LOGIC

1
LOGIC MAS The pilot project conducted at the
Research Laboratory of Intelligent Systems
(LabIS http//labis.vsb.cz/ )

• The main goal
• Research of information technologies needed for
  coordination of autonomous intelligent agents in
  extraordinary or emergency situations

2
The project (Logic MAS)

• Performed by
• The Faculty of Electric Engineering and Computer
  Science
• Department of Computer Science
• The Faculty of Mining and Geology
• Institute of Geographic Information Systems
• The team consists of
• 8 senior researches (Professors Docents)
• 6 PhD students
• 3 technical staff people

3
Logic MAS the structure
4
The project (Logic MAS)

• Motto A Good Theory is
• the Best for Practice!
• Gains in addition to MAS classics
• Theoretical Background Transparent Intensional
  Logic
• Fine-grained Logical Analysis of Natural language
• Fine-grained Knowledge Representation
• Knowledge data management
• Involving space and time (Geoinformatics)
• Reasoning under vague / incomplete knowledge,
  fuzzy approach

5
Theoretical background of the Logic MAS
project

• Transparent Intensional Logic (TIL)
• Founded by the late Professor Pavel Tichý
  (Charles university till 1968, then the
  University of Dunedin)
• Professor Pavel Materna (Masaryk university of
  Brno, Czech Academy of Sciences)
• Pavel Cmorej, Fero Gahér, Bjorn Jespersen, Marie
  Duží, Jarek Müller,
• and many others .
• http//www.phil.muni.cz/fil/logika/til/

6
A fragment of TIL bibliography

• P. Tichý The Foundations of Freges Logic. De
  Gruyter 1988
• P. Tichý Svoboda, Jespersen, Cheyne, eds. Pavel
  Tichys Collected Papers in Logic and Philosophy.
  Filosofia, Prague University of Otago Press
  2004
• P. Materna, M. Duží Parmenides Principle,
  Philosophia (Israel) 32, 1-4, 155-180, 2005.
• M. Duží, P. Materna Logical Form. Essays on
  the Foundations of Mathematics and logic, G. Sica
  (ed.). Polimetrica, Monza Italy, 115-153
• M. Duží Intensional Logic and the irreducible
  contrast between de dicto and de re. ProFil,
  Vol. 5, No 1, 2004, 1-34. http//profil.muni.cz/01
  _2004/duzi_de_dicto_de_re.pdf
• B. Jespersen Explicit Intensionalization,
  Anti-Actualism, and How Smiths Murderer Might
  Not Murdered Smith, Dialectica, Vol.59, No 3,
  2005, 285-314.

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Transparent Intensional Logic (TIL)

• (a brief overview of the relevant portions of TIL
  from the MAS viewpoint)
• Logical analysis of natural language
• Knowledge representation

8
TIL (natural) languages

• Crucial question
• What is the meaning of an expression E?
• Alonzo Church Concepts are structured meanings
• Procedures structured from the algorithmic point
  of view, known as TIL constructions
• Meaning of a sentence mode of presentation
  (construction) of the denoted proposition

9
Formalizing reasoning of autonomous agents using
TIL

• A rational agent in a multi-agent world is
• able to reason about the world (what holds true),
  and
• about its own cognitive state, and
• about that of other agents
• The theory has to be able to
• talk about and quantify over the objects of
  propositional attitudes structured meanings of
  the embedded clauses
• iterate attitudes of distinct agents
• express self-referential statements
• respect different inferential abilities of
  resource bounded agents

10
TIL - Procedural theory of meanings

• Obviously, any set-theoretical theory (predicate
  logics, modal, intensional logics, epistemic
  logics, deontic, etc., etc.) is not able to
  handle structured meanings
• there is nothing about a set in virtue of which
  it may be said to present something (Zalta)
• each (general) concept is in such a theory
  identified with the respective set.
• We need to distinguish between a concept of an
  entity and the entity itself
• meaning algorithmically structured procedure
  TIL construction

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Procedural Theory of Meaning

