Knowledge-based proof planning - Jörg Siekmann
... technical advances in representational techniques (see for indexing [42]), systems have gained considerable strength and they can prove nontrivial open mathematical problems, such as the Robbins Algebra Conjecture [79], whose proofs are often unintuitive and therefore tricky for humans. In general, ...
... technical advances in representational techniques (see for indexing [42]), systems have gained considerable strength and they can prove nontrivial open mathematical problems, such as the Robbins Algebra Conjecture [79], whose proofs are often unintuitive and therefore tricky for humans. In general, ...
Semantic Tableau Proof System for First-Order Logic
... obtained. Next, for each non-quasiuniversal node X of τ2i+1, extend each open path containing X by applying the appropriate tableau rule. Again, let τ2i+2 be the finite tableau so obtained. In this construction, a closed path is never extended, so all closed paths of τ∞ are finite. In addition, the ...
... obtained. Next, for each non-quasiuniversal node X of τ2i+1, extend each open path containing X by applying the appropriate tableau rule. Again, let τ2i+2 be the finite tableau so obtained. In this construction, a closed path is never extended, so all closed paths of τ∞ are finite. In addition, the ...
Computers and the Sociology of Mathematical Proof Donald
... Mathematicians, too, also turned to the computer for proof, though not, in general, to automated theorem-provers: their use of the computer was rather for ad hoc assistance in particular parts of complicated proofs. The best-known, and most debated, instance is the use of computer analysis by Kenne ...
... Mathematicians, too, also turned to the computer for proof, though not, in general, to automated theorem-provers: their use of the computer was rather for ad hoc assistance in particular parts of complicated proofs. The best-known, and most debated, instance is the use of computer analysis by Kenne ...
PDF
... students have brought up this issue themselves, but under more realistic conditions it is more likely that the teacher would have to raise this point. Mathematical case study 3: Lagrange’s theorem Before we move on with a generic proof, we bring a brief mathematical introduction of Lagrange’s theore ...
... students have brought up this issue themselves, but under more realistic conditions it is more likely that the teacher would have to raise this point. Mathematical case study 3: Lagrange’s theorem Before we move on with a generic proof, we bring a brief mathematical introduction of Lagrange’s theore ...
An Efficient Scheme for Proving a Shuffle
... obtain such {ai } and a if we generate {g̃, g̃1 , . . . g̃n } randomly[Br93]. This way it also suffices the requirement that the verifier should not know log g g̃ for zeroknowledge property. We require the prover to perform the same permutation on the set of fixed basis as he did on the input cipher ...
... obtain such {ai } and a if we generate {g̃, g̃1 , . . . g̃n } randomly[Br93]. This way it also suffices the requirement that the verifier should not know log g g̃ for zeroknowledge property. We require the prover to perform the same permutation on the set of fixed basis as he did on the input cipher ...
On Mathematical Proving1
... case of asynchronous Web-based proving (Stefaneas, Vandoulakis, 2012)), but they are in communication. A human prover may experience an insight (intention) that something in mathematics is true and produce an item (using some semiotic code) to communicate his experience. This item may be not a proof ...
... case of asynchronous Web-based proving (Stefaneas, Vandoulakis, 2012)), but they are in communication. A human prover may experience an insight (intention) that something in mathematics is true and produce an item (using some semiotic code) to communicate his experience. This item may be not a proof ...
SECTION I(g) PROOFS
... This would have made the proof longer and really gain nothing from this inclusion. Generally in a mathematical proof we might have to cover a number of choices but the proof is the same for each of these selections. In this case it is smarter to say “without loss of generality assume…” Without loss ...
... This would have made the proof longer and really gain nothing from this inclusion. Generally in a mathematical proof we might have to cover a number of choices but the proof is the same for each of these selections. In this case it is smarter to say “without loss of generality assume…” Without loss ...
Verifiable Shuffling
... (b) Reliance on such a basis “hides complexity”. In typical application, the construction of these elements involves exponentiations in order to project into the encoding subgroup. These exponentiations are typically “large,” costing a few times that of a exponentiation in the protocol itself. One c ...
... (b) Reliance on such a basis “hides complexity”. In typical application, the construction of these elements involves exponentiations in order to project into the encoding subgroup. These exponentiations are typically “large,” costing a few times that of a exponentiation in the protocol itself. One c ...
