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Unformatted text preview: Computation
Part II
Introduction to Cognitive and Information Sciences
April 21, 2011 Sym Sys 100: Computation, Part II 1 Overview of This Week
Last Time: Sym Sys 100: Computation, Part II 2 Overview of This Week
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Introduction to Turing Machines Sym Sys 100: Computation, Part II 2 Overview of This Week
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Introduction to Turing Machines Examples of Coding / Universal Turing Machines Sym Sys 100: Computation, Part II 2 Overview of This Week
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Introduction to Turing Machines Examples of Coding / Universal Turing Machines ChurchTuring Thesis Sym Sys 100: Computation, Part II 2 Overview of This Week
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Introduction to Turing Machines Examples of Coding / Universal Turing Machines ChurchTuring Thesis Incomputability Sym Sys 100: Computation, Part II 2 Overview of This Week
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Introduction to Turing Machines Examples of Coding / Universal Turing Machines ChurchTuring Thesis Incomputability This Time: Sym Sys 100: Computation, Part II 2 Overview of This Week
Last Time:
Introduction to Turing Machines Examples of Coding / Universal Turing Machines ChurchTuring Thesis Incomputability This Time:
Computability and Complexity Sym Sys 100: Computation, Part II 2 Overview of This Week
Last Time:
Introduction to Turing Machines Examples of Coding / Universal Turing Machines ChurchTuring Thesis Incomputability This Time:
Computability and Complexity Other (Equivalent) Models of Computation Sym Sys 100: Computation, Part II 2 Overview of This Week
Last Time:
Introduction to Turing Machines Examples of Coding / Universal Turing Machines ChurchTuring Thesis Incomputability This Time:
Computability and Complexity Other (Equivalent) Models of Computation The Computational Theory of Mind Sym Sys 100: Computation, Part II 2 Overview of This Week
Last Time:
Introduction to Turing Machines Examples of Coding / Universal Turing Machines ChurchTuring Thesis Incomputability This Time:
Computability and Complexity Other (Equivalent) Models of Computation The Computational Theory of Mind Representation and Symbols Sym Sys 100: Computation, Part II 2 Overview of This Week
Last Time:
Introduction to Turing Machines Examples of Coding / Universal Turing Machines ChurchTuring Thesis Incomputability This Time:
Computability and Complexity Other (Equivalent) Models of Computation The Computational Theory of Mind Representation and Symbols Don’t worry too much about details today, try to focus on the big picture.
Sym Sys 100: Computation, Part II 2 Complexity In Tuesday’s lecture, we were primarily concerned with the abstract
notion of computability, and therefore time and space were assumed
to be unbounded (though time and space used in any single
computation is always ﬁnite). Sym Sys 100: Computation, Part II 3 Complexity In Tuesday’s lecture, we were primarily concerned with the abstract
notion of computability, and therefore time and space were assumed
to be unbounded (though time and space used in any single
computation is always ﬁnite). In other words, as far as computability is concerned, all Turing
Machines that compute a given function are created equal. Sym Sys 100: Computation, Part II 3 Complexity In Tuesday’s lecture, we were primarily concerned with the abstract
notion of computability, and therefore time and space were assumed
to be unbounded (though time and space used in any single
computation is always ﬁnite). In other words, as far as computability is concerned, all Turing
Machines that compute a given function are created equal. When it comes to computing in the real world, we do not have this
luxury. Time and space are of the essence. Sym Sys 100: Computation, Part II 3 Complexity Consider, as an example, diﬀerent sorting algorithms. Sym Sys 100: Computation, Part II 4 Complexity Consider, as an example, diﬀerent sorting algorithms. An instance of the sorting problem is the following:
Given some ﬁnite list of numbers as input, produce as
output that same list in order from lowest to highest. Sym Sys 100: Computation, Part II 4 Complexity Consider, as an example, diﬀerent sorting algorithms. An instance of the sorting problem is the following:
Given some ﬁnite list of numbers as input, produce as
output that same list in order from lowest to highest. For example, with input 4, 2, 7, 1, 5, we must give output 1, 2, 4, 5, 7. Sym Sys 100: Computation, Part II 4 Complexity Consider, as an example, diﬀerent sorting algorithms. An instance of the sorting problem is the following:
Given some ﬁnite list of numbers as input, produce as
output that same list in order from lowest to highest. For example, with input 4, 2, 7, 1, 5, we must give output 1, 2, 4, 5, 7. This is a central task in numerous applications. Sym Sys 100: Computation, Part II 4 Complexity Consider, as an example, diﬀerent sorting algorithms. An instance of the sorting problem is the following:
Given some ﬁnite list of numbers as input, produce as
output that same list in order from lowest to highest. For example, with input 4, 2, 7, 1, 5, we must give output 1, 2, 4, 5, 7. This is a central task in numerous applications.