• Pavel Tichý (1968) Sense and Procedure, later
  in Intensions in terms of Turing machines
  formulated the idea of structured meanings
• Pavel Tichý (1988) The Foundations of Freges
  Logic (TIL)
• Pavel Materna (1998) Concepts and Objects
  concept is a closed construction, (2004)
  Conceptual Systems
• Y. Moschovakis (1994, 2003) Sense and
  denotation as algorithm and value
• P. Tichý Svoboda, Jespersen, Cheyne, eds. Pavel
  Tichys Collected Papers in Logic and Philosophy.
  Filosofia, Prague University of Otago Press 2004

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TIL constructions

• Abstract procedures instructions on the way to
  the output entity
• Example Goldbach hypothesis
• Every even number is a sum of two primes
• We know the procedure (instruction) to be
  performed. However, we do not know the output
  (True?, False?).
• Understanding concerns meanings, i.e.,
  constructions (procedures)
• and only derivatively performing the procedures,
  thus obtaining the outputs (if any) e.g., a
  truth-value, or a possible-world proposition
• Notation Montague-like ?-term denoting not a
  function, but the mode of presentation of it,
  i.e., construction of the function
• Barendregt the meaning of a ?-term is the
  respective algorithm!

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Ontology of TIL

• Constructions are objects of knowing (believing
  )
• To be able to talk about the objects of attitudes
  (meanings of embedded clauses), we need not only
  to use constructions, but also to mention them
• Rich ontology entities organized in a
  two-dimensional infinite ramified hierarchy of
  types
• any entity of any type of any order (even a
  construction) can be mentioned within the theory
  without generating paradox.

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Two-dimensional infinite hierarchy
1st order no constructions, set-theoretical entities Atomic ?, ?, ?, ?, Molecular (set of partial functions) (??), (((??)?)?),
2nd order constructions of 1st order entities Atomic ?1 , Molecular (??), (((??)?)?), (??1), (((???1)?)?),
3rd order constructions of 1st and 2nd order entities Atomic ?2 , Molecular (??), (???1), (((??)?)?), (??2), (((???2)?)?),
And so on ad infinity

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Ramified hierarchy of types

• 1st order algorithmically non-structured
  (set-theoretical entities)
• Atomic types ? True, False ?
  individuals (universal universe of
  discourse) ? time points (real numbers) ?
  possible worlds (consistent maximum
  sets of facts)
• Functional types sets of partial functions
  (mappings) (?1,,?n) ? ? denoted (? ?1?n).
• Rule increasing mereological complexity
  (horizontal)
• If ?1,,?n, ? are types of order n, then (?
  ?1?n) is a type of order n.
• (?-)sets are mapped by characteristic functions
  (??).

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Remark

• TIL is an open-ended system. The above epistemic
  base ?,?,?,? was chosen, because it is apt for
  natural-language analysisbut in the case of
  mathematics, a (partially) distinct base would be
  appropriate for instance, the base consisting of
  natural numbers, of type ?, and truth-valuesof
  type ? ?, ?

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Ramified hierarchy of types

• nth order Constructions
• Variables x, y, z ... ? any type (not only
  individuals!)
• Trivialisation 0X ? object X
• Closure ? x1 ... xn C ? Function / (?
  ?1...?n) ?1 ?n ?
• Composition C X1 Xn ? Value of the
  function (? ?1...?n) ?1
  ?n ?
• molecular types (horizontal rule) sets of
  partial functions involving constructions (?1
  ?2?n), ?i ?1

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Ramified hierarchy of types

• 3rd order
• Atomic type ?2 ? constructions of 1st or 2nd
  order entities (all of them belong to type ?2)
  and
• horizontal rule of creatingMolecular types
  (?1 ?2?n), ?i ?2
• And so on ?3, ?4, ..., ?n, ..., (? ?n), (? ?m
  ?n),
• Example
• The set of improper constructions of order n
  Impropern / (? ?n) an object of the type of
  order n1

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Examples 1st-order constructions

• The function is not a construction.
• It is a mapping of type (? ??), i.e., a set of
  triples, the first two members of which are
  natural numbers, while the third member is their
  sum.
• 0 ?the simplest construction (primitive
  concept) of
• Composition 0 x 01 ?v (v-constructs) a
  successor of any number x
• Closure ?x 0 x 01 ? the successor function.
• The composition of this closure with 05, i.e.,
  ?x 0 x 01 05 ? the number 6.