Knowledge of proofs*
... Still, there is a case for (10). The antecedent of (10) involves quantification over an infinite totality, viz. all future times. Hence it is not possible, i.e. not for finite beings, to verify it by checking for each time that (5) has not been verified. Neither could we really know that it will not ...
... Still, there is a case for (10). The antecedent of (10) involves quantification over an infinite totality, viz. all future times. Hence it is not possible, i.e. not for finite beings, to verify it by checking for each time that (5) has not been verified. Neither could we really know that it will not ...
Proof in a nutshell - Benjamin
... We now need two assumptions – two general rules about angles and parallel lines that we will assume are always true. If we wanted to prove the assumptions, we could do it later or let someone else prove them. However, you cannot get to a stage where there are no assumptions. All mathematical theorem ...
... We now need two assumptions – two general rules about angles and parallel lines that we will assume are always true. If we wanted to prove the assumptions, we could do it later or let someone else prove them. However, you cannot get to a stage where there are no assumptions. All mathematical theorem ...
Mathematical Review - USC Upstate: Faculty
... Proof by Contraposition Used when we have to prove a statement of the form P Q. Instead of proving P Q, we prove its equivalent ~Q ~P Example: Prove that if the square of an integer is odd then the integer is odd We can prove using direct proof the statement: If an integer is even then its sq ...
... Proof by Contraposition Used when we have to prove a statement of the form P Q. Instead of proving P Q, we prove its equivalent ~Q ~P Example: Prove that if the square of an integer is odd then the integer is odd We can prove using direct proof the statement: If an integer is even then its sq ...
On Designatedly Verified (Non-interactive) Watermarking Schemes
... a signer to construct a proof that will convince only a designated verifier. The designated verifier cannot present the proof to convince any third party because he is fully capable of generating the same proof by himself. A protocol is ‘oblivious’ when it does not leak any information whether given ...
... a signer to construct a proof that will convince only a designated verifier. The designated verifier cannot present the proof to convince any third party because he is fully capable of generating the same proof by himself. A protocol is ‘oblivious’ when it does not leak any information whether given ...
Implicit Learning of Common Sense for Reasoning
... The means of acquisition and use of common sense knowledge is a central question of Artificial Intelligence, starting with McCarthy’s work [1959]. As a starting point, it is helpful to distinguish between common sense knowledge that is learned from typical experiences, and knowledge that is (at leas ...
... The means of acquisition and use of common sense knowledge is a central question of Artificial Intelligence, starting with McCarthy’s work [1959]. As a starting point, it is helpful to distinguish between common sense knowledge that is learned from typical experiences, and knowledge that is (at leas ...
On Correctness of Mathematical Texts from a Logical and Practical
... in other words, an affirmation which is not meant to be proved. A formula F is logically correct in view of a sequence of sections Γ (the logical context of F ), denoted Γ F , whenever F can be deduced in the classical first-order predicate calculus from the formula images of sections from Γ . Logica ...
... in other words, an affirmation which is not meant to be proved. A formula F is logically correct in view of a sequence of sections Γ (the logical context of F ), denoted Γ F , whenever F can be deduced in the classical first-order predicate calculus from the formula images of sections from Γ . Logica ...
Math 3000 Section 003 Intro to Abstract Math Homework 6
... hypothesis is stated correctly, and the reader is alerted about the subsequent inductive step. However, although the (correct) idea of the following computation is apparent, its presentation is not acceptable. In particular, the fifth sentence is logically false: from the stated assumptions, it cann ...
... hypothesis is stated correctly, and the reader is alerted about the subsequent inductive step. However, although the (correct) idea of the following computation is apparent, its presentation is not acceptable. In particular, the fifth sentence is logically false: from the stated assumptions, it cann ...
Proof - Piazza
... Create a table in which each row is a perfect cube between 1 and 1000, and each column is a perfect cube between 1 and 1000. Fill the table with all possible ways of adding a row value and a column value. Check that the table (that has 100 entries) does not have 1000 in it. In the next slide, we wil ...
... Create a table in which each row is a perfect cube between 1 and 1000, and each column is a perfect cube between 1 and 1000. Fill the table with all possible ways of adding a row value and a column value. Check that the table (that has 100 entries) does not have 1000 in it. In the next slide, we wil ...
