Notes on the ACL2 Logic
... But what we are after is reasoning about programs, and while propositional logic will play an important role, we need more powerful logics. To see why, let’s simplify things for a moment and consider conjectures involving numbers and arithmetic operations. Consider the conjecture: 1. a+b = ba What d ...
... But what we are after is reasoning about programs, and while propositional logic will play an important role, we need more powerful logics. To see why, let’s simplify things for a moment and consider conjectures involving numbers and arithmetic operations. Consider the conjecture: 1. a+b = ba What d ...
Introduction Into Functional Programming
... 2.2 If the searched word is “less” then the shown ones, then 2.2.1 continue search within the left half of the book; otherwise 2.2.2 continue search within with the right half of the book. ...
... 2.2 If the searched word is “less” then the shown ones, then 2.2.1 continue search within the left half of the book; otherwise 2.2.2 continue search within with the right half of the book. ...
Goal-directed Proof Theory
... The concept of goal directed computation we adopt can also be seen as a generalization of the notion of uniform proof as introduced in [Miller et al. 91]. As far as we know, a goal-directed presentation have been given of (fragments of) intuitionistic logic [Gabbay and Reyle 84],[Miller 89], [McCart ...
... The concept of goal directed computation we adopt can also be seen as a generalization of the notion of uniform proof as introduced in [Miller et al. 91]. As far as we know, a goal-directed presentation have been given of (fragments of) intuitionistic logic [Gabbay and Reyle 84],[Miller 89], [McCart ...
A Pebble Weighted Automata and Weighted Logics
... when the last author was visiting LSV as a full time Inria researcher. This work was partly supported by ANR 2010 BLAN 0202 01 FREC, ARCUS Île-de-France–Inde, and LIA ...
... when the last author was visiting LSV as a full time Inria researcher. This work was partly supported by ANR 2010 BLAN 0202 01 FREC, ARCUS Île-de-France–Inde, and LIA ...
KURT GÖDEL - National Academy of Sciences
... the axioms that the range of the variables in them constitutes a countable collection, contradicting the theorem of Cantor by which the subsets of {0, 1, 2, ...} (which are among the sets for his theory) constitute an uncountable collection. This is Skolem's paradox (1923). It is not a direct contra ...
... the axioms that the range of the variables in them constitutes a countable collection, contradicting the theorem of Cantor by which the subsets of {0, 1, 2, ...} (which are among the sets for his theory) constitute an uncountable collection. This is Skolem's paradox (1923). It is not a direct contra ...
TR-14-06 - Ynot - Harvard University
... The commands !τ M and M :=τ N are used to read and write memory respectively. The index τ is the type of the value being read or written. Note that unlike ML and most statically-typed languages, HTT supports strong updates. That is, if x is a location holding a nat, then we can update the contents o ...
... The commands !τ M and M :=τ N are used to read and write memory respectively. The index τ is the type of the value being read or written. Note that unlike ML and most statically-typed languages, HTT supports strong updates. That is, if x is a location holding a nat, then we can update the contents o ...
Cut-elimination for provability logics and some results in display logic
... A syntactic proof of cut-elimination yields a procedure to eliminate every instance of the cut-rule from a derivation in the sequent calculus thus leading to a cutfree derivation. This is a central result in the proof-theory of a logic. In 1983, Valentini [71] presented a syntactic proof of cut-elim ...
... A syntactic proof of cut-elimination yields a procedure to eliminate every instance of the cut-rule from a derivation in the sequent calculus thus leading to a cutfree derivation. This is a central result in the proof-theory of a logic. In 1983, Valentini [71] presented a syntactic proof of cut-elim ...
Lecture Notes
... Example 2.4 For binary relations R and S on A we define their composition (denoted R ◦ S) as follows. R ◦ S = {(a, c) | for some b ∈ A, (a, b) ∈ R and (b, c) ∈ S} We may extend this binary relational composition to an n-fold composition of a single relation R as follows. Basis. R1 = R Induction step ...
... Example 2.4 For binary relations R and S on A we define their composition (denoted R ◦ S) as follows. R ◦ S = {(a, c) | for some b ∈ A, (a, b) ∈ R and (b, c) ∈ S} We may extend this binary relational composition to an n-fold composition of a single relation R as follows. Basis. R1 = R Induction step ...