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Examples 1st-order constructions

• The composition 0 x 00 does not v-construct
  anything for any valuation of x it is improper.
• The closure ?x 0 x 00 ? (??) is not improper
• it constructs something, even though it is only a
  degenerate function (one undefined at all its
  arguments).
• Members of ?1
• 0, 0 x 01, ?x 0 x 01, ?x 0 x 01 05,
  0 x 00, ?x 0 x 00,

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Examples of higher-order constructions

• A member of ?2
• 0IMPROPER 00 x 00 ? True
• The constituent 00 x 00 / ?2 is an atomic
  proper construction 00 x 00 ? 0 x 00 / ?1.
  
• It is atomic, because the construction 0 x 00
  is not used here as a constituent but only
  mentioned as an input object.

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Intensions vs. Extensions

• ?-intension a member of a type (??)
• frequently ((??)?), denoted ???
• ?-extension not a function from ?
• Examples of intensions
• student / (??)?? ? property of individuals
• the president of CR / ??? ? individual office
• Charles is a student / ??? ? proposition
• age of / (??)?? ? attribute (empirical function)
• calculate / (???n)?? ? relation-in-intension

23
Example of Analysis

• Notation
• variables w ? ?, t ? ?
• composition C w t ? Cwt
• President of CR ? ??? (an individual office
  constructed by)
• ?w?t 0Presidentwt 0CR
• (? ???) ???
• 
  ?
• Abstr. over w,t ???

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TIL semantics Shifting Frege-Church semantic
scheme

• The president of CR (Empirical) expression
•   conceptual level
• ?w?t 0Presidentwt 0CR (how) meaning
  construction
•   ontological level
• office / ??? (what) intension (
  denotation)
• a priori level
• Nobody in Feb.2003 Reference Value of
  the
• Vaclav Klaus (now) intension (in w,t)
• Empirical level, out of the scope of logic
  result of empirical information retrieval

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Method of analysing expressions

• consists of the following three steps
• Type-theoretical analysis Assign types to the
  objects talked about, i.e. only to those that are
  denoted by sub-expressions of E besides, try not
  to omit any semantically self-contained
  sub-expression of E (to use all of them).
• Synthesis Compose constructions of these objects
  so as to construct the object D denoted by E.
• Type checking Use the assigned types for control
  so as to check whether the various types are
  compatible and, furthermore, produce the right
  type of object in the intended manner.

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Examples

• The highest mountain is in Asia
• Mountain/ (??)??, BAsia/ (??)??, Highest/ (?
  (??))??, HMA / ???
• A possible analysis is also a trivialisation of
  the denoted proposition 0HMA (no good, of
  course)
• A more fine-grained analysis (but not the
  best)combining constructions of Mountain, BAsia,
  Highest as follows
• ?w ?t 0BAsia wt 0Highest wt
  0Mountain wt
• ( ? ? ) (? (? ?)) ( ? ? )
• ?
• ?
• abstraction over t (??)
• abstraction over w ((??)?), abbreviated ???
  (proposition)
• Abbreviated ?w ?t 0BAsiawt 0Highestwt
  0Mountainwt

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Example (the highest ) continued

• The highest mountain is in Asia
• Mountain/ (??)??, BAsia/ (??)??, Highest/ (?
  (??))??, HMA / ???
• Asia is the greatest continent
• We cannot deduce that
• The highest mountain is in the greatest continent
• We need to refine the analysis is (in) has a
  self-contained meaning Is / ((? ? ?))?? , Asia /
  ???
• The most adequate analysis (relative to a CS)
• ?w?t 0Iswt ?w?t 0Highestwt 0Mountainwtwt
  0Asiawt

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The (most adequate) analysis

• Relative to a given set of primitive concepts--
  ontology (conceptual system)
• enables us to infer just (all and only) the
  logical consequences of the assumptions so that
• the inference machine should not
• over infer (i.e., infer something that does
  not follow) or
• under infer (i.e., not to infer something that
  does follow)

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Example

• John Kerry wanted to become the President of USA
• The President of USA knows that John Kerry wanted
  to become the President of USA
• The President of USA is George W. Bush
• --------------------------------------------------
  --------------------------Hence what ???
• George W. Bush knows that John Kerry wanted to
  become George W. Bush ?
• How to block such an undesirable substitutions?
  Over-inferring a nonsense?
• By using TIL hyper-intensional expressive
  semantics

