Summary of Grammar
Results
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Decidability
Everything about regular
Membership in CFLs and CSLs
CKY for CFLs
Emptiness for CFLs
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UCF (Charles E.
Sample Question#7
7.
Present a Context-Free Grammar,
G, such that L(G) = cfw_ aibjck | i<k or
j<k .
What is max(L(G)?
What is min(L(G)?
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11
Sample Question#8
8.
Assuming
Recursion Theorem
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ClickReproducibility subtitle style
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Simple Form
Theorem: There is an index e, such
that
x e(x) = e
G
This means that we have a function
that always produc
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Rules to Abide By
Do Your Own Work
Late Assignments
When you turn in an assignment, you are implicitly
telling me that these are the fruits of your labor. D
Context Free # 1
Leftmost versus rightmost derivations.
Parse trees, A-trees. Definition of a parse tree and
proof that A =>* X iff there exists an A-tree with
X as its yield.
Ambiguity and leftmost (
Hilberts Belief
All mathematics could be developed
within a formal system that allowed
the mechanical creation and
checking of proofs.
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Gdel
In 1931 he showed that an
How They relate
Let S be an arbitrary subset of some universe U.
The predicate S over U may be defined by:
S(x) = true if and only if x
S
S is called the characteristic function of S.
Let K be some
Hilberts Tenth
Diophantine Equations are style
Click to edit Master subtitleSemi-decidable
One Variable Diophantine Equations are Solvable
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Hilberts 10th is SemiDecidable
Consider over one vari
Useful Macros # 1
GOTO A
Z Z+1
IF Z = 0 GOTO A
IF V = 0 GOTO A
IF V = 0 GOTO E
GOTO A
[E]
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11
Useful Macros # 2
V
[A]
IF V = 0 GOTO E
V V-1
GOTO A
[E]
V
k+1 / assume we
Register Machines
2nd to edit
ClickModel Master subtitle style
Feels Like Assembly Language
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Register Machine Concepts
A register machine consists of a finite length
program, each of whose inst
Final Exam Topics
Exclusions as Master subtitle style
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Material is from 111 on.
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Exclusions
No explicit Turing Machines to write
No explicit FRSs to write
No exp
First Order Predicate
Calculus
Undecidability and subtitle style
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First Order Primitive
Symbols
Universe of discourse: U
Variables: x, x1, x2, , y, y1, y2,
Undecidability of Finite
Convergence for Operators
on Formal Languages
Relation to Master subtitle style
Click to editReal-Time
(Constant Time) Execution
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Simple Operators
Concatenation
A
bB
In
Some Known Results
Regular languages, A and B
Ah
A
A h B is regular
A * B is not necessarily regular
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k ] B is regular, for all k>0
Deciding whether or not A
an open problem
UCF (Charl
Exam#2 Review
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Material
You are responsible for material that
was covered on Exam#1 and the
next few days (reducibility and Rices
Theorem).
Study notes throu
Term Rewriting
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Types of Rewriting
String rewriting is just grammars and the variety
of rewriting systems posed by Post. In fact. Lsystems are a form of conc
Lindenmayer systems
Grammars and Biology
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Modeling Plants
Massively inspired by
Prusinkiewicz & Lindenmayer
The algorithmic beauty of plants, 1990, Springer Verlag
A
Bibliography
Prusinkiewicz & Lindenmayer
The algorithmic beauty of plants, 1990, Springer Verlag
Prusinkiewicz et al.
L-systems and beyond, Siggraph 2003 course notes
Both available online at:
http:/a
Consequences
We cannot decide if the set of valid
terminating traces of an arbitrary
machine M are finite.
Put differently, we cannot decide if
there is an upper bound on the
length of any valid trace
Diadic PIPC
Diadic limits us to two variables
PIPC means Partial Implicational
Propositional Calculus, and limits us
to implication as only connective
Partial just means we get a fragment
Problems
Is
Factor Replacement
Systems
3rd Model
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Deceptively Simple
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Factor Replacement
Concepts
A factor replacement system (FRS) consists of a
finite (ordered) sequenc
Variants of Petri Nets
A Petri Net is not computationally complete. In fact, its
halting and word problems are decidable. However, its
containment problem (are the markings of one net
contained in tho
Recursive Functions
Primitive and -Recursive
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Primitive Recursive
An Incomplete Model
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Basis of PRFs
The primitive
COT 5310
Fall 2006
Midterm#1 Sample
Name:_
Generally useful information. The notation z = <x,y> denotes the pairing function with inverses x = <z>1 and y = <z>2. The minimization notation y [P(.,y)] m
COT 5310
Fall 2006
Midterm#1 Sample
Name:
KEY
16 1. Choosing from among (REC) recursive, (RE) re non-recursive, (NR) non-re, categorize each of the sets in a) through d). Justify your answer by showin
COT 5310 Homework 2 Key
(a) Present a Register Machine that computes FIB. Assume R1 = x; at termination set R2 = 1 if x is a member of the Fibonacci sequence and 0 if not. 1. 2. 3. 4. 5. 6. 7. 8. 9. 1
COT 5310 Homework 3 Key
3. Show that prfs are closed under mutual recursion. That is, assuming F1 , F2 and G1 and G2 are pr, show that H1 and H2 are, where H1 (0, x) = F1 (x) H1 (y + 1, x) = G1 (y, x,
COT 5310: Assignment 4
1. Present a Turing Machine to do MAX of n non-zero arguments, n>=0. You know you've run out of arguments when you encounter the value 0, represented by two successive 0's (blan
COT 5310 Homework 5 Key Fall 2007 1. Let INF = cfw_f : dom(f ) is infinite and NE = cfw_f : there is a y such that f (y) converges. Show that NE m INF. Present the mapping and then explain why it work