Many algorithms exists, but some are better than others: Sym Sys 100: Computation, Part II 4 Complexity Consider, as an example, diﬀerent sorting algorithms. An instance of the sorting problem is the following:
Given some ﬁnite list of numbers as input, produce as
output that same list in order from lowest to highest. For example, with input 4, 2, 7, 1, 5, we must give output 1, 2, 4, 5, 7. This is a central task in numerous applications.
Many algorithms exists, but some are better than others: • Bubble Sort: Compare ﬁrst two numbers; swap if in wrong order.
Then compare second two, third two, etc. until the end. Then
repeat from the beginning, until nothing left to swap. Sym Sys 100: Computation, Part II 4 Complexity Consider, as an example, diﬀerent sorting algorithms. An instance of the sorting problem is the following:
Given some ﬁnite list of numbers as input, produce as
output that same list in order from lowest to highest. For example, with input 4, 2, 7, 1, 5, we must give output 1, 2, 4, 5, 7. This is a central task in numerous applications.
Many algorithms exists, but some are better than others: • Bubble Sort: Compare ﬁrst two numbers; swap if in wrong order.
Then compare second two, third two, etc. until the end. Then
repeat from the beginning, until nothing left to swap.
How long will this take in the worst case scenario? Sym Sys 100: Computation, Part II 4 Complexity Consider, as an example, diﬀerent sorting algorithms. An instance of the sorting problem is the following:
Given some ﬁnite list of numbers as input, produce as
output that same list in order from lowest to highest. For example, with input 4, 2, 7, 1, 5, we must give output 1, 2, 4, 5, 7. This is a central task in numerous applications.
Many algorithms exists, but some are better than others: • Bubble Sort: Compare ﬁrst two numbers; swap if in wrong order.
Then compare second two, third two, etc. until the end. Then
repeat from the beginning, until nothing left to swap.
How long will this take in the worst case scenario?
• Merge sort: Use a simple merging operation to merge individual
elements, and then groups of elements until the entire list is merged. Sym Sys 100: Computation, Part II 4 Complexity Consider, as an example, diﬀerent sorting algorithms. An instance of the sorting problem is the following:
Given some ﬁnite list of numbers as input, produce as
output that same list in order from lowest to highest. For example, with input 4, 2, 7, 1, 5, we must give output 1, 2, 4, 5, 7. This is a central task in numerous applications.
Many algorithms exists, but some are better than others: • Bubble Sort: Compare ﬁrst two numbers; swap if in wrong order.
Then compare second two, third two, etc. until the end. Then
repeat from the beginning, until nothing left to swap.
How long will this take in the worst case scenario?
• Merge sort: Use a simple merging operation to merge individual
elements, and then groups of elements until the entire list is merged.
How expensive is this in the worst case? Sym Sys 100: Computation, Part II 4 Complexity But what does this have to do with Turing Machines? Sym Sys 100: Computation, Part II 5 Complexity But what does this have to do with Turing Machines? From Papadimitriou’s classic textbook, Computational Complexity:
Any reasonable attempt to model mathematically
computer algorithms and their performance is bound to end
up with a model of computation and associated time cost
that is equivalent to Turing machines within a polynomial. Sym Sys 100: Computation, Part II 5 Complexity But what does this have to do with Turing Machines? From Papadimitriou’s classic textbook, Computational Complexity:
Any reasonable attempt to model mathematically
computer algorithms and their performance is bound to end
up with a model of computation and associated time cost
that is equivalent to Turing machines within a polynomial. Papadimitriou calls this the Quantitative ChurchTuring Thesis. Sym Sys 100: Computation, Part II 5 Complexity But what does this have to do with Turing Machines? From Papadimitriou’s classic textbook, Computational Complexity:
Any reasonable attempt to model mathematically
computer algorithms and their performance is bound to end
up with a model of computation and associated time cost
that is equivalent to Turing machines within a polynomial. Papadimitriou calls this the Quantitative ChurchTuring Thesis. Sym Sys 100: Computation, Part II 5 Models of Computation Example #1: Standard Programming Languages
In general we say a computational model is Turingcomplete if every
input/output function deﬁnable in one is also deﬁnable in the other.
