RuleML2015 The Herbrand Manifesto - Thinking Inside the Box RuleML
The traditional semantics for First Order Logic (sometimes called Tarskian semantics) is based on the notion of interpretations of constants. Herbrand semantics is an alternative semantics based directly on truth assignments for ground sentences rather than interpretations of constants. Herbrand semantics is simpler and more intuitive than Tarskian semantics; and, consequently, it is easier to teach and learn. Moreover, it is more expressive. For example, while it is not possible to finitely axiomatize integer arithmetic with Tarskian semantics, this can be done easily with Herbrand Semantics. The downside is a loss of some common logical properties, such as compactness and completeness. However, there is no loss of inferential power. Anything that can be proved according to Tarskian semantics can also be proved according to Herbrand semantics. In this presentation, we define Herbrand semantics; we look at the implications for research on logic and rules systems and automated reasoning; and and we assess the potential for popularizing logic.
presupposition
types of presuppostion
properties of presupposition
implicature
types of implicature
properties of implicature
Grice's theory of implicature
Coperative principle
conversational Maxims
Relevance theory
RuleML2015 The Herbrand Manifesto - Thinking Inside the Box RuleML
The traditional semantics for First Order Logic (sometimes called Tarskian semantics) is based on the notion of interpretations of constants. Herbrand semantics is an alternative semantics based directly on truth assignments for ground sentences rather than interpretations of constants. Herbrand semantics is simpler and more intuitive than Tarskian semantics; and, consequently, it is easier to teach and learn. Moreover, it is more expressive. For example, while it is not possible to finitely axiomatize integer arithmetic with Tarskian semantics, this can be done easily with Herbrand Semantics. The downside is a loss of some common logical properties, such as compactness and completeness. However, there is no loss of inferential power. Anything that can be proved according to Tarskian semantics can also be proved according to Herbrand semantics. In this presentation, we define Herbrand semantics; we look at the implications for research on logic and rules systems and automated reasoning; and and we assess the potential for popularizing logic.
presupposition
types of presuppostion
properties of presupposition
implicature
types of implicature
properties of implicature
Grice's theory of implicature
Coperative principle
conversational Maxims
Relevance theory
SYNTACTIC ANALYSIS BASED ON MORPHOLOGICAL CHARACTERISTIC FEATURES OF THE ROMA...kevig
This paper gives complete guidelines for authors submitting papers for the AIRCC Journals. This paper refers to the syntactic analysis of phrases in Romanian, as an important process of natural language processing. We will suggest a real-time solution, based on the idea of using some words or groups of words that indicate grammatical category; and some specific endings of some parts of sentence. Our idea is based on some characteristics of the Romanian language, where some prepositions, adverbs or some specific endings can provide a lot of information about the structure of a complex sentence. Such characteristics can be found in other languages, too, such as French. Using a special grammar, we developed a system (DIASEXP) that can perform a dialogue in natural language with assertive and interogative sentences about a “story” (a set of sentences describing some events from the real life).
SYNTACTIC ANALYSIS BASED ON MORPHOLOGICAL CHARACTERISTIC FEATURES OF THE ROMA...kevig
This paper gives complete guidelines for authors submitting papers for the AIRCC Journals. This paper refers to the syntactic analysis of phrases in Romanian, as an important process of natural language processing. We will suggest a real-time solution, based on the idea of using some words or groups of words that indicate grammatical category; and some specific endings of some parts of sentence. Our idea is based on some characteristics of the Romanian language, where some prepositions, adverbs or some specific endings can provide a lot of information about the structure of a complex sentence. Such characteristics can be found in other languages, too, such as French. Using a special grammar, we developed a system (DIASEXP) that can perform a dialogue in natural language with assertive and interogative sentences about a “story” (a set of sentences describing some events from the real life).
1Week 3 Section 1.4 Predicates and Quantifiers As.docxjoyjonna282
1
Week 3: Section 1.4 Predicates and Quantifiers
Assume that the universe of discourse is all the people who are participating in
this course. Also, let us assume that we know each person in the course. Consider the
following statement: “She/he is over 6 feet tall”. This statement is not a proposition
since we cannot say that it either true or false until we replace the variable (she/he) by a
person’s name. The statement “She/he is over 6 feet tall” may be denoted by the symbol
P(n) where n stands for the variable and P, the predicate, “is over six feet tall”. The
symbol P (or lower case p) is used because once the variable is replaced (by a person’s
name in this case) the above statement becomes a proposition.
For example, if we know that Jim is over 6 feet tall, the statement “Jim is over six
feet tall” is a (true) proposition. The truth set of a predicate is all values in the domain
that make it a true statement. Another example, consider the statement, “for all real
numbers x, x2 –5x + 6 = (x - 2) (x – 3)”. We could let Q(x) stand for x2 –5x + 6 = (x - 2)
(x – 3). Also, we note that the truth values of Q(x) are indeed all real numbers.