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Example - solution

• ?w?t 0Wantwt 0Kerry ?w?t 0Becomewt 0Kerry ?w?t
  0Pres 0USA,
• ?w?t 0 ?w?t 0Preswt 0USAwt 0Bush
• ?w?t 0Knowwt ?w?t 0Preswt 0USAwt
• 0?w?t 0Wantwt 0Kerry ?w?t 0Becomewt
  0Kerry ?w?t 0Preswt 0USA
• Types Want / (? ? ???), Become / (? ? ???), Know
  / (? ? ?1)??
• Now we can substitute 0Bush for ?w?t 0Preswt
  0USAwt thus deducing that G.W. Bush knows that
  John Kerry wanted to become the President of USA,
  
• but not that he wanted to become G.W.Bush.
• The undesirable substitution of 0Bush for the
  latter occurrence of the construction ?w?t
  0Preswt 0USA is blocked.

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Computationally intractable?

• We first need to know
• And only afterwards to infer from the assumptions
• Why recursive axiomatisation first ?
• Communication of agents must not fall into
  inconsistencies
• AI the struggle for consistency

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TIL approach to knowledge representation in a
multi-agent world

• Goal
• Particular agents have to communicate in a
  (pseudo-) natural language, in order to
  understand each other, and to provide relevant
  information whenever and where-ever needed to
  whom-ever.
• Three kinds of knowledge
• implicit (which leads to an explosion of
  knowledge and the paradox of omniscience)
• explicit (which deprives an agent of any
  inferential capabilities)
• inferable (of a realistic agent with some
  inferential capabilities, who, however, is not
  logically omniscient).

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The problem of Knowledge management

• PWS propositions as objects of knowledge
• An approach apt for handling implicit knowledge
• Intensional semantics of epistemic (modal) logics
• Kripke and Montague-Scott structures
• Montague logic
• does not have the desirable Church-Rosser
  property (cap and cup just imitate a proper
  handling of the de dicto / de re distinction)
• cannot handle inferable or explicit knowledge in
  a proper way - the problem of logical
  omniscience cannot be avoided.- the tightest
  restriction omniscience can be restricted to
  equivalence, since equivalent formulas are
  indistinguishable.

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The problem of Knowledge management

• Formulae as objects of knowledge
• Syntactic approaches
• Apt for handling Explicit knowledge
• but they are prone to inconsistency (stemming
  from self-referential statements and the
  necessity to mention formulas within the theory)
  when disquoting formulas (Tarski the
  impossibility to define a universal Truth)
• However, an agent is not related to a piece of
  syntax, but to its meaning
• Neither Montague / Kripke nor syntactic
  approaches are usable when modelling a
  multi-agent system of resource-bounded
  intelligent autonomous agents which act, but are
  not omniscient

35
TIL approach to knowledge representation in a
multi-agent world

• TIL constructions of propositions
  (hyper-intensions) as objects of knowledge
• Appropriate for handling all the three kinds of
  knowing, in particular Inferable knowledge
• However, has to meet the problem of a (in a way)
  too fine-grained individuation of knowledge
• Technically as fine-grained as the syntactic
  approach Two major distinctions

36
TIL approach to knowledge representation

• 1. an agent is not related to a formula, but to
  the meaning of the embedded clause, the
  respective construction
• ?w?t 0Knowwt 0A 0?w?t 0Card ?x 0Inhwt x
  0Prague 01048576
• ?1
• Know / (? ? ?1)??
• Agent does not have to know that the number of
  inhabitants in Prague is equal to hexa 100 000
  (165)
• 2. does not restrict the set of formulas the
  agent is said to know, instead we compute the
  inferable knowledge relative to the inference
  rule(s) the agent is able to use

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Computing inferable knowledge

• Infa / ((? ?n) (? ?n))?? ? an agents
  inferential abilities
• b a set of constructions knowledge of a
• c inferred construction inferable piece of
  knowledge
• b ? (? ?n), c ? ?n, d ? ?n
• r derivation according to the rule(s) r that a
  masters (is able to use)
• Infa ?w ?t ?b ?c b c ? ?r (b r c)

