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... Thus, changing the environment can affect what properties a program satisfies. Programming logics usually axiomatize program behavior under certain assumptions about the environment. Logics to reason about real-time, for example, axiomatize assumptions about how time advances while the program execu ...

... Thus, changing the environment can affect what properties a program satisfies. Programming logics usually axiomatize program behavior under certain assumptions about the environment. Logics to reason about real-time, for example, axiomatize assumptions about how time advances while the program execu ...

How to Write a 21st Century Proof

... One mistake I made in the earlier article was to advocate making proofs both easier to read and more rigorous. Learning both a new way to write proofs and how to be more precise and rigorous was too high a barrier for most mathematicians. I try here to separate the two goals. I hope that structurin ...

... One mistake I made in the earlier article was to advocate making proofs both easier to read and more rigorous. Learning both a new way to write proofs and how to be more precise and rigorous was too high a barrier for most mathematicians. I try here to separate the two goals. I hope that structurin ...

Understanding SPKI/SDSI Using First-Order Logic

... against a set of SPKI/SDSI statements, together with an entailment relation that determines whether a query follows from a set of SPKI/SDSI statements. A good formal semantics should achieve the following four goals. First, the class of queries supported by the semantics should be large and include ...

... against a set of SPKI/SDSI statements, together with an entailment relation that determines whether a query follows from a set of SPKI/SDSI statements. A good formal semantics should achieve the following four goals. First, the class of queries supported by the semantics should be large and include ...

CS243: Discrete Structures Mathematical Proof Techniques

... Proof by cases: Exhaustively enumerate different possibilities, and prove the theorem for each case ...

... Proof by cases: Exhaustively enumerate different possibilities, and prove the theorem for each case ...

Effectively Polynomial Simulations

... polynomial-time in m truth-preserving transformation and produces an A-proof of φ. If Q produces not an Afrom (encodings of) boolean formulas to (encodings of) proof but a proof in some other proof system B, then A boolean formulas, R(f, m) such that when m is at least is said to be weakly automatiz ...

... polynomial-time in m truth-preserving transformation and produces an A-proof of φ. If Q produces not an Afrom (encodings of) boolean formulas to (encodings of) proof but a proof in some other proof system B, then A boolean formulas, R(f, m) such that when m is at least is said to be weakly automatiz ...

Euclidian Roles in Description Logics

... I satisfies Eucl(R) if for all x, y, z ∈ ∆I , {hx, yi, hx, zi} ⊆ RI → hy, zi ∈ RI . As with transitive role axioms, in order to efficiently handle Euclidian roles we have to understand the Euclidian closure of a relation R and handle properly value restrictions, ∀R.C. In Figure 1 (a) we can see the ...

... I satisfies Eucl(R) if for all x, y, z ∈ ∆I , {hx, yi, hx, zi} ⊆ RI → hy, zi ∈ RI . As with transitive role axioms, in order to efficiently handle Euclidian roles we have to understand the Euclidian closure of a relation R and handle properly value restrictions, ∀R.C. In Figure 1 (a) we can see the ...

The Complexity of Satisfiability Problems: Refining Schaefer`s

... In 1978, Schaefer classified the Boolean constraint satisfaction problem and showed that, depending on the allowed relations in a propositional formula, the problem is either in P or is NP-complete [Sch78]. This famous “dichotomy theorem” does not consider the fact that different problems in P have ...

... In 1978, Schaefer classified the Boolean constraint satisfaction problem and showed that, depending on the allowed relations in a propositional formula, the problem is either in P or is NP-complete [Sch78]. This famous “dichotomy theorem” does not consider the fact that different problems in P have ...

ppt - UBC Computer Science

... x D, P(x) or y D, Q(y)? If you know x D, P(x), you can say that for any d in D that P(d) is true P(d) is true for any particular d in D or for an arbitrary one. If you know y D, Q(y), you can say that for some d in D, Q(d) is true, but you don’t know which one So, assume nothing ...

... x D, P(x) or y D, Q(y)? If you know x D, P(x), you can say that for any d in D that P(d) is true P(d) is true for any particular d in D or for an arbitrary one. If you know y D, Q(y), you can say that for some d in D, Q(d) is true, but you don’t know which one So, assume nothing ...

Lecture Slides

... x D, P(x) or y D, Q(y)? If you know x D, P(x), you can say that for any d in D that P(d) is true P(d) is true for any particular d in D or for an arbitrary one. If you know y D, Q(y), you can say that for some d in D, Q(d) is true, but you don’t know which one So, assume nothing ...

... x D, P(x) or y D, Q(y)? If you know x D, P(x), you can say that for any d in D that P(d) is true P(d) is true for any particular d in D or for an arbitrary one. If you know y D, Q(y), you can say that for some d in D, Q(d) is true, but you don’t know which one So, assume nothing ...

