Uses of Greek Letters and Other Symbols in CMPSCI 601
letter name typical uses in CMPSCI 601
α
alpha
formula, etc.
β
beta
formula
γ
gamma formula
Γ
Gamma alphabet, vocabulary, set of formulas
δ
delta
formula, transition function
∆
Delta
change = new - old
epsilon empty string, small positive real number
ζ
zeta
constant zero function
η
eta
mapping
θ
theta
formula
Θ
Theta
ι
iota
κ
kappa cardinal number, program counter
λ
lambda function abstraction, e.g., λx(x2 )
µ
mu
interpretation function on terms, minimization function
ν
nu
ξ
xi
o
omicron
π
pi
3.14159265 · · · , prime number
Π
Pi
set of predicate symbols, proof, product
ρ
rho
σ
sigma successor function, symbol in Σ
Σ
Sigma alphabet, vocabulary, set of formulas, sum
τ
tau
Υ
Upsilon
ϕ
phi
formula
Φ
Phi
set of formulas, SO formula, set of function symbols
χ
chi
characteristic function
ψ
psi
formula
Ψ
Psi
set of formulas, second-order formula
ω
omega formula
Ω
Omega lower bound Ω(f (n))
1
symbol
#
?
≤
∩
∪
⊥
·
◦
⇒⇐
∼
=
↓
∅
≡
,→
∃
∀
d·e
b·c
iff
∧
∨
¬
→
f :A→B
7→
⇒
↔
⇔
log
|=
`
%
⊕
∼
℘
t
⊆
⊂
6=
>
.
name
number sign
star
less than or equal
intersection
union
bottom
box
cdot
circ
Contradiction
iso
downarrow
emptyset
equiv
is an abbreviation for
exists
forall
ceiling
floor
iff
land
lor
lnot
rightarrow
rightarrow
mapsto
Rightarrow
leftrightarrow
Leftrightarrow
log
models
proves
nearrow
oplus
sim
power set
sqcup
subseteq
psubset
top
triangleright
typical meaning or uses in CMPSCI 601
separator, #a (w) = number of a’s in w
Σ? = set of all finite words from Σ
less than or equal; is reducible to; substructure of
intersection
union
“false” in a logical formula
end of proof, definition, etc.
indicates place for an argument, multiplication
composition or concatenation
contradiction in informal (metamathematical) statement
isomorphic
M (w)↓ means M converges on input w
emptyset
equivalent, semantically equivalent, elementarily equivalent
e.g., (α → β) ,→ (¬α ∨ β)
there exists
for all
smallest integer greater than or equal to
largest integer less than or equal to
if and only if
logical and
logical or
logical not
implies in a logical formula
f is a function from A to B
a 7→ b means that the map takes a to b
implies in informal (metamathematical) statement
iff in a logical formula
iff in informal (metamathematical) statement
log base 2
A |= ϕ means “ϕ is true in A”
Γ ` ϕ means “ϕ can be proved from Γ”
M (w) =% means M diverges on input w
exclusive or, sum mod 2
has samecardinality,
is
equivalent to
℘(S) = A A ⊆ S
space symbol on TM tape
subset or equal to
proper subset of
“true” in a logical formula
left marker on TM tape
2
other letters name
A
cal A
B
cal B
C
cal C
L
cal L
N
bf N
Q
bf Q
R
bf R
Z
bf Z
ℵ0
aleph 0
typical meaning or uses in CMPSCI 601
logical structure
logical structure
complexity class
language, L(M ) = language accepted by M
the set of natural numbers, N = {0, 1, 2, . . .}
the set of rational numbers
the set of real numbers
the set of integers, Z = {. . . , −2, −1, 0, 1, 2, . . .}
cardinality of N
name
DSPACE
NSPACE
DTIME
NTIME
ASPACE
ATIME
name
r.e.
co-r.e.
Recursive
Primitive Recursive
EXPTIME
PSPACE
PH
NP
P
NL
L
CFL
Regular
Complexity Measures
deterministic space
nondeterministic space
deterministic time
nondeterministic time
alternating space
alternating time
Complexity Classes
recursively enumumerable sets
sets whose complements are r.e.
recursive sets
primitive recursive sets
O(1)
exponential time DTIME[2n ]
polynomial space DSPACE[nO(1) ]
polynomial-time hierarchy
nondeterministic polynomial time, NTIME[nO(1) ]
polynomial time, DTIME[nO(1) ]
nondeterministic logspace, NSPACE[log n]
logspace, DSPACE[log n]
context-free languages
regular languages
3