(We are now back in the realm of abstract computability.) Sym Sys 100: Computation, Part II 6 Models of Computation Example #1: Standard Programming Languages
In general we say a computational model is Turingcomplete if every
input/output function deﬁnable in one is also deﬁnable in the other.
(We are now back in the realm of abstract computability.) Typical programming languages like Java, C++, Pascal, Lisp, and so
on, give us our ﬁrst examples of Turingcomplete models of
computation. Sym Sys 100: Computation, Part II 6 Models of Computation Example #1: Standard Programming Languages
In general we say a computational model is Turingcomplete if every
input/output function deﬁnable in one is also deﬁnable in the other.
(We are now back in the realm of abstract computability.) Typical programming languages like Java, C++, Pascal, Lisp, and so
on, give us our ﬁrst examples of Turingcomplete models of
computation.
• Thus, for example, to show that Java is Turingcomplete, we would
have to show that for any Turing Machine, we could construct a
Java program that computed the same function (and vice versa). Sym Sys 100: Computation, Part II 6 Models of Computation Example #1: Standard Programming Languages
In general we say a computational model is Turingcomplete if every
input/output function deﬁnable in one is also deﬁnable in the other.
(We are now back in the realm of abstract computability.) Typical programming languages like Java, C++, Pascal, Lisp, and so
on, give us our ﬁrst examples of Turingcomplete models of
computation.
• Thus, for example, to show that Java is Turingcomplete, we would
have to show that for any Turing Machine, we could construct a
Java program that computed the same function (and vice versa).
Crucial features:
Sym Sys 100: Computation, Part II Finite set of symbols;
While loops;
Functional composition;
... 6 Models of Computation Example #1: Standard Programming Languages
In general we say a computational model is Turingcomplete if every
input/output function deﬁnable in one is also deﬁnable in the other.
(We are now back in the realm of abstract computability.) Typical programming languages like Java, C++, Pascal, Lisp, and so
on, give us our ﬁrst examples of Turingcomplete models of
computation.
• Thus, for example, to show that Java is Turingcomplete, we would
have to show that for any Turing Machine, we could construct a
Java program that computed the same function (and vice versa).
Crucial features:
Finite set of symbols;
While loops;
Functional composition;
... Random Access Memory (RAM) is also Turingcomplete. Sym Sys 100: Computation, Part II 6 Models of Computation Example #2: λCalculus
The λCalculus was invented by Turing’s supervisor, Alonzo Church,
and was the main inspiration for functional programming languages. Sym Sys 100: Computation, Part II 7 Models of Computation Example #2: λCalculus
The λCalculus was invented by Turing’s supervisor, Alonzo Church,
and was the main inspiration for functional programming languages.
Computation in the lambda calculus amounts to evaluating
functions represented by λterms. For example, λx (x + 1) is the
λterm representing the successor function. Sym Sys 100: Computation, Part II 7 Models of Computation Example #2: λCalculus
The λCalculus was invented by Turing’s supervisor, Alonzo Church,
and was the main inspiration for functional programming languages.
Computation in the lambda calculus amounts to evaluating
functions represented by λterms. For example, λx (x + 1) is the
λterm representing the successor function.
An operation called βreduction corresponds to making a single
computation step. Sym Sys 100: Computation, Part II 7 Models of Computation Example #2: λCalculus
The λCalculus was invented by Turing’s supervisor, Alonzo Church,
and was the main inspiration for functional programming languages.
Computation in the lambda calculus amounts to evaluating
functions represented by λterms. For example, λx (x + 1) is the
λterm representing the successor function.
An operation called βreduction corresponds to making a single
computation step.
For example, we can apply λx (x + 1) to a single argument, say 4,
β and applying βreduction gives us: (λx (x + 1))4 → (4 + 1) = 5. Sym Sys 100: Computation, Part II 7 Models of Computation Example #2: λCalculus
The λCalculus was invented by Turing’s supervisor, Alonzo Church,
and was the main inspiration for functional programming languages.
Computation in the lambda calculus amounts to evaluating
functions represented by λterms. For example, λx (x + 1) is the
λterm representing the successor function.
An operation called βreduction corresponds to making a single
computation step.
For example, we can apply λx (x + 1) to a single argument, say 4,
β and applying βreduction gives us: (λx (x + 1))4 → (4 + 1) = 5.
The power of the λcalculus comes from the ability to deﬁne
functions that take functions as arguments; plus the ability to deﬁne
functions recursively by means of the “Ycombinator”:
λy (λx (y )(x )x )λx (y )(x )x Sym Sys 100: Computation, Part II 7 Models of Computation Example #2: λCalculus
The λCalculus was invented by Turing’s supervisor, Alonzo Church,
and was the main inspiration for functional programming languages.