38
Example using disjunctive syllogism

• ?w ?t ?b ?c ?d b d ? b c,d?w?t ?(2d)wt ?
  (2c)wt
• Roughly if there is a d?b and (?d ? c)?b, then
  inferable c
• (the agent masters also substitution, and 20c ?
  c)
• c,d?w?t ?(2d)wt ? (2c)wt stands for
• 0Sub 0Tr c 0c 0Tr d 0d 0?w?t ?(2d)wt ?
  (2c)wt
• b ?v ?w?t 0Baldwt 0Charles, , ?w?t
  ?0Baldwt 0Charles ? 0Kingwt 0Charles,
  
• c ?v ?w?t 0Kingwt 0Charles

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example continued

• c ?v ?w?t 0Kingwt 0Charles
• d ?v ?w?t 0Baldwt 0Charles actual values (of
  formal parameters c, d)
• 0Sub 0Tr c 0c 0Tr d 0d 0?w?t ?(2d)wt ?
  (2c)wt ?v
• ?w?t ?20?w?t 0Baldwt 0Charleswt ? 20?w?t
  0Kingwt 0Charleswt ,
• which is equivalent to?w?t ?0Baldwt
  0Charles ? 0Kingwt 0Charles

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Computational semantics in details

• There are technical problems here we need to
  mention the construction by trivialising it
• calling a sub-procedure with formal parameters
  c, d.
• To release variables c, d bound by
  trivialisation, we have to use special
  substitution functions SUB
• substituting actual values for formal parameters.
  
• The upper index c,d is a notational abbreviation
  of these facilities (talking about the objects
  of attitudes)
• double-executing variables ranging over
  constructions of propositions, 2c, 2d, returns
  the respective propositions.
• (2d)wt intentional descent constructs a
  truth-value

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The function Inf(a) is postulated to be

• subclassical if ? is derived from a stock of
  knowledge ?, then ? is entailed by ? (in any
  w,t) if ? ? 0Inf(a)wt ? , then ? ?
• reflexive ? ? 0Inf(a)wt ? in any w,t.
  (The agent a does not forget what a already
  knows.)
• if the function Inf(a) is subclassical and
  reflexive, then it is monotonic if ? ? ?
  then 0Inf(a)wt ? ? 0Inf(a)wt ? .
• The function Inf(a) is not idempotent 0Inf(a)wt
  0Inf(a)wt A is not a subset of 0Inf(a)wt A
  in any w,t.

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Computing the Inferable Knowledge Epistemic
Closure

• Recursive definition (omitting trivialisations
  if used)
• K0awt Kexpawt
• Kn1awt Infawt Knawt
• Nothing other Kinfawt ?j Kjawt
• Kinfawt Infawt Kinfawt fixed point of
  Infa
• Monotonic reasoning the least fixed point
• Kinfawt ??x Infawt x ? Kexpawt

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Is the Epistemic Closure valid ?

• No, not in the non-restricted version
• Yes if defined as the least fixed-point of the
  function Infa containing an agents explicit
  knowledge relativized to agents inferential
  abilities
• Kexpawt ? Kinfawt ? Kimpawt
• idiot a masters genius some
  rules
• The logic is proposed as the logic that
  accommodates the philosophical desiderata that
  should be met in a multi-agent world

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Conclusion

• Given an agent furnished with a stock of
  recursively enumerable explicit knowledge and a
  flawless command of only some rules of inference,
  there is an upper limit to the new knowledge it
  would be logically possible for the agent to
  derive from the agents old knowledge the
  Closure.
• Logical omniscience vanquished ! (?) Epistemic
  closure vindicated ! (?)
• The theory does not take into account complexity
  problems

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Further research

• TIL Inference machine specification
• Involving complexity problems (time and space
  limitations)
• Doxastic logic of Beliefs (managing hypotheses)
• Belief revision and knowledge base update
• Non-monotonic reasoning

46
Main References

• Pavel Tichý The Foundations of Frege Logic de
  Gruyter, 1988
• Pavel Materna Concepts and Objects Acta
  Philosophica Fennica 63, 1998
• Pavel Materna Conceptual Systems 2004
• Svoboda, Jespersen, Cheyne, eds. Pavel Tichys
  Collected Papers in Logic and Philosophy.
  Filosofia, Prague University of Otago Press
  2004
• TIL home page http//www.phil.muni.cz/fil/logika/
  til
• or http//www.cs.vsb.cz/duzi