Horn formula minimization - RIT Scholar Works
... Given two Boolean functions f and g, if for every possible truth assignment, f (p1 , . . . , pn ) = true implies g(p1 , . . . , pn ) = true, then we denote this relationship by f → g, meaning f implies g. Likewise, for two clauses C and C , we say that C → C if C implies C . For example, given ...
... Given two Boolean functions f and g, if for every possible truth assignment, f (p1 , . . . , pn ) = true implies g(p1 , . . . , pn ) = true, then we denote this relationship by f → g, meaning f implies g. Likewise, for two clauses C and C , we say that C → C if C implies C . For example, given ...
DIPLOMAMUNKA
... An expression is an atomic formula if it is of the form t1 = t2 , where t1 and t2 are terms. An expression is a formula if it belongs to the following recursively defined set of expressions: F1 Every atomic formula is a formula. F2 If ϕ and ψ are formulas, then ¬ϕ and (ϕ → ψ) are both formulas. F3 I ...
... An expression is an atomic formula if it is of the form t1 = t2 , where t1 and t2 are terms. An expression is a formula if it belongs to the following recursively defined set of expressions: F1 Every atomic formula is a formula. F2 If ϕ and ψ are formulas, then ¬ϕ and (ϕ → ψ) are both formulas. F3 I ...
Informal Proceedings of the 30th International Workshop on
... protocol for key exchange and then encryption with derived keys. For human users this is most visible as transport layer security (TLS) used by all web browsers. History has shown that developing such protocols is an error-prone process, and attacks have been found even after protocols were in wides ...
... protocol for key exchange and then encryption with derived keys. For human users this is most visible as transport layer security (TLS) used by all web browsers. History has shown that developing such protocols is an error-prone process, and attacks have been found even after protocols were in wides ...
Chiron: A Set Theory with Types, Undefinedness, Quotation, and
... and reasoning about functions and undefinedness is usually performed in the metalogic instead of in the logic itself. Chiron is a set theory that has a much higher level of practical expressivity than traditional set theories. It is intended to be a general-purpose logic that, unlike traditional logi ...
... and reasoning about functions and undefinedness is usually performed in the metalogic instead of in the logic itself. Chiron is a set theory that has a much higher level of practical expressivity than traditional set theories. It is intended to be a general-purpose logic that, unlike traditional logi ...
Pebble weighted automata and transitive - LSV
... from the grammar ϕ ::= k | α | ¬ϕ | ϕ ∨ ϕ | ϕ ∧ ϕ, with k ∈ K and α ∈ L. In particular, quantifications are only allowed in formulas α ∈ L. The following lemma shows in particular that an L-step formula assumes a finite number of values, each of which corresponds to an L-definable language. Lemma W ...
... from the grammar ϕ ::= k | α | ¬ϕ | ϕ ∨ ϕ | ϕ ∧ ϕ, with k ∈ K and α ∈ L. In particular, quantifications are only allowed in formulas α ∈ L. The following lemma shows in particular that an L-step formula assumes a finite number of values, each of which corresponds to an L-definable language. Lemma W ...
Kripke completeness revisited
... As mathematics progresses, notions that were obscure and perplexing become clear and straightforward, sometimes even achieving the status of “obvious.” Then hindsight can make us all wise after the event. But we are separated from the past by our knowledge of the present, which may draw us into “see ...
... As mathematics progresses, notions that were obscure and perplexing become clear and straightforward, sometimes even achieving the status of “obvious.” Then hindsight can make us all wise after the event. But we are separated from the past by our knowledge of the present, which may draw us into “see ...
7._Relational_Proposition - abuad lms
... founder. In this example “is the university’s founder,” is the predicate • EXAMPLE 2: Afe Babalola is the Chancellor of Afe Babalola University Ado-Ekiti. In this example “is the Chancellor” is the predicate ...
... founder. In this example “is the university’s founder,” is the predicate • EXAMPLE 2: Afe Babalola is the Chancellor of Afe Babalola University Ado-Ekiti. In this example “is the Chancellor” is the predicate ...
Insights into Modal Slash Logic and Modal Decidability
... assignment ~x 7→ ~c if there is a winning strategy for player E (player A respectively) in game G(φ, M, ~c). If there is no winning strategy for either player, the formula φ is said to be non-determined in M under the assignment ~x 7→ ~c. If φ is a sentence, we say that φ is true (false, non-determi ...
... assignment ~x 7→ ~c if there is a winning strategy for player E (player A respectively) in game G(φ, M, ~c). If there is no winning strategy for either player, the formula φ is said to be non-determined in M under the assignment ~x 7→ ~c. If φ is a sentence, we say that φ is true (false, non-determi ...
Equality in the Presence of Apartness: An Application of Structural
... In classical logic, each universal axiom can be converted, through reduction to conjunctive normal form, to a conjunction of regular formulas, and therefore all universal axioms can be converted into regular rules. By using intuitionistic logic, we have instead a limitation to those axioms that are ...
... In classical logic, each universal axiom can be converted, through reduction to conjunctive normal form, to a conjunction of regular formulas, and therefore all universal axioms can be converted into regular rules. By using intuitionistic logic, we have instead a limitation to those axioms that are ...
Written
... reflexive; (b) symmetric; (c) reflexive and symmetric; (d) reflexive and contain (1, 2); (e) symmetric and contain (1, 2); (f) anti-symmetric; (g) anti-symmetric and contain (1, 2); (h) symmetric and anti-symmetric; (i) reflexive, symmetric and anti-symmetric. a) Each of these relations must contain ...