Computation in the lambda calculus amounts to evaluating
functions represented by λterms. For example, λx (x + 1) is the
λterm representing the successor function.
An operation called βreduction corresponds to making a single
computation step.
For example, we can apply λx (x + 1) to a single argument, say 4,
β and applying βreduction gives us: (λx (x + 1))4 → (4 + 1) = 5.
The power of the λcalculus comes from the ability to deﬁne
functions that take functions as arguments; plus the ability to deﬁne
functions recursively by means of the “Ycombinator”:
λy (λx (y )(x )x )λx (y )(x )x Theorem
The functions deﬁnable in λcalculus are exactly the functions
computable by a Turing Machine.
Sym Sys 100: Computation, Part II 7 Models of Computation Example #3: Neural Networks
Certain kinds of neural networks can also be seen as implementing
input/output functions. Sym Sys 100: Computation, Part II 8 Models of Computation Example #3: Neural Networks
Certain kinds of neural networks can also be seen as implementing
input/output functions. Sym Sys 100: Computation, Part II 8 Models of Computation Example #3: Neural Networks
Certain kinds of neural networks can also be seen as implementing
input/output functions. Under certain assumptions about the activation functions and
architecture of the network, it can be shown that these models can
only implement Turingcomputable functions, provided that the
values of weights are restricted to rational numbers. Sym Sys 100: Computation, Part II 8 Models of Computation Example #3: Neural Networks
Certain kinds of neural networks can also be seen as implementing
input/output functions. Under certain assumptions about the activation functions and
architecture of the network, it can be shown that these models can
only implement Turingcomputable functions, provided that the
values of weights are restricted to rational numbers.
If we allow arbitrarily precise real number valued weights, then we
can compute nonTuring computable functions. Sym Sys 100: Computation, Part II 8 Models of Computation Example #3: Neural Networks
Certain kinds of neural networks can also be seen as implementing
input/output functions. Under certain assumptions about the activation functions and
architecture of the network, it can be shown that these models can
only implement Turingcomputable functions, provided that the
values of weights are restricted to rational numbers.
If we allow arbitrarily precise real number valued weights, then we
can compute nonTuring computable functions.
Could this be a challenge to the ChurchTuring Thesis?
Sym Sys 100: Computation, Part II 8 Computational Theory of Mind Computation and the Mind
We now turn to the question of what computation might have to do
with the workings of the human mind. Sym Sys 100: Computation, Part II 9 Computational Theory of Mind Computation and the Mind
We now turn to the question of what computation might have to do
with the workings of the human mind. One of the central goals of cognitive science and AI is to build
computational models, including actual implementations, of mental
phenomena. What is the status of these models? Sym Sys 100: Computation, Part II 9 Computational Theory of Mind Computation and the Mind
We now turn to the question of what computation might have to do
with the workings of the human mind. One of the central goals of cognitive science and AI is to build
computational models, including actual implementations, of mental
phenomena. What is the status of these models? We have seen that almost all of these models are equivalent on one
level: in terms of their input/output behavior. Sym Sys 100: Computation, Part II 9 Computational Theory of Mind Computation and the Mind
We now turn to the question of what computation might have to do
with the workings of the human mind. One of the central goals of cognitive science and AI is to build
computational models, including actual implementations, of mental
phenomena. What is the status of these models? We have seen that almost all of these models are equivalent on one
level: in terms of their input/output behavior.
The Computational Theory of Mind subsumes all of these. Sym Sys 100: Computation, Part II 9 Computational Theory of Mind Computation and the Mind
We now turn to the question of what computation might have to do
with the workings of the human mind. One of the central goals of cognitive science and AI is to build
computational models, including actual implementations, of mental
phenomena. What is the status of these models? We have seen that almost all of these models are equivalent on one
level: in terms of their input/output behavior.
The Computational Theory of Mind subsumes all of these. At the same time, they make very diﬀerent claims about the
processing that leads from input to output. What does this
diﬀerence come down to? Sym Sys 100: Computation, Part II 9 Computational Theory of Mind Implementing a Computation
Quotation from the assigned paper by D. Chalmers:
A physical system implements a given computation when there
exists a grouping of physical states of the system into
statetypes and a onetoone mapping from formal states of the
computation to physical statetypes, such that formal states
related by an abstract statetransition relation are mapped onto
physical statetypes related by a corresponding causal
statetransition relation. Sym Sys 100: Computation, Part II 10 Computational Theory of Mind Implementing a Computation
Quotation from the assigned paper by D. Chalmers:
A physical system implements a given computation when there
exists a grouping of physical states of the system into
statetypes and a onetoone mapping from formal states of the
computation to physical statetypes, such that formal states
related by an abstract statetransition relation are mapped onto
physical statetypes related by a corresponding causal
statetransition relation.
