth-first search explores graph states by distance from a start. The specification should say whether to return one witness, every witness or proof of absence. A procedure that eventually finds every reachable witness may still fail to terminate when none exists in an infinite space. Representation and available structure determine which candidates can safely be ruled out without inspection.
Search frontier 2026-10-25
The generated but unexpanded search nodes maintained by a search algorithm. Best-first search selects the next expansion from this frontier using a priority queue or an equivalent ordering operation. A lower-bound argument about a frontier node on an optimal path is central to A-star search correctness.
State-space search 2026-11-05
A state-space search solves a problem by exploring a transition system: a state records the current situation, a transition relation specifies legal moves, and a goal state specifies success. The answer is usually a path in a transition system, rather than just the final configuration. This separates modeling the problem from choosing the search algorithm.
For a sliding-tile puzzle, the state is the arrangement of tiles and the empty square; one move slides a neighboring tile into that square. Breadth-first search finds a solution using the fewest moves when all moves have equal cost. Depth-first search can use less working storage but need not find a shortest solution. Remembering visited configurations prevents repeatedly exploring a cycle. The same formulation applies to route finding and automated planning.[ref]
Artificial Intelligence: A Modern Approach by Stuart Russell and Peter Norvig. Chapter 3, problem formulation and search strategies
Stuart J. Russell Created 2026-10-24 Updated 2026-11-05
Stuart J. Russell is a computer scientist whose work concerns artificial intelligence and the design of systems that choose actions from available information. With Peter Norvig, he wrote Artificial Intelligence: A Modern Approach by Stuart Russell and Peter Norvig. Its agent perspective connects search algorithms, first-order logic, probabilistic reasoning, decision theory and machine learning.[ref]
Artificial Intelligence: A Modern Approach by Stuart Russell and Peter Norvig. Author byline and chapters 2–22 in the fourth-edition contents
His research and lectures also examine uncertainty about an intelligent system's objectives. That question differs from uncertainty about its observations: even a system that predicts its environment accurately can need information about what outcome its user wants.[ref]
Stuart Russell's Berkeley research profile. Research areas; Uncertainty in Objectives and Human Compatible links
Tree search 2026-10-24
A search algorithm that explores action histories in a search tree. Two histories ending in the same state remain distinct nodes unless an additional repetition rule identifies them. Consequently a finite state space can generate an infinite search tree: a legal stay action at a nonterminal state already suffices. A graph search instead records explored states to control repetitions. Confusing these models changes termination and completeness claims.