Handout
... Σ = (Ω, Π), fixes an alphabet of non-logical symbols, where • Ω is a set of function symbols f with arity n ≥ 0, written arity(f ) = n, • Π is a set of predicate symbols p with arity m ≥ 0, written arity(p) = m. ...
... Σ = (Ω, Π), fixes an alphabet of non-logical symbols, where • Ω is a set of function symbols f with arity n ≥ 0, written arity(f ) = n, • Π is a set of predicate symbols p with arity m ≥ 0, written arity(p) = m. ...
term rewriting.
... to show that some statements (often called a conjecture, goal, conclusion) is a logical consequence of a set of statements (often called hypothesis, assumptions or axioms). The ATP in CafeOBJ is based on (Prover9). Most of the logic problems on this course use the ATP. We set up problems such as Por ...
... to show that some statements (often called a conjecture, goal, conclusion) is a logical consequence of a set of statements (often called hypothesis, assumptions or axioms). The ATP in CafeOBJ is based on (Prover9). Most of the logic problems on this course use the ATP. We set up problems such as Por ...
GENERATORS AND RELATIONS FOR n-QUBIT CLIFFORD
... In this paper, we define a normal form for Clifford circuits, and we prove that every Clifford operator has a unique normal form. Moreover, we present a rewrite system by which any Clifford circuit can be reduced to normal form. This yields a presentation of Clifford operators in terms of generators ...
... In this paper, we define a normal form for Clifford circuits, and we prove that every Clifford operator has a unique normal form. Moreover, we present a rewrite system by which any Clifford circuit can be reduced to normal form. This yields a presentation of Clifford operators in terms of generators ...
CPEN 214 Digital Logic Design Fall 2003
... Binary number system The binary number system. – Base is 2 - symbols (0,1) - Binary Digits (Bits) – For Numbers > 1, add more significant digits in position to the left, e.g. 10>1. – Each position carries a weight (using decimal). ...
... Binary number system The binary number system. – Base is 2 - symbols (0,1) - Binary Digits (Bits) – For Numbers > 1, add more significant digits in position to the left, e.g. 10>1. – Each position carries a weight (using decimal). ...
PDF
... Immerman suggested a new logic FO + IFP + C extending FO + IFP with a counting construct (Immerman (1986)). But even this logic fails to capture PTIME, as has been proved by Cai et al. (1992). Abiteboul and Vianu de ned another extension of FO + IFP, called FO + IFP + W, which has a nondeterministic ...
... Immerman suggested a new logic FO + IFP + C extending FO + IFP with a counting construct (Immerman (1986)). But even this logic fails to capture PTIME, as has been proved by Cai et al. (1992). Abiteboul and Vianu de ned another extension of FO + IFP, called FO + IFP + W, which has a nondeterministic ...
Canonicity and representable relation algebras
... Here, graphs are undirected and loop-free: G = (V, E) where E ⊆ V × V is irreflexive and symmetric. For k ≥ 3, a cycle of length k in G is a sequence v1 , . . . , vk ∈ V of distinct nodes with (v1 , v2 ), (v2 , v3 ), . . . , (vk , v1 ) ∈ E. A subset X ⊆ V is independent if E ∩ (X × X) = ∅. For k < ω ...
... Here, graphs are undirected and loop-free: G = (V, E) where E ⊆ V × V is irreflexive and symmetric. For k ≥ 3, a cycle of length k in G is a sequence v1 , . . . , vk ∈ V of distinct nodes with (v1 , v2 ), (v2 , v3 ), . . . , (vk , v1 ) ∈ E. A subset X ⊆ V is independent if E ∩ (X × X) = ∅. For k < ω ...
Logic Synthesis of MEM Relay Circuits
... As CMOS scaling begins to reach its fundamental limits, micro-electro-mechanical (MEM) relays provide an attractive option for improvements in energy efficiency due to their low leakage and near ideal I-V characteristics. However, mechanical actuation of MEM relays introduces significantly more dela ...