In other words there is assumed to be an isomorphism between the
abstract structure of the computational model and the physical
causal structure of the world. Sym Sys 100: Computation, Part II 10 Computational Theory of Mind Implementing a Computation
Quotation from the assigned paper by D. Chalmers:
A physical system implements a given computation when there
exists a grouping of physical states of the system into
statetypes and a onetoone mapping from formal states of the
computation to physical statetypes, such that formal states
related by an abstract statetransition relation are mapped onto
physical statetypes related by a corresponding causal
statetransition relation.
In other words there is assumed to be an isomorphism between the
abstract structure of the computational model and the physical
causal structure of the world.
This is meant to apply to: Sym Sys 100: Computation, Part II 10 Computational Theory of Mind Implementing a Computation
Quotation from the assigned paper by D. Chalmers:
A physical system implements a given computation when there
exists a grouping of physical states of the system into
statetypes and a onetoone mapping from formal states of the
computation to physical statetypes, such that formal states
related by an abstract statetransition relation are mapped onto
physical statetypes related by a corresponding causal
statetransition relation.
In other words there is assumed to be an isomorphism between the
abstract structure of the computational model and the physical
causal structure of the world.
This is meant to apply to:
• digital computers, Sym Sys 100: Computation, Part II 10 Computational Theory of Mind Implementing a Computation
Quotation from the assigned paper by D. Chalmers:
A physical system implements a given computation when there
exists a grouping of physical states of the system into
statetypes and a onetoone mapping from formal states of the
computation to physical statetypes, such that formal states
related by an abstract statetransition relation are mapped onto
physical statetypes related by a corresponding causal
statetransition relation.
In other words there is assumed to be an isomorphism between the
abstract structure of the computational model and the physical
causal structure of the world.
This is meant to apply to:
• digital computers,
• brains, Sym Sys 100: Computation, Part II 10 Computational Theory of Mind Implementing a Computation
Quotation from the assigned paper by D. Chalmers:
A physical system implements a given computation when there
exists a grouping of physical states of the system into
statetypes and a onetoone mapping from formal states of the
computation to physical statetypes, such that formal states
related by an abstract statetransition relation are mapped onto
physical statetypes related by a corresponding causal
statetransition relation.
In other words there is assumed to be an isomorphism between the
abstract structure of the computational model and the physical
causal structure of the world.
This is meant to apply to:
• digital computers,
• brains,
• and anything else that we might think of as computing. Sym Sys 100: Computation, Part II 10 Computational Theory of Mind Implementing a Computation
Quotation from the assigned paper by D. Chalmers:
A physical system implements a given computation when there
exists a grouping of physical states of the system into
statetypes and a onetoone mapping from formal states of the
computation to physical statetypes, such that formal states
related by an abstract statetransition relation are mapped onto
physical statetypes related by a corresponding causal
statetransition relation.
In other words there is assumed to be an isomorphism between the
abstract structure of the computational model and the physical
causal structure of the world.
