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COM2003 Automata, Logic & Computation Logical Analysis & Inference: Introduction Mark Stevenson Department of Computer Science University of Sheffield

Mark Stevenson (U. Sheffield)

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Course Details — Semester 2 • Instructor:

Mark Stevenson (m.stevenson@dcs.shef.ac.uk)

• Module homepage: http://staffwww.dcs.shef.ac.uk/people/M.Stevenson/campus only/com2003/

 link from my personal homepage • Consult the homepage for:  all key course details  announcements  lecture slides  tutorial / lab exercises Mark Stevenson (U. Sheffield)

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Course Aims & Topics This module teaches the mathematical and logical tools for modelling and analysing computing systems: state machines, formal languages, logics, proof systems, and logic programming. Semester 2 heading:

Logic, Proof and Computation

• logical analysis and inference • propositional and predicate logic: syntax and semantics • translating from English to logic  both sentences, and whole arguments

• deduction and proof  both ‘natural’ and computational approaches

• computation through logic  look at Prolog as example of a logic programming language

Mark Stevenson (U. Sheffield)

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to uncovering such a situation, and metaheuristic search techniques are then employed to seek failures in the implementation. As a simple example, take the wrapping counter function of Figure 28. This function implements a counter, which takes an integer value between 0 and 10, and returns the increment. If the input is 10, the counter wraps round to 0. The pre-condition for this function is simply:

Relevance within Computer Science

n ≥ 0 ∧ n ≤ 10

• Required to understand research papers and background material The post-condition is:

(n < 10 → r = n + 1) ∨ (n = 10 → r = 0) where n is the input value and r is the return value.

• Leads to better understanding constructs A constraint system is then derivedoftoprogramming describe conditions of implementation

non-conformance by taking the pre-condition in conjunction with the negated

• post-condition: Used in many areas of CS, including Semantic Web and Natural Language Processing

n ≥ 0 ∧ n ≤ 10 ∧ ¬((n < 10 → r = n + 1) ∨ (n = 10 → r = 0))

(1)

obj(n ≥ 0) + obj(n ≤ 10)+ min((obj(n < 10) + obj(r &= n + 1)), (obj(n = 10) + obj(r &= 0)))

(2)

AnEncounter applications in some level 3 how and “close” 4 courses objective function is derived to indicate failure is .

This is constructed from the above constraint system using the rules in Tables 2 and 3:

It was(U. found that Mark Stevenson Sheffield)

the COM2003: landscapes of the objective functions derived from such Automata, Logic & Computation

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Inference and Logical Form • Logic is the Science of Reasoning (one view, at least!) • Domain includes study of argumentation. • Argument — idealised to be:  Sequence of sentences (n), beginning with some premises (n − 1), and ending with a conclusion , i.e. is of form:

Premise1 , . . . Premisen−1 therefore Conclusion

• May be written:

Premise1 .. . Premisen−1 Conclusion

Mark Stevenson (U. Sheffield)

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Inference & Logical Form (contd)

â&#x20AC;˘ Sometimes, we can immediately see that an argument is correct or incorrect:

e.g. (1) All humans are mortal Socrates is a human Socrates is mortal e.g. (2) All humans are mortal Fido is mortal Fido is a human

Mark Stevenson (U. Sheffield)

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Inference & Logical Form (contd) e.g. (1) All humans are mortal Socrates is a human Socrates is mortal

â&#x20AC;˘ Argument (1) is logically valid. â&#x20AC;˘ Valid inference is truth-preserving i.e. if the premises are true, then the conclusion must also be true

â&#x20AC;˘ We say that . . .

conclusion follows from the premises conclusion is a logical consequence of the premises premises logically entail the conclusion

Mark Stevenson (U. Sheffield)

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Inference & Logical Form (contd) e.g. (2) All humans are mortal Fido is mortal Fido is a human

â&#x20AC;˘ Argument (2) is not logically valid.

 truth of its conclusion is uncertain, even if the premises are true  in this simple case, can even imagine situation where premises are true and conclusion false

â&#x20AC;˘ Logical validity does not depend on:  specific content

e.g. human vs. dog vs. politician vs. alien

 the factual truth of what is stated

â&#x20AC;˘ An argument is valid as a consequence of its logical form Mark Stevenson (U. Sheffield)

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Inference & Logical Form (contd) . . . an argument is valid as a consequence of its logical form.

â&#x20AC;˘ Following is of the same form as (1), and valid in same way: e.g. (3) All thargs are drekken Kretz is a tharg Kretz is drekken

â&#x20AC;˘ Likewise (4) has the same form, and is again valid: e.g. (4) All cats are reptiles Tiddles is a cat Tiddles is a reptile

 (4) shows a valid argument may have a false conclusion i.e. if any of the premises are false Mark Stevenson (U. Sheffield)

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Argument Schemata • Can represent common aspects of form of valid and invalid arguments by using argument schemata

• Argument (1) has schematic form:

All P are Q a is P a is Q

 always produces a valid argument • Likewise, (2) has schematic form:

All P are Q a is Q a is P

 always produces an invalid argument Mark Stevenson (U. Sheffield)

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Sentences & Propositions • A key term: PROPOSITION  used to designate what a sentence says about the world

• Synonymous sentences express the same proposition e.g. Paris is the capital of France France’s capital is Paris.

vs.

• Ambiguous sentences express more than one proposition e.g. Flying planes can be dangerous

• Proposition expressed may depend on context:

i.e. when, where, and by whom sentence is uttered e.g. I am hungry.

• For logic, it’s the proposition expressed by a sentence that matters Mark Stevenson (U. Sheffield)

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Levels of Logical Analysis • Logic addresses the logical form of sentences and arguments, and its role in determining logical validity and truth

• One approach is to find a manner of expression or notation which mirrors the relevant aspects of logical form

• Such a formal logic has three components:  a syntax:

• rules determining which strings of symbols count as formulas (‘sentences’) of the logic’s language

 a semantics:

• rules determining the meaning of formulas in the language

 a deductive system:

• rules for constructing proofs that some conclusion follows from given premises

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Levels of Logical Analysis (contd) • The analysis of logical form can be pursued at different levels or granularities

• Propositional logic studies inter -sentential relations i.e. relations between simple or compound sentences

 it ignores internal structure of simple sentences

• Example:

Either Charles is clever or Charles is rich Charles is not clever Charles is rich

Can be viewed as having underlying propositional form: (either) P or Q not P

where:

P = “Charles is clever” Q = “Charles is rich”

Q Mark Stevenson (U. Sheffield)

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Levels of Logical Analysis (contd) • Propositional logic is not fine-grained enough for all forms of reasoning

• Sometimes, need also to reason about entities and properties e.g.

All humans are mortal Socrates is a human Socrates is mortal

• For this we need predicate logic • More generally, there are a range of different logics  have different granularities  make different assumptions  are suited to different domains Mark Stevenson (U. Sheffield)

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