... As CMOS scaling begins to reach its fundamental limits, micro-electro-mechanical (MEM) relays provide an attractive option for improvements in energy efficiency due to their low leakage and near ideal I-V characteristics. However, mechanical actuation of MEM relays introduces significantly more dela ...
Lecture slides for week 6 - Department of Computer Science and
... %5d place an integer right justified in a field of 5 characters %7.2f place a floating point number with two digits after the ...
... %5d place an integer right justified in a field of 5 characters %7.2f place a floating point number with two digits after the ...
Non-Commutative Arithmetic Circuits with Division
... and presence of inversion gates. Combined with Reutenauer’s theorem, this implies that the inverse of an n × n matrix cannot be computed by a formula smaller than 2Ω(n) . In circuit complexity, one keeps searching for properties that would imply that a function is hard to compute. For a polynomial f ...
... and presence of inversion gates. Combined with Reutenauer’s theorem, this implies that the inverse of an n × n matrix cannot be computed by a formula smaller than 2Ω(n) . In circuit complexity, one keeps searching for properties that would imply that a function is hard to compute. For a polynomial f ...
CPEN 214 Digital Logic Design Fall 2003
... 10’s complement of N : 10n– N (N is a decimal #) 1’s complement of N : (2n-1) – N (N is a binary #) 1’s complement can be formed by changing 1’s to 0’s and 0’s to 1’s 2’s complement of a number is obtained by leaving all least significant 0’s and the first 1 unchanged, and replacing 1’s with 0’s and ...
... 10’s complement of N : 10n– N (N is a decimal #) 1’s complement of N : (2n-1) – N (N is a binary #) 1’s complement can be formed by changing 1’s to 0’s and 0’s to 1’s 2’s complement of a number is obtained by leaving all least significant 0’s and the first 1 unchanged, and replacing 1’s with 0’s and ...
CH8B
... – Low true signal names are followed by ‘(L)’ to indicate low true – High true signal names are followed by ‘(H)’ to indicate low true ...
... – Low true signal names are followed by ‘(L)’ to indicate low true – High true signal names are followed by ‘(H)’ to indicate low true ...
Digital Systems
... • Binary logic or Boolean algebra deals with variables that take on two discrete values. • Binary logic consists of binary variables (e.g. A,B,C, x,y,z and etc.) that can be 1 or 0 and logical operations such as: AND: x.y=z or zy=z (see AND truth table) OR: x + y =z (see OR truth table) NOT: x’ =z ( ...
... • Binary logic or Boolean algebra deals with variables that take on two discrete values. • Binary logic consists of binary variables (e.g. A,B,C, x,y,z and etc.) that can be 1 or 0 and logical operations such as: AND: x.y=z or zy=z (see AND truth table) OR: x + y =z (see OR truth table) NOT: x’ =z ( ...
A Crevice on the Crane Beach: Finite-Degree
... / AC0 , but assert that “contrary to [their] original complexity. Some expressiveness results were also derived hope, [their] Ehrenfeucht-Fraïssé game arguments are not from Crane Beach Properties, for instance Lee [16] shows that simpler than classical lower bounds.” More recent promising FO[+] is ...
... / AC0 , but assert that “contrary to [their] original complexity. Some expressiveness results were also derived hope, [their] Ehrenfeucht-Fraïssé game arguments are not from Crane Beach Properties, for instance Lee [16] shows that simpler than classical lower bounds.” More recent promising FO[+] is ...
Non-commutative arithmetic circuits with division
... phenomenon of nested inversion provides a new invariant not present in the commutative setting - the height of a rational function. The height is the minimum number of nested inversions in a formula computing this rational function. For a long time, it was not even clear that the height is unbounde ...
... phenomenon of nested inversion provides a new invariant not present in the commutative setting - the height of a rational function. The height is the minimum number of nested inversions in a formula computing this rational function. For a long time, it was not even clear that the height is unbounde ...