This is meant to apply to:
• digital computers,
• brains,
• and anything else that we might think of as computing. Chalmers goes on to argue that mind just is physical implementation
of a computation. We can remain agnostic about this further claim. Sym Sys 100: Computation, Part II 10 Computational Theory of Mind Levels of Analysis
In principle, this allows for diﬀerent levels of analysis, or diﬀerent
levels of computation. Sym Sys 100: Computation, Part II 11 Computational Theory of Mind Levels of Analysis
In principle, this allows for diﬀerent levels of analysis, or diﬀerent
levels of computation. Consider the case of digital computers: Sym Sys 100: Computation, Part II 11 Computational Theory of Mind Levels of Analysis
In principle, this allows for diﬀerent levels of analysis, or diﬀerent
levels of computation. Consider the case of digital computers:
•
•
•
• Highlevel programming languages (Java, C++, etc.);
Lowlevel machine code;
Electrical circuits;
... Sym Sys 100: Computation, Part II 11 Computational Theory of Mind Levels of Analysis
In principle, this allows for diﬀerent levels of analysis, or diﬀerent
levels of computation. Consider the case of digital computers:
•
•
•
• Highlevel programming languages (Java, C++, etc.);
Lowlevel machine code;
Electrical circuits;
... Inspired by the case of digital computers, many have argued that the
mind can also be studied at a high level, analogous to highlevel
programming languages. Sym Sys 100: Computation, Part II 11 Computational Theory of Mind Levels of Analysis
In principle, this allows for diﬀerent levels of analysis, or diﬀerent
levels of computation. Consider the case of digital computers:
•
•
•
• Highlevel programming languages (Java, C++, etc.);
Lowlevel machine code;
Electrical circuits;
... Inspired by the case of digital computers, many have argued that the
mind can also be studied at a high level, analogous to highlevel
programming languages. This is related to D. Marr’s idea of diﬀerent levels of analysis:
1. Implementational level;
2. Algorithmic level;
3. Computational level. Sym Sys 100: Computation, Part II 11 Symbols and Representations It is ultimately an empirical question what kinds of computational
models work to explain what kinds of behavior. This is a matter of
subjecting models to experiments. Sym Sys 100: Computation, Part II 12 Symbols and Representations It is ultimately an empirical question what kinds of computational
models work to explain what kinds of behavior. This is a matter of
subjecting models to experiments. One of the conceptual diﬃculties is that it is unclear what the
elements of the relevant computations are supposed to be. Sym Sys 100: Computation, Part II 12 Symbols and Representations It is ultimately an empirical question what kinds of computational
models work to explain what kinds of behavior. This is a matter of
subjecting models to experiments. One of the conceptual diﬃculties is that it is unclear what the
elements of the relevant computations are supposed to be.
In Chalmers’ terminology, we need to know what the physical objects
are that we suppose are involved in the relevant causal relations. Sym Sys 100: Computation, Part II 12 Symbols and Representations It is ultimately an empirical question what kinds of computational
models work to explain what kinds of behavior. This is a matter of
subjecting models to experiments. One of the conceptual diﬃculties is that it is unclear what the
elements of the relevant computations are supposed to be.
In Chalmers’ terminology, we need to know what the physical objects
are that we suppose are involved in the relevant causal relations.
• Single ﬁrings of single neurons? Sym Sys 100: Computation, Part II 12 Symbols and Representations It is ultimately an empirical question what kinds of computational
models work to explain what kinds of behavior. This is a matter of
subjecting models to experiments. One of the conceptual diﬃculties is that it is unclear what the
elements of the relevant computations are supposed to be.
In Chalmers’ terminology, we need to know what the physical objects
are that we suppose are involved in the relevant causal relations.
• Single ﬁrings of single neurons?
• Patterns of ﬁring among clusters of neurons? Sym Sys 100: Computation, Part II 12 Symbols and Representations It is ultimately an empirical question what kinds of computational
models work to explain what kinds of behavior. This is a matter of
subjecting models to experiments. One of the conceptual diﬃculties is that it is unclear what the
elements of the relevant computations are supposed to be.
In Chalmers’ terminology, we need to know what the physical objects
are that we suppose are involved in the relevant causal relations.
• Single ﬁrings of single neurons?
• Patterns of ﬁring among clusters of neurons?
• Something more abstract? Sym Sys 100: Computation, Part II 12 Symbols and Representations It is ultimately an empirical question what kinds of computational
models work to explain what kinds of behavior. This is a matter of
subjecting models to experiments. One of the conceptual diﬃculties is that it is unclear what the
elements of the relevant computations are supposed to be.
In Chalmers’ terminology, we need to know what the physical objects
are that we suppose are involved in the relevant causal relations.
• Single ﬁrings of single neurons?
• Patterns of ﬁring among clusters of neurons?
• Something more abstract? The question is, what are the basic building blocks of the mind? Sym Sys 100: Computation, Part II 12 Symbols and Representations Interlude: Representation
Repeated from the Stanford Bulletin entry for Symbolic Systems:
The observation that both human beings and computers can
manipulate symbols lies at the heart of Symbolic Systems, an
interdisciplinary program focusing on the relationship between
natural and artiﬁcial systems that represent, process, and act
on information. Sym Sys 100: Computation, Part II 13 Symbols and Representations Interlude: Representation
Repeated from the Stanford Bulletin entry for Symbolic Systems:
The observation that both human beings and computers can
manipulate symbols lies at the heart of Symbolic Systems, an
interdisciplinary program focusing on the relationship between
natural and artiﬁcial systems that represent, process, and act
on information.