Quantifiers:
There are two quantifiers used in mathematics: “for all” and “there exists”. The
symbol used “for all” is an upside down A, namely, . The symbol used for “there
exists” is a backwards E, namely, . We realize that the standard, every day usage of the
English language does not necessarily coincide with the Mathematical usage of English,
so we have to clarify what we mean by the two quantifiers.
For all For every For each For any
There exists at least one There exists There is Some
The table indicates that the mathematical meaning of the universal quantifier, for
all, coincides with our everyday usage of this term. However, the mathematical meaning
of the existential quantifier does not. When we use the word “some” in everyday
language we ordinarily mean two or more; yet, in mathematics the word “some” means at
least one, which is true when there is exactly one.
The Negation of the “For all “Quantifier:
Consider the statement “All people in this course are over 6 feet tall.” Assume it
is false (I am not over six feet tall). How do we prove it is false? All we have to do is to
point to one person to prove the statement is false. That is, all we need to do is give one
counterexample. We need only show that there exists at least one person in this class
who is not over 6 feet tall. Here is a more formal procedure.
Example 1:
Let P(n)stand for “people in this course are over 6 feet tall”, then the sentence
“All people in this course are over 6 feet tall” can be written as: “ n P(n)”. The negative,
“ ( n P(n))”, is equivalent to: “ n( P(n))”. So, in English the negative is, “There is
(there is at least one/ there exists/ some) a person in this room who is not over 6 feet tall.”
2
Example 2:
How w ...
Knowledge Based Reasoning: Agents, Facets of Knowledge. Logic and Inferences: Formal Logic,
Propositional and First Order Logic, Resolution in Propositional and First Order Logic, Deductive
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How to Make a Field invisible in Odoo 17Celine George
It is possible to hide or invisible some fields in odoo. Commonly using “invisible” attribute in the field definition to invisible the fields. This slide will show how to make a field invisible in odoo 17.
Operation “Blue Star” is the only event in the history of Independent India where the state went into war with its own people. Even after about 40 years it is not clear if it was culmination of states anger over people of the region, a political game of power or start of dictatorial chapter in the democratic setup.
The people of Punjab felt alienated from main stream due to denial of their just demands during a long democratic struggle since independence. As it happen all over the word, it led to militant struggle with great loss of lives of military, police and civilian personnel. Killing of Indira Gandhi and massacre of innocent Sikhs in Delhi and other India cities was also associated with this movement.
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This slides describes the basic concepts of ICT, basics of Email, Emerging Technology and Digital Initiatives in Education. This presentations aligns with the UGC Paper I syllabus.
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This presentation provides a briefing on how to upload submissions and documents in Google Classroom. It was prepared as part of an orientation for new Sainik School in-service teacher trainees. As a training officer, my goal is to ensure that you are comfortable and proficient with this essential tool for managing assignments and fostering student engagement.
1. About what cannot be said.
Maarten Marx
University of Amsterdam
May 15, 2014
2. Abstract
It is not easy to think of things that you cannot express. Language seems
limitless. On the other hand, we often use expressions like ”I cannot find
the words to thank you/say how much I love you/express my sadness” etc.
For artificial languages, it is often possible to give semantic
characterizations of their expressive power. We can use these to show that
some concept is not expressible. This is a powerful tool. Without it, we
can only say that until now we did not find a way to express the concept in
the language.But we do not know if this is because of the limits of our
knowledge or because of the limits of the language. In this presentation we
make these notions precise and show with examples how they can be used
in concrete situations.
3. Outline
1 Natural Languages
2 Expressing things succinctly
Numbers
Still used in some Georgian shops: tallying
Roman Numbers
What we all know and use: decimal system
Numbers in computers: zeros and ones
Propositional logic
Basis of all reasoning
3 basic types of sentences
Spotting contradictions
3 Limits of expressivity
First order logic and numbers
Integers and Rationals
Rationals and Reals
4 Take home message
5. Natural Languages
For natural languages, linguists have given bounds on the expressivity
of the morphology and the grammar.
E.g., Chomsky claimed that English morphology can be generated by
a regular language and English grammar by a context-free language.
Chomsky has/had the idea that people are born with some kind of
language or grammar-instinct.
But then grammars cannot be arbitrarily difficult/expressive.
It is a sport among linguists to find counterexamples to proposed
bounds.
7. Expressing things succinctly
Our aim
Show that different languages can express the same, but using less
resources.
Less resources??? That just means ”easier or shorter”
Our method
We first look at representing numbers.
Then at the basis of all reasoning: propositional logic.
8. Expressing quantities
We want to create a notation for expressing positive quantities:
1, 2, 3, 4, . . .
We evaluate our notation on how much work it is to write down a
number.