Verification of a Cryptographic Primitive: SHA-256 ANDREW W. APPEL
... might profitably be rewritten in functional languages with clean proof theories for effective verification. But cryptographic primitives are not written in these languages; if we want to verify a well established widely used open-source cryptographic implementation, we need tooling for C. Synthesis ...
... might profitably be rewritten in functional languages with clean proof theories for effective verification. But cryptographic primitives are not written in these languages; if we want to verify a well established widely used open-source cryptographic implementation, we need tooling for C. Synthesis ...
View raw file - aaa
... exp : type. lam : (exp -> exp) -> exp. app : exp -> exp -> exp. check : exp -> t -> type. check/lam : check (lam M) (A arrow B) <- {x:exp} (check x A -> check (M x) B). ...
... exp : type. lam : (exp -> exp) -> exp. app : exp -> exp -> exp. check : exp -> t -> type. check/lam : check (lam M) (A arrow B) <- {x:exp} (check x A -> check (M x) B). ...
Abella: A System for Reasoning about Relational Specifications
... Types in Abella are the simple types; such types are either primitive types or built from two types using the arrow type constructor →. The type constructor → associates to the right, so every type in Abella can be written in the form τ1 → · · · → τn → b (for n ≥ 0) where b is an atomic type that is ...
... Types in Abella are the simple types; such types are either primitive types or built from two types using the arrow type constructor →. The type constructor → associates to the right, so every type in Abella can be written in the form τ1 → · · · → τn → b (for n ≥ 0) where b is an atomic type that is ...
Formale Methoden der Softwaretechnik Formal methods of software
... The problem with this proof is step 8. In this step we have used step 3, a step that occurs within an earlier subproof. But it turns out that this sort of justification—one that reaches back inside a subproof that has already ended—is not legitimate. To understand why it’s not legitimate, we need to ...
... The problem with this proof is step 8. In this step we have used step 3, a step that occurs within an earlier subproof. But it turns out that this sort of justification—one that reaches back inside a subproof that has already ended—is not legitimate. To understand why it’s not legitimate, we need to ...
logic for the mathematical
... we shall quite freely use methods of proof such as contradiction and induction, and the student should not find this troubling. Not having appealed to something like the axiom of choice could of course be important if and when you move on and wish to use this material to study foundations. The secon ...
... we shall quite freely use methods of proof such as contradiction and induction, and the student should not find this troubling. Not having appealed to something like the axiom of choice could of course be important if and when you move on and wish to use this material to study foundations. The secon ...
Constraint propagation
... There exists a reasoning technique, such that for any theory T and formula F, such that T |= F, the reasoning technique proves T |= F. ...
... There exists a reasoning technique, such that for any theory T and formula F, such that T |= F, the reasoning technique proves T |= F. ...
A Second Edition: Verification of a Cryptographic Primitive: SHA-256 ANDREW W. APPEL
... SHA-256, the Secure Hash Algorithm with 256-bit digests, is not an encryption algorithm, but it is used in encryption protocols. The methods I discuss in this paper can be applied to the same issues that appear in ciphers such as AES: interpretation of standards documents, big-endian protocols imple ...
... SHA-256, the Secure Hash Algorithm with 256-bit digests, is not an encryption algorithm, but it is used in encryption protocols. The methods I discuss in this paper can be applied to the same issues that appear in ciphers such as AES: interpretation of standards documents, big-endian protocols imple ...
A Judgmental Reconstruction of Modal Logic
... An alternative way to understand local completeness is to reconsider our meaning explanation of conjunction. We have said that a verification of A ∧ B consists of a verification of A and a verification of B. Local completeness entails that it is always possible to bring the verification of A ∧ B int ...
... An alternative way to understand local completeness is to reconsider our meaning explanation of conjunction. We have said that a verification of A ∧ B consists of a verification of A and a verification of B. Local completeness entails that it is always possible to bring the verification of A ∧ B int ...
A joint logic of problems and propositions, a modified BHK
... theoretical truths, one can systematize schemes of solutions of problems — for example, of geometric construction problems. [...] Thus, in addition to theoretical logic, a certain new calculus of problems arises. [...] Surprisingly, the calculus of problems coincides in form with Brouwer’s intuition ...
... theoretical truths, one can systematize schemes of solutions of problems — for example, of geometric construction problems. [...] Thus, in addition to theoretical logic, a certain new calculus of problems arises. [...] Surprisingly, the calculus of problems coincides in form with Brouwer’s intuition ...