A number of questions: Sym Sys 100: Computation, Part II 13 Symbols and Representations Interlude: Representation
Repeated from the Stanford Bulletin entry for Symbolic Systems:
The observation that both human beings and computers can
manipulate symbols lies at the heart of Symbolic Systems, an
interdisciplinary program focusing on the relationship between
natural and artiﬁcial systems that represent, process, and act
on information.
A number of questions:
What is information? Sym Sys 100: Computation, Part II 13 Symbols and Representations Interlude: Representation
Repeated from the Stanford Bulletin entry for Symbolic Systems:
The observation that both human beings and computers can
manipulate symbols lies at the heart of Symbolic Systems, an
interdisciplinary program focusing on the relationship between
natural and artiﬁcial systems that represent, process, and act
on information.
A number of questions:
What is information? What are symbols? Sym Sys 100: Computation, Part II 13 Symbols and Representations Interlude: Representation
Repeated from the Stanford Bulletin entry for Symbolic Systems:
The observation that both human beings and computers can
manipulate symbols lies at the heart of Symbolic Systems, an
interdisciplinary program focusing on the relationship between
natural and artiﬁcial systems that represent, process, and act
on information.
A number of questions:
What is information? What are symbols? What is it for one thing to represent something else? Sym Sys 100: Computation, Part II 13 Symbols and Representations Interlude: Representation
Repeated from the Stanford Bulletin entry for Symbolic Systems:
The observation that both human beings and computers can
manipulate symbols lies at the heart of Symbolic Systems, an
interdisciplinary program focusing on the relationship between
natural and artiﬁcial systems that represent, process, and act
on information.
A number of questions:
What is information? What are symbols? What is it for one thing to represent something else? Sym Sys 100: Computation, Part II 14 Symbols and Representations Symbolic Representation
By virtue of what does a word like “squirrel” stand for squirrels? Sym Sys 100: Computation, Part II 15 Symbols and Representations Symbolic Representation
By virtue of what does a word like “squirrel” stand for squirrels? Presumably it has something to do with a convention that has been
set up, by some speech community. Sym Sys 100: Computation, Part II 15 Symbols and Representations Symbolic Representation
By virtue of what does a word like “squirrel” stand for squirrels? Presumably it has something to do with a convention that has been
set up, by some speech community. There is nothing inherent to the word “squirrel”, in either its
phonological or orthographic representation, that connects it
essentially to squirrels. Sym Sys 100: Computation, Part II 15 Symbols and Representations Symbolic Representation
By virtue of what does a word like “squirrel” stand for squirrels? Presumably it has something to do with a convention that has been
set up, by some speech community. There is nothing inherent to the word “squirrel”, in either its
phonological or orthographic representation, that connects it
essentially to squirrels. We might call this symbolic representation. It is very much
characteristic of Turing Machines, λcalculus, computer programs,
and any other “purely formal” models of computation. Sym Sys 100: Computation, Part II 15 Symbols and Representations Symbolic Representation
By virtue of what does a word like “squirrel” stand for squirrels? Presumably it has something to do with a convention that has been
set up, by some speech community. There is nothing inherent to the word “squirrel”, in either its
phonological or orthographic representation, that connects it
essentially to squirrels. We might call this symbolic representation. It is very much
characteristic of Turing Machines, λcalculus, computer programs,
and any other “purely formal” models of computation. Question: In what ways, if any, could the mind be said to compute
over symbolic representations? Sym Sys 100: Computation, Part II 15 Symbols and Representations Symbolic Representation
By virtue of what does a word like “squirrel” stand for squirrels? Presumably it has something to do with a convention that has been
set up, by some speech community. There is nothing inherent to the word “squirrel”, in either its
phonological or orthographic representation, that connects it
essentially to squirrels. We might call this symbolic representation. It is very much
characteristic of Turing Machines, λcalculus, computer programs,
and any other “purely formal” models of computation. Question: In what ways, if any, could the mind be said to compute
over symbolic representations? Recall Searle’s Chinese Room thought experiment from the reading
in the ﬁrst week. Do you ﬁnd it convincing? Sym Sys 100: Computation, Part II 15 Symbols and Representations
What are the alternatives? Sym Sys 100: Computation, Part II 16 Symbols and Representations
What are the alternatives?
What does the following represent, and how? Sym Sys 100: Computation, Part II 16 Symbols and Representations
What are the alternatives?
What does the following represent, and how? Some representations seem to be iconic or imagistic, in the sense
that the relation holds by virtue of similarity or direct analogy. Sym Sys 100: Computation, Part II 16 Symbols and Representations
What are the alternatives?
