Proving lower bounds on the amount of resources needed to compute specific functions is one of th... more Proving lower bounds on the amount of resources needed to compute specific functions is one of the most active branches of theoretical computer science. Significant progress has been made recently in proving lower bounds in two restricted models of Boolean circuits. One is the model of small depth circuits, and in this book Johan Torkel Hastad has developed very powerful techniques for proving exponential lower bounds on the size of small depth circuits' computing functions.The techniques described in "Computational Limitations for Small Depth Circuits" can be used to demonstrate almost optimal lower bounds on the size of small depth circuits computing several different functions, such as parity and majority. The main tool used in the proof of the lower bounds is a lemma, stating that any AND of small fanout OR gates can be converted into an OR of small fanout AND gates with high probability when random values are substituted for the variables.Hastad also applies this tool to relativized complexity, and discusses in great detail the computation of parity and majority in small depth circuits.Contents: Introduction. Small Depth Circuits. Outline of Lower Bound Proofs. Main Lemma. Lower Bounds for Small Depth Circuits. Functions Requiring Depth k to Have Small Circuits. Applications to Relativized Complexity. How Well Can We Compute Parity in Small Depth? Is Majority Harder than Parity? Conclusions.John Hastad is a postdoctoral fellow in the Department of Mathematics at MIT C"omputational Limitations of Small Depth Circuits" is a winner of the 1986 ACM Doctoral Dissertation Award.
... 180 Stuart Kurtz, Stephen R. Mahaney, James S. Royer, and Janos Simon The computational techn... more ... 180 Stuart Kurtz, Stephen R. Mahaney, James S. Royer, and Janos Simon The computational technologies that we describe in this paper are very different from Adleman's, not only in the particular biochemical systems involved but also in the nature of the computations each ...
For a classical mathematician, mathematics consists of the discovery of preexisting mathematical ... more For a classical mathematician, mathematics consists of the discovery of preexisting mathematical truth. This understanding of mathematics is captured in Paul Erdös’s notion of “God’s Book of Mathematics,” which contains the best mathematical definitions, theorems, and proofs, and from which fortunate mathematicians are occasionally permitted read a page. Intuitionism takes the position that mathematical objects are mental constructions. Intuitionistic epistemology centers on proof, rather than truth. Thus, intuitionists analyze propositional combinations of mathematical statements in terms of what it takes to prove them, and a proof of φ∧ψ consists of a proof of φ together with a proof of ψ, a proof of φ ∨ ψ consists of a proof of φ or a proof of ψ, while a proof of φ ⇒ ψ consists of an algorithm that converts proofs of φ into proofs of ψ. For an intuitionist, a propositional formula is a tautology if it can be proven, e.g., α ⇒ α is an intuitionistic tautology because to convert a ...
The Collatz problem, widely known as the 3x + 1 problem, asks whether or not a certain simple ite... more The Collatz problem, widely known as the 3x + 1 problem, asks whether or not a certain simple iterative process halts on all inputs. We build on earlier work by J. H. Conway, and show that a natural generalization of the Collatz problem is recursively undecidable. 1
In this paper we introduce a new semantic definition for predicate circumscription. We argue that... more In this paper we introduce a new semantic definition for predicate circumscription. We argue that the new definition is the most general one possible that captures the intuition, i.e., minimizing the objects that satisfy a given property. Yet, we can prove that it does not lead to inconsistency: If the original theory is consistent, then the circumscribed theory is also consistent. We also investigate quasiminimality as a midpoint between abstract minimality and the new definition to provide further comparison. We provide examples of consistent theories that have inconsistent set of consequences under the original and abstract definitions of circumscription. These theories have consistent consequences under the feasible commitment predicate circumscription. Both the feasible commitment predicate circumscription and quasiminimality coincide with circumscription based on abstract minimality whenever it produces sensible results. Therefore, these new definitions also coincid...
Early papers on biological computing focussed on combinatorial and algorithmic issues, and worked... more Early papers on biological computing focussed on combinatorial and algorithmic issues, and worked with intentionally oversimpliied chemical models. In this paper, we reintroduce complexity to the chemical model by considering the eeect problem size has on the initial concentrations of reactants, and the eeect this has in turn on the rate of production and quantity of nal reaction products. We give a sobering preliminary analyses of Adleman's technique for solving Hamiltonian path. Even on the simplest problems, the annealling phase of Adleman's technique requires time (n 2) rather than the O(log n) complexity given by a computationally inspired but chemically naive analysis. On more diicult problems, not only does the rate of production of witnessing molecules drop exponentially in problem size, the nal yield also drops exponentially. These issues are not objections to biological computing per se, but rather diiculties to be overcome in its development as a viable technology...