Incompleteness in the finite domain

... Alternatively, we can view the proposed conjectures as axioms. In fact, NP 6= P has been treated as an axiom “to prove” hardness of various problems. In many cases NP 6= P does not suﬃce and therefore a number of stronger hypotheses have been proposed. For example, the Exponential Time Hypothesis of ...

... Alternatively, we can view the proposed conjectures as axioms. In fact, NP 6= P has been treated as an axiom “to prove” hardness of various problems. In many cases NP 6= P does not suﬃce and therefore a number of stronger hypotheses have been proposed. For example, the Exponential Time Hypothesis of ...

Proofs

... [0,1] to B= [0, 2]. Proof: We build two injections and conclude there must be a bijection without ever exhibiting the bijection. Let f be the identity map from A to B. Then f is an injection (and we conclude that | A | | B | ). Define the function g from B to A as g(x) = x/4. Then g is an injectio ...

... [0,1] to B= [0, 2]. Proof: We build two injections and conclude there must be a bijection without ever exhibiting the bijection. Let f be the identity map from A to B. Then f is an injection (and we conclude that | A | | B | ). Define the function g from B to A as g(x) = x/4. Then g is an injectio ...

Chapter X: Computational Complexity of Propositional Fuzzy Logics

... some complexity class), the situation is analogous to the classical case: satisfiability is NP-complete, while tautologousness and consequence (hence, theoremhood and provability) are coNP-complete. One might ask why consequence relation comes out no more difficult than tautologousness. This chapter ...

... some complexity class), the situation is analogous to the classical case: satisfiability is NP-complete, while tautologousness and consequence (hence, theoremhood and provability) are coNP-complete. One might ask why consequence relation comes out no more difficult than tautologousness. This chapter ...

Incompleteness in the finite domain

... and bounded arithmetic seem to follow a general pattern. For example, as we noted above, polynomial time computations are associated with the theory S21 by a witnessing theorem. If we take S22 , which we believe is a stronger theory, then the corresponding function class is PNP ,2 which we believe i ...

... and bounded arithmetic seem to follow a general pattern. For example, as we noted above, polynomial time computations are associated with the theory S21 by a witnessing theorem. If we take S22 , which we believe is a stronger theory, then the corresponding function class is PNP ,2 which we believe i ...

pdf

... The idea here is to search through sub-formulas of the given formulas that might be TRUE simultaneously. For example, if is TRUE, then must be TRUE and must be FALSE. Starting with the input formula, we build a tree of possible models based on subformulas and derive a contradiction in each br ...

... The idea here is to search through sub-formulas of the given formulas that might be TRUE simultaneously. For example, if is TRUE, then must be TRUE and must be FALSE. Starting with the input formula, we build a tree of possible models based on subformulas and derive a contradiction in each br ...

pdf [local copy]

... the rewrite system for the Hydra battle (Moser, 2009; Fleischer, 2007), since the terms one obtains are simpler in some specifiable sense. It turns out that in the present situation the crux is, as becomes clear from Kripke’s further remarks, that he considers the case where one chooses at each elim ...

... the rewrite system for the Hydra battle (Moser, 2009; Fleischer, 2007), since the terms one obtains are simpler in some specifiable sense. It turns out that in the present situation the crux is, as becomes clear from Kripke’s further remarks, that he considers the case where one chooses at each elim ...

pdf

... e.g. for the rewrite system for the Hydra battle [Mos09, Fle07], since the terms one obtains are simpler in some specifiable sense. It turns out that in the present situation the crux is, as becomes clear from Kripke’s further remarks, that he considers the case where one chooses at each elimination ...

... e.g. for the rewrite system for the Hydra battle [Mos09, Fle07], since the terms one obtains are simpler in some specifiable sense. It turns out that in the present situation the crux is, as becomes clear from Kripke’s further remarks, that he considers the case where one chooses at each elimination ...

Chapter 2

... For example, if the context is number theory, and we are asked to prove that the product of two even integers is also even, we can use knowledge about number theory. In particular, we could use the fact that an even integer is divisible by 2, or that an even integer m can be rewritten as 2k for some ...

... For example, if the context is number theory, and we are asked to prove that the product of two even integers is also even, we can use knowledge about number theory. In particular, we could use the fact that an even integer is divisible by 2, or that an even integer m can be rewritten as 2k for some ...

Optimal acceptors and optimal proof systems

... The existence of an optimal (or p-optimal) proof system is a major open question for many languages including TAUT. Optimality would imply p-optimality for any system and any language if and only if the natural proof system for SAT (satisfying assignments) is p-optimal; the existence of optimal syst ...