9. Still used in some Georgian shops: tallying
Definition: tallying
I, II, III, . . . , IIII, IIII I, . . . , IIII IIII
Evaluation
Clearly works, every number can be expressed
Uses really only one symbol: I
Size of the expression equals the quantity which is expressed
This is not handy for very rich people. . .
10. Roman Numbers
Definition
I, II, III, IIII, IV , . . . , IX, . . . , D, . . . , C, . . . , L, . . . , M
Evaluation
Clearly works, every number can be expressed
Uses several different symbols
Value of a symbol depends on its position: compare eg IX and XI.
Size of the expression much smaller than the quantity which is
expressed
except for really large numbers (as M is the ”largest” symbol)
11. Decimal system
Definition
1, 2, 3, . . . , 10, 11, . . . , 100, . . . , 1.000, . . . , 1.000.000, . . .
Evaluation
Uses 10 different symbols
Value of a symbol depends on its position:
In 538,
the 5 means 500,
the 3 means 30,
and the 8 means 8
To express a number n we need less than log10(n) + 1 digits.
For example, we express 1000 with 4 digits.
To express the enormous wealth of Ivanishvili we need only 10 or 11
digits.
Very handy and efficient
12. Numbers in computers: zeros and ones
Definition
1, 10, 11, 100, 101, 110, 111, 1000, . . .
Evaluation
Used in every computer.
Value of a symbol depends on its position: In 101 stands the first 1
for 22 and the last 1 for 20, and thus 101 means five.
With tallying, we used 1 symbol. In our decimal system we use 10
symbols. Here we use just 2 symbols.
Question Is the zero-one system more like tallying or more like the
decimal system in terms of the size of the expressions?
13. Answer: more like the decimal system
Number of inhabitants of Georgia
Tallying: IIII IIII IIII IIII . . . IIII IIII IIII IIII IIII IIII (very very big)
Decimal: 4.000.000
Binary string with length at most log2(4.000.000), which is
log2(4 ∗ 1.000 ∗ 1.000) = log2(4) + log2(1.000) + log2(1.000) =
2 + 10 + 10 = 22
Number of inhabitants of the world
Tallying: 8 billion: 2.000 times as long as that for Georgia
Decimal: 8 billion uses only 10 digits, about 1.5 times the number of
digits for Georgia
Binary log2(Georgia) + log2(2.000) = 22 + 11 = 33. Also just about
1.5 times longer.
14. Conclusion
Different ways of expressing the same notions
can differ very much in the ease or costs needed to express something.
15. Propositional logic
Our aim
Show that checking whether you state something that makes sense can be
very time consuming.
Our method
Employ elementary reasoning
16. Propositional logic: Basis of all reasoning
Atomic propositions
”It is hot.”
”This talk is too long.”
Represent them by symbols p, q, r, . . ..
Can be true or false. They are atoms. We do not look inside.
Connectives
not φ, φ and ψ, φ or ψ
Examples
p and not q
not ( p or q )
p or not (q and p)
p and p
17. 3 basic types of sentences
Tautology
Is always true. Example
18. 3 basic types of sentences
Tautology
Is always true. Example p or not p
Contingency
Can be true and can be false. Example
19. 3 basic types of sentences
Tautology
Is always true. Example p or not p
Contingency
Can be true and can be false. Example p or q
Contradiction
Is always false. Example
20. 3 basic types of sentences
Tautology
Is always true. Example p or not p
Contingency
Can be true and can be false. Example p or q
Contradiction
Is always false. Example p and not p
22. Ex falso quodlibet
We better be careful with stating something, because,
when we state a contradiction,
everything follows from it.
23. Example
Assume you have as true
A This talk is great
B This talk is not great
We show that ”Santa Claus exists” follows.
from A we can derive
C This talk is great or Santa Claus exists.
Combining B and C we get,
D Santa Claus exists.
Don’t worry, we can also deduce that Santa Clause does not exist.
26. Easy when formula justs lists possible worlds
Special shape
not only in front of atoms
formula is a disjunction of conjunctions
(p and q and not r) or ( p and q and not p) or (not q and . . . ) or . . .
Spotting contradictions
Is very easy and quick:
for each disjunct, check if it contains an atom and its negation;
if all are like that, the formula is a contradiction
otherwise it is not.
How long does that take?
We just walk through the formula from left to right
and store at most one disjunct in our memory.
27. But life is not always as simple
Rewrite formulas
Every formula φ can be rewritten into this special form,
without changing its meaning.
φ is thus a contradiction precisely if its special form is one.
Rewriting is expensive
In bad cases, the equivalent of φ in special form
is exponentially larger than φ.
Example: a φ of 400 symbols leads then to a special form of 2400
symbols.
This is very much. A gigabyte is 230 bytes. A terrabyte is 240 bytes.