Non-commutative arithmetic circuits with division
... phenomenon of nested inversion provides a new invariant not present in the commutative setting - the height of a rational function. The height is the minimum number of nested inversions in a formula computing this rational function. For a long time, it was not even clear that the height is unbounde ...
... phenomenon of nested inversion provides a new invariant not present in the commutative setting - the height of a rational function. The height is the minimum number of nested inversions in a formula computing this rational function. For a long time, it was not even clear that the height is unbounde ...
homogeneous polynomials with a multiplication theorem
... in less than n variables, make a linear transformation on the variables x< so that /i(x) is a polynomial in X\, • • • , xni, in which nx is as small as possible. When we make the induced transformation on the basal numbers of the algebra, the modulus of the algebra is not in general ei and so the co ...
... in less than n variables, make a linear transformation on the variables x< so that /i(x) is a polynomial in X\, • • • , xni, in which nx is as small as possible. When we make the induced transformation on the basal numbers of the algebra, the modulus of the algebra is not in general ei and so the co ...
Inequalities with variables on both sides
... A-Plus Advertising charges a fee of $24 plus $0.10 per flyer to print and deliver flyers. Print and More charges $0.25 per flyer. For how many flyers is the cost at A-Plus Advertising less than the cost of Print and More? Let f represent the number of flyers printed. A-Plus Advertising plus fee of $ ...
... A-Plus Advertising charges a fee of $24 plus $0.10 per flyer to print and deliver flyers. Print and More charges $0.25 per flyer. For how many flyers is the cost at A-Plus Advertising less than the cost of Print and More? Let f represent the number of flyers printed. A-Plus Advertising plus fee of $ ...
Logic in Computer Science
... In a sense, propositional logic (PL) is the coarsest logic: PL is domain-independent. Statements are only distinguished with respect to their truth values, e.g., there is no difference between the sentences “2 + 3 = 5” and “Konstanz is situated on Lake Constance” as both statements are true. So, sta ...
... In a sense, propositional logic (PL) is the coarsest logic: PL is domain-independent. Statements are only distinguished with respect to their truth values, e.g., there is no difference between the sentences “2 + 3 = 5” and “Konstanz is situated on Lake Constance” as both statements are true. So, sta ...
Encoding Knowledge with Predicate Logic
... Resolution Resolution is used to prove that a given set of clauses is inconsistent. (i.e., it cannot have a model). Each time a resolvent is formed, it is added to the set of clauses. This does not change the consistency of the set. However, if the null clause is ever added to the set, then it becom ...
... Resolution Resolution is used to prove that a given set of clauses is inconsistent. (i.e., it cannot have a model). Each time a resolvent is formed, it is added to the set of clauses. This does not change the consistency of the set. However, if the null clause is ever added to the set, then it becom ...
BOOLEAN ALGEBRA Boolean algebra, or the algebra of logic, was
... "+" (Boolean "addition"—here corresponding to inclusive disjunction), “ i ” (Boolean "multiplication") and "–" (called Boolean complementation: note that the operation sending x, y to x i (–y) corresponds to Boole's "subtraction") are known as Boolean operations. The elements 0 and 1 are called the ...
... "+" (Boolean "addition"—here corresponding to inclusive disjunction), “ i ” (Boolean "multiplication") and "–" (called Boolean complementation: note that the operation sending x, y to x i (–y) corresponds to Boole's "subtraction") are known as Boolean operations. The elements 0 and 1 are called the ...
presentation source
... Resolution Resolution is used to prove that a given set of clauses is inconsistent. (i.e., it cannot have a model). Each time a resolvent is formed, it is added to the set of clauses. This does not change the consistency of the set. However, if the null clause is ever added to the set, then it becom ...
... Resolution Resolution is used to prove that a given set of clauses is inconsistent. (i.e., it cannot have a model). Each time a resolvent is formed, it is added to the set of clauses. This does not change the consistency of the set. However, if the null clause is ever added to the set, then it becom ...