What does the following represent, and how? Some representations seem to be iconic or imagistic, in the sense
that the relation holds by virtue of similarity or direct analogy.
N.B., many representations, like that above or like the word
“chickadee”, are something in between symbolic and iconic. Sym Sys 100: Computation, Part II 16 Symbols and Representations
What are the alternatives?
What does the following represent, and how? Some representations seem to be iconic or imagistic, in the sense
that the relation holds by virtue of similarity or direct analogy.
N.B., many representations, like that above or like the word
“chickadee”, are something in between symbolic and iconic.
Are there representations in our mind that are somehow imagistic in
character? And how could demonstrate this? Sym Sys 100: Computation, Part II 16 Symbols and Representations Shepard’s Mental Rotation Experiments Sym Sys 100: Computation, Part II 17 Symbols and Representations Shepard’s Mental Rotation Experiments Subjects are asked to determine whether two ﬁgures are identical by
“mentally rotating” one until it ﬁts the shape of the second. Sym Sys 100: Computation, Part II 17 Symbols and Representations Shepard’s Mental Rotation Experiments
Subjects are asked to determine whether two ﬁgures are identical by
“mentally rotating” one until it ﬁts the shape of the second.
The amount of time it takes to ﬁnd the solution correlates very
closely with the degree of rotation between the ﬁgures, suggesting
that there is some reality to these “mental images”. Sym Sys 100: Computation, Part II 17 Symbols and Representations Shepard’s Mental Rotation Experiments
Subjects are asked to determine whether two ﬁgures are identical by
“mentally rotating” one until it ﬁts the shape of the second.
The amount of time it takes to ﬁnd the solution correlates very
closely with the degree of rotation between the ﬁgures, suggesting
that there is some reality to these “mental images”.
Some have taken this to show there are representations in the mind
or brain with pictorial or imagistic properties. Sym Sys 100: Computation, Part II 17 Symbols and Representations Cognition without Representation?
Recently some researchers have challenged the computational theory
of mind from other directions. Sym Sys 100: Computation, Part II 18 Symbols and Representations Cognition without Representation?
Recently some researchers have challenged the computational theory
of mind from other directions. Some think the very idea of representation is misguided, and with it
goes computation (so they argue). Sym Sys 100: Computation, Part II 18 Symbols and Representations Cognition without Representation?
Recently some researchers have challenged the computational theory
of mind from other directions. Some think the very idea of representation is misguided, and with it
goes computation (so they argue). Perhaps human behavior is better modeled as a complex dynamical
system, the way we analyze other physical systems? Sym Sys 100: Computation, Part II 18 Symbols and Representations Summary
Main points from today: Sym Sys 100: Computation, Part II 19 Symbols and Representations Summary
Main points from today:
Within the realm of computable procedures and programs,
complexity is an important measure of feasibility and tractability. Sym Sys 100: Computation, Part II 19 Symbols and Representations Summary
Main points from today:
Within the realm of computable procedures and programs,
complexity is an important measure of feasibility and tractability. There are many diﬀerent models of computation, which on a very
abstract level are all equivalent to the Turing Machine. Sym Sys 100: Computation, Part II 19 Symbols and Representations Summary
Main points from today:
Within the realm of computable procedures and programs,
complexity is an important measure of feasibility and tractability. There are many diﬀerent models of computation, which on a very
abstract level are all equivalent to the Turing Machine. To compare them and understand how each one works, we have to
understand how they compute and what they compute over. Sym Sys 100: Computation, Part II 19 Symbols and Representations Summary
Main points from today:
Within the realm of computable procedures and programs,
complexity is an important measure of feasibility and tractability. There are many diﬀerent models of computation, which on a very
abstract level are all equivalent to the Turing Machine. To compare them and understand how each one works, we have to
understand how they compute and what they compute over. This brought us to the topic of representation and the relationship
between computation and the mind. What is “the stuﬀ of thought”? Sym Sys 100: Computation, Part II 19 Symbols and Representations Summary
Main points from today:
Within the realm of computable procedures and programs,
complexity is an important measure of feasibility and tractability. There are many diﬀerent models of computation, which on a very
abstract level are all equivalent to the Turing Machine. To compare them and understand how each one works, we have to
understand how they compute and what they compute over. This brought us to the topic of representation and the relationship
between computation and the mind. What is “the stuﬀ of thought”? Next week:
Logic and Reasoning Sym Sys 100: Computation, Part II 19 ...
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 Spring '09

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