... The existence of an optimal (or p-optimal) proof system is a major open question for many languages including TAUT. Optimality would imply p-optimality for any system and any language if and only if the natural proof system for SAT (satisfying assignments) is p-optimal; the existence of optimal syst ...

full text (.pdf)

... coinductive step analogous to the inductive step in proofs by induction. What is missing is the ﬁnal argument that the proof is a valid application of the coinduction principle; but it is not necessary to include this step for the same reason that it is not necessary to argue with every inductive pr ...

... coinductive step analogous to the inductive step in proofs by induction. What is missing is the ﬁnal argument that the proof is a valid application of the coinduction principle; but it is not necessary to include this step for the same reason that it is not necessary to argue with every inductive pr ...

Chapter 9: Initial Theorems about Axiom System AS1

... number has a given property Ã, one proves that 0 has Ã, and one proves that if a number n has Ã, then so does its successor n+. This is known as weak induction. Recall that there is also the method of strong induction. According to this method, if one wants to prove that every number has Ã, one prov ...

... number has a given property Ã, one proves that 0 has Ã, and one proves that if a number n has Ã, then so does its successor n+. This is known as weak induction. Recall that there is also the method of strong induction. According to this method, if one wants to prove that every number has Ã, one prov ...

Single tree grammars

... Proof. It suffices to show that in the derivation SA ⇒ x, every time the production A → A1 A2 · · · An is used, Ai → will be used later for all but at most one of the Ai ’s. We prove this by contradiction. Assume that i and j are the smallest integers for which the productions Ai → di Aui and Aj → ...

... Proof. It suffices to show that in the derivation SA ⇒ x, every time the production A → A1 A2 · · · An is used, Ai → will be used later for all but at most one of the Ai ’s. We prove this by contradiction. Assume that i and j are the smallest integers for which the productions Ai → di Aui and Aj → ...

Indirect Proofs - Stanford University

... Proof: By contrapositive; we prove that if n is odd, then n2 is odd. Since n is odd, there is some integer k such that n = 2k + 1. Squaring both sides of this equality and simplifying gives the following: n2 = (2k + 1)2 n2 = 4k2 + 4k + 1 n2 = 2(2k2 + 2k) + 1. From this, we see that there is an integ ...

... Proof: By contrapositive; we prove that if n is odd, then n2 is odd. Since n is odd, there is some integer k such that n = 2k + 1. Squaring both sides of this equality and simplifying gives the following: n2 = (2k + 1)2 n2 = 4k2 + 4k + 1 n2 = 2(2k2 + 2k) + 1. From this, we see that there is an integ ...

pdf

... objects and both properties P and Q express relationships between these two objects. Note too that while Theorem 1.8 appears to be about only one object (the lattice of recursively enumerable sets), the other three theorems are about an unspecified number of objects. In fact, our Theorem 1.1 is abou ...

... objects and both properties P and Q express relationships between these two objects. Note too that while Theorem 1.8 appears to be about only one object (the lattice of recursively enumerable sets), the other three theorems are about an unspecified number of objects. In fact, our Theorem 1.1 is abou ...

Algebraic Laws for Nondeterminism and Concurrency

... programs is far from clear and in this paper we put forth one possible definition. The essenceof our approach is that the behavior of a program is determined by how it communicates with an observer.We begin by assumingthat every program action is observablein this way; later we allow that some actio ...

... programs is far from clear and in this paper we put forth one possible definition. The essenceof our approach is that the behavior of a program is determined by how it communicates with an observer.We begin by assumingthat every program action is observablein this way; later we allow that some actio ...

# Kolmogorov complexity

In algorithmic information theory (a subfield of computer science and mathematics), the Kolmogorov complexity (also known as descriptive complexity, Kolmogorov–Chaitin complexity, algorithmic entropy, or program-size complexity) of an object, such as a piece of text, is a measure of the computational resources needed to specify the object. It is named after Andrey Kolmogorov, who first published on the subject in 1963.For example, consider the following two strings of 32 lowercase letters and digits:abababababababababababababababab4c1j5b2p0cv4w1x8rx2y39umgw5q85s7The first string has a short English-language description, namely ""ab 16 times"", which consists of 11 characters. The second one has no obvious simple description (using the same character set) other than writing down the string itself, which has 32 characters.More formally, the complexity of a string is the length of the shortest possible description of the string in some fixed universal description language (the sensitivity of complexity relative to the choice of description language is discussed below). It can be shown that the Kolmogorov complexity of any string cannot be more than a few bytes larger than the length of the string itself. Strings, like the abab example above, whose Kolmogorov complexity is small relative to the string's size are not considered to be complex.The notion of Kolmogorov complexity can be used to state and prove impossibility results akin to Cantor's diagonal argument, Gödel's incompleteness theorem, and Turing's halting problem.