A petabyte 250
2400 is roughly a 1 with 120 zeros, which is roughly the number of
atoms in the entire universe. . .
28. What to learn from this all
Spotting contradictions is hard
In eassence, nobody knows if there is a truely better way of spotting
contradictions than the one just sketched.
Artificial Intelligence is all about trying to do things faster in special
cases.
Expressivity
Different languages may both express the same thing
but one can be much more efficient/handy/succinct than the other
No we will look at real limits of expressive power.
29. Showing that a language has limits
Can express and cannot express
In general it is harder to show that you (and nobody else too) cannot
express something in a language
than to show that you can
in the latter case, you just do it
Example
Express ”exclusive or” in propositional logic.
p xor q is true precisly if exactly one of p and q is true
30. Showing that a language has limits
Can express and cannot express
In general it is harder to show that you (and nobody else too) cannot
express something in a language
than to show that you can
in the latter case, you just do it
Example
Express ”exclusive or” in propositional logic.
p xor q is true precisly if exactly one of p and q is true
(p and not q) or (not p and q)
31. First order logic and numbers
Talk about individuals and relations among them
Add constants, variables, quantifiers, and relations to propositional
logic.
Example
There is no largest number
32. First order logic and numbers
Talk about individuals and relations among them
Add constants, variables, quantifiers, and relations to propositional
logic.
Example
There is no largest number
not∃x∀y. y ≤ x
Every donkey has a tail
33. First order logic and numbers
Talk about individuals and relations among them
Add constants, variables, quantifiers, and relations to propositional
logic.
Example
There is no largest number
not∃x∀y. y ≤ x
Every donkey has a tail
∀ x. if DONKEY(x) then ∃ y. TAIL(x,y).
This wants Facebook: all friends of your friends are your own friends
if x Friend y and y Friend z then x Friend z.
34. Integers and Rationals
Integers Z: 1,2,3,4,. . . . Rationals Q: . . . , 1
2, . . . , 2
3, . . . , 1, . . .
We speak about them in First Order Logic using only the relation <
(smaller than).
Question: can we express the difference between integers and
rationals in FOL?
Yes!
35. Integers and Rationals
Integers Z: 1,2,3,4,. . . . Rationals Q: . . . , 1
2, . . . , 2
3, . . . , 1, . . .
We speak about them in First Order Logic using only the relation <
(smaller than).
Question: can we express the difference between integers and
rationals in FOL?
Yes!
Between any two different numbers there is another number.
36. Integers and Rationals
Integers Z: 1,2,3,4,. . . . Rationals Q: . . . , 1
2, . . . , 2
3, . . . , 1, . . .
We speak about them in First Order Logic using only the relation <
(smaller than).
Question: can we express the difference between integers and
rationals in FOL?
Yes!
Between any two different numbers there is another number.
∀x∀y. x < y ⇒ ∃z(x < z & z < y)
37. Difference between finite and infinite
Question
Can we express the difference between all integers and a finite number
of integers?
38. Difference between finite and infinite
Question
Can we express the difference between all integers and a finite number
of integers?
Yes
39. Difference between finite and infinite
Question
Can we express the difference between all integers and a finite number
of integers?
Yes
”there is a largest number”
∃x. not∃y. x < y
The same question but only using the equality relation.
We need to know how many elements there are in the finite set. Say n
∃x1∃x2 . . . ∃xn+1(x1 = x2&x1 = x3& . . .)
Note Formula is not general, and can be very long.
40. Rationals and Reals
Reals
Reals, R: all numbers which you can write using decimal expansion.
Thus all rationals, but also
√
2, π, . . .
Question
Can we express the difference between these two in FOL?
41. Rationals and Reals
Reals
Reals, R: all numbers which you can write using decimal expansion.
Thus all rationals, but also
√
2, π, . . .
Question
Can we express the difference between these two in FOL?
No!
To show that we need to play games.
42. Model comparison games
Rules
Two players (Spoiler and Duplicator) play on two models
Spoiler starts and picks a number in one model
Dupicator answers by picking a number in the other model.
these numbers are connected by a line.
and so on
Duplicator must always answer so that the lines do not cross.
if she cannot answer she lost, if she can continue forever she wins.
Games and Logic
Two models make the same formulas true if Duplicator can win any
game played on them.
43. What we learn from this
If you want to express the difference between R and Q you need a
language stronger than FOL.
The difference is:
in R every upward bounded set of numbers has a least upper bound.
The least upper bound of {n ∈ Q | n < π} is π, but π does not exists
in Q.
Note that we quantified over SETS, not just over elements.
44. Take home message
Expressivity of languages
Every language (artificial and natural) has limits to what it can
express.
These limits say something about the complexity of the language.
Know your limits
If you cannot express what you want in some language, then
it is either your fault, or
the ”fault” (the limits) of the language.
If you know these limits, you can either search further, or accept
reality and be in peace.