Name: MATH 250 : LINEAR ALGEBRA
... (b) Adapt the approach from problem 1.1 to give a different proof that 7 6∈ Q. 1.3 Given α, β 6∈ Q and q ∈ Q. (a) Must it be true that q + α 6∈ Q? If so, prove it. If not, give a counterexample (i.e. give explicit choices of q and α such that q + α ∈ Q). (b) Must it be true that qα 6∈ Q? If so, prov ...
... (b) Adapt the approach from problem 1.1 to give a different proof that 7 6∈ Q. 1.3 Given α, β 6∈ Q and q ∈ Q. (a) Must it be true that q + α 6∈ Q? If so, prove it. If not, give a counterexample (i.e. give explicit choices of q and α such that q + α ∈ Q). (b) Must it be true that qα 6∈ Q? If so, prov ...
Proof Techniques - Dartmouth Math Home
... For example, you might be asked to prove that some formula, written in terms of n, holds for all n ∈ Z. If you are lucky, you might be able to prove it directly. However, it’s possible to envision, at least in principle, a systematic way of writing down such a proof. You could prove the result for n ...
... For example, you might be asked to prove that some formula, written in terms of n, holds for all n ∈ Z. If you are lucky, you might be able to prove it directly. However, it’s possible to envision, at least in principle, a systematic way of writing down such a proof. You could prove the result for n ...
Geo 2.6 Proving SuppandComp
... Utah State Core Standard and Indicators Geometry Standard 2, 3 Process Standards 2, 3 Summary This activity uses students’ prior knowledge of algebra to introduce geometric proof. Students prove a solution to an equation using algebraic properties, and write their own statements to be proved. After ...
... Utah State Core Standard and Indicators Geometry Standard 2, 3 Process Standards 2, 3 Summary This activity uses students’ prior knowledge of algebra to introduce geometric proof. Students prove a solution to an equation using algebraic properties, and write their own statements to be proved. After ...
System for Automated Deduction (SAD): a tool
... term properties for the term occurrences in the formula image of A. Term properties are literals that tell us something important about a given term occurrence. A literal (i.e. an atomic formula or its negation) L is considered to be a property of a term t in a context Γ , whenever t is a subterm of ...
... term properties for the term occurrences in the formula image of A. Term properties are literals that tell us something important about a given term occurrence. A literal (i.e. an atomic formula or its negation) L is considered to be a property of a term t in a context Γ , whenever t is a subterm of ...
MathematicalProofsLP
... To prove a statement, one must use generalizations to show the statement holds for all instances satisfying the property. ...
... To prove a statement, one must use generalizations to show the statement holds for all instances satisfying the property. ...
Zero-knowledge proof
In cryptography, a zero-knowledge proof or zero-knowledge protocol is a method by which one party (the prover) can prove to another party (the verifier) that a given statement is true, without conveying any information apart from the fact that the statement is indeed true.If proving the statement requires knowledge of some secret information on the part of the prover, the definition implies that the verifier will not be able to prove the statement in turn to anyone else, since the verifier does not possess the secret information. Notice that the statement being proved must include the assertion that the prover has such knowledge (otherwise, the statement would not be proved in zero-knowledge, since at the end of the protocol the verifier would gain the additional information that the prover has knowledge of the required secret information). If the statement consists only of the fact that the prover possesses the secret information, it is a special case known as zero-knowledge proof of knowledge, and it nicely illustrates the essence of the notion of zero-knowledge proofs: proving that one has knowledge of certain information is trivial if one is allowed to simply reveal that information; the challenge is proving that one has such knowledge without revealing the secret information or anything else.For zero-knowledge proofs of knowledge, the protocol must necessarily require interactive input from the verifier, usually in the form of a challenge or challenges such that the responses from the prover will convince the verifier if and only if the statement is true (i.e., if the prover does have the claimed knowledge). This is clearly the case, since otherwise the verifier could record the execution of the protocol and replay it to someone else: if this were accepted by the new party as proof that the replaying party knows the secret information, then the new party's acceptance is either justified – the replayer does know the secret information – which means that the protocol leaks knowledge and is not zero-knowledge, or it is spurious – i.e. leads to a party accepting someone's proof of knowledge who does not actually possess it.Some forms of non-interactive zero-knowledge proofs of knowledge exist, but the validity of the proof relies on computational assumptions (typically the assumptions of an ideal cryptographic hash function).