Proving lower bounds on the amount of resources needed to compute specific functions is one of th... more Proving lower bounds on the amount of resources needed to compute specific functions is one of the most active branches of theoretical computer science. Significant progress has been made recently in proving lower bounds in two restricted models of Boolean circuits. One is the model of small depth circuits, and in this book Johan Torkel Hastad has developed very powerful techniques for proving exponential lower bounds on the size of small depth circuits' computing functions.The techniques described in "Computational Limitations for Small Depth Circuits" can be used to demonstrate almost optimal lower bounds on the size of small depth circuits computing several different functions, such as parity and majority. The main tool used in the proof of the lower bounds is a lemma, stating that any AND of small fanout OR gates can be converted into an OR of small fanout AND gates with high probability when random values are substituted for the variables.Hastad also applies this tool to relativized complexity, and discusses in great detail the computation of parity and majority in small depth circuits.Contents: Introduction. Small Depth Circuits. Outline of Lower Bound Proofs. Main Lemma. Lower Bounds for Small Depth Circuits. Functions Requiring Depth k to Have Small Circuits. Applications to Relativized Complexity. How Well Can We Compute Parity in Small Depth? Is Majority Harder than Parity? Conclusions.John Hastad is a postdoctoral fellow in the Department of Mathematics at MIT C"omputational Limitations of Small Depth Circuits" is a winner of the 1986 ACM Doctoral Dissertation Award.
... 180 Stuart Kurtz, Stephen R. Mahaney, James S. Royer, and Janos Simon The computational techn... more ... 180 Stuart Kurtz, Stephen R. Mahaney, James S. Royer, and Janos Simon The computational technologies that we describe in this paper are very different from Adleman's, not only in the particular biochemical systems involved but also in the nature of the computations each ...
For a classical mathematician, mathematics consists of the discovery of preexisting mathematical ... more For a classical mathematician, mathematics consists of the discovery of preexisting mathematical truth. This understanding of mathematics is captured in Paul Erdös’s notion of “God’s Book of Mathematics,” which contains the best mathematical definitions, theorems, and proofs, and from which fortunate mathematicians are occasionally permitted read a page. Intuitionism takes the position that mathematical objects are mental constructions. Intuitionistic epistemology centers on proof, rather than truth. Thus, intuitionists analyze propositional combinations of mathematical statements in terms of what it takes to prove them, and a proof of φ∧ψ consists of a proof of φ together with a proof of ψ, a proof of φ ∨ ψ consists of a proof of φ or a proof of ψ, while a proof of φ ⇒ ψ consists of an algorithm that converts proofs of φ into proofs of ψ. For an intuitionist, a propositional formula is a tautology if it can be proven, e.g., α ⇒ α is an intuitionistic tautology because to convert a ...
The Collatz problem, widely known as the 3x + 1 problem, asks whether or not a certain simple ite... more The Collatz problem, widely known as the 3x + 1 problem, asks whether or not a certain simple iterative process halts on all inputs. We build on earlier work by J. H. Conway, and show that a natural generalization of the Collatz problem is recursively undecidable. 1
In this paper we introduce a new semantic definition for predicate circumscription. We argue that... more In this paper we introduce a new semantic definition for predicate circumscription. We argue that the new definition is the most general one possible that captures the intuition, i.e., minimizing the objects that satisfy a given property. Yet, we can prove that it does not lead to inconsistency: If the original theory is consistent, then the circumscribed theory is also consistent. We also investigate quasiminimality as a midpoint between abstract minimality and the new definition to provide further comparison. We provide examples of consistent theories that have inconsistent set of consequences under the original and abstract definitions of circumscription. These theories have consistent consequences under the feasible commitment predicate circumscription. Both the feasible commitment predicate circumscription and quasiminimality coincide with circumscription based on abstract minimality whenever it produces sensible results. Therefore, these new definitions also coincid...
Early papers on biological computing focussed on combinatorial and algorithmic issues, and worked... more Early papers on biological computing focussed on combinatorial and algorithmic issues, and worked with intentionally oversimpliied chemical models. In this paper, we reintroduce complexity to the chemical model by considering the eeect problem size has on the initial concentrations of reactants, and the eeect this has in turn on the rate of production and quantity of nal reaction products. We give a sobering preliminary analyses of Adleman's technique for solving Hamiltonian path. Even on the simplest problems, the annealling phase of Adleman's technique requires time (n 2) rather than the O(log n) complexity given by a computationally inspired but chemically naive analysis. On more diicult problems, not only does the rate of production of witnessing molecules drop exponentially in problem size, the nal yield also drops exponentially. These issues are not objections to biological computing per se, but rather diiculties to be overcome in its development as a viable technology...
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