... reflexive; (b) symmetric; (c) reflexive and symmetric; (d) reflexive and contain (1, 2); (e) symmetric and contain (1, 2); (f) anti-symmetric; (g) anti-symmetric and contain (1, 2); (h) symmetric and anti-symmetric; (i) reflexive, symmetric and anti-symmetric. a) Each of these relations must contain ...
On Rosser sentences and proof predicates
... Löb’s theorem seem to tell the whole story of Pr . Indeed, the result on possible non-uniqueness of Rosser sentences is the first requiring more than these conditions, together with “the usual” ordering of proofs, for a settlement. It is also clear that “the usual” ordering and “the usual” proof pr ...
... Löb’s theorem seem to tell the whole story of Pr . Indeed, the result on possible non-uniqueness of Rosser sentences is the first requiring more than these conditions, together with “the usual” ordering of proofs, for a settlement. It is also clear that “the usual” ordering and “the usual” proof pr ...
No Syllogisms for the Numerical Syllogistic
... expression of either of the forms p or p̄, where p is an atom. A literal which is an atom is said to be positive; all other literals are said to be negative. If l = p̄ is a negative literal, then we take l̄ to denote the positive literal p. An N † -formula is an expression of either of the forms (≤ ...
... expression of either of the forms p or p̄, where p is an atom. A literal which is an atom is said to be positive; all other literals are said to be negative. If l = p̄ is a negative literal, then we take l̄ to denote the positive literal p. An N † -formula is an expression of either of the forms (≤ ...
1. Propositional Logic 1.1. Basic Definitions. Definition 1.1. The
... behave very differently: there can be many assumptions, but only one consequence, and while rules can add or remove formulas from the assumptions, they can only modify the conclusion. In the sequent calculus, we will allow both sides of a sequent to be sets of formulas (although we will later study ...
... behave very differently: there can be many assumptions, but only one consequence, and while rules can add or remove formulas from the assumptions, they can only modify the conclusion. In the sequent calculus, we will allow both sides of a sequent to be sets of formulas (although we will later study ...
Sequent Calculus in Natural Deduction Style
... principal in at least the right premiss of cut. In addition, the cut formula can be principal somewhere higher up in the derivation of the right premiss of cut, and the cut is permuted up there in one step. For all other cases of cut, we prove that the cut formula is a subformula of the conclusion. ...
... principal in at least the right premiss of cut. In addition, the cut formula can be principal somewhere higher up in the derivation of the right premiss of cut, and the cut is permuted up there in one step. For all other cases of cut, we prove that the cut formula is a subformula of the conclusion. ...
PDF
... that are inherently vacuous by model but are not inherently vacuous by mutation. For example, consider the formula ϕ = p ∨ q. Every deterministic Kripke structure that satisfies ϕ has its (single) initial state labeled either by p or by q or by both, and thus it satisfies ϕ vacuously. On the other h ...
... that are inherently vacuous by model but are not inherently vacuous by mutation. For example, consider the formula ϕ = p ∨ q. Every deterministic Kripke structure that satisfies ϕ has its (single) initial state labeled either by p or by q or by both, and thus it satisfies ϕ vacuously. On the other h ...
Constructing Cut Free Sequent Systems With Context Restrictions
... rules which works uniformly for classical and intuitionistic logics. The rules so constructed are by construction sound and complete (in the presence of cut) and give rise to unlabelled sequent systems that are amenable to saturation under cuts between rules. In case the resulting rules fulfil our c ...
... rules which works uniformly for classical and intuitionistic logics. The rules so constructed are by construction sound and complete (in the presence of cut) and give rise to unlabelled sequent systems that are amenable to saturation under cuts between rules. In case the resulting rules fulfil our c ...
Deep Sequent Systems for Modal Logic
... disjunction, and the turnstile as implication. Hypersequents are also pure. A labelled sequent, on the other hand, does not generally have an equivalent modal formula. Hein, Stewart and Stouppa use the calculus of structures [3, 4] to give pure systems for modal logics. This formalism is based on de ...
... disjunction, and the turnstile as implication. Hypersequents are also pure. A labelled sequent, on the other hand, does not generally have an equivalent modal formula. Hein, Stewart and Stouppa use the calculus of structures [3, 4] to give pure systems for modal logics. This formalism is based on de ...
Logic 1 Lecture Notes Part I: Propositional Logic
... A note on use versus mention: most of the time, language is used to talk about nonlinguistic entities and states of affairs, such as dogs, cats and football matches. However, sometimes languages is not used but rather mentioned, as in the observation that ‘cat’ is a 3 letter word. In the context of ...
... A note on use versus mention: most of the time, language is used to talk about nonlinguistic entities and states of affairs, such as dogs, cats and football matches. However, sometimes languages is not used but rather mentioned, as in the observation that ‘cat’ is a 3 letter word. In the context of ...
Godel`s Proof
... faithfully replicated. The point of these two examples (and I could give many more) is that human thinking in all its flexible and fallible glory can in principle be modeled by a “fixed set of directives,” provided one is liberated from the preconception that computers, built on arithmeti- ...
... faithfully replicated. The point of these two examples (and I could give many more) is that human thinking in all its flexible and fallible glory can in principle be modeled by a “fixed set of directives,” provided one is liberated from the preconception that computers, built on arithmeti- ...
Document
... into variables (x,y,z,…), constants (0,1,…,a,b,c,…), and function symbols (f,g,h,…) with fixed arity, the number of arguments they take. A term is constructed as follows. 1) A variable and a constant are terms. 2) If f is a function symbol of arity n and if t1 ,…, tn are terms, then f(t1,…,tn) is a ...
... into variables (x,y,z,…), constants (0,1,…,a,b,c,…), and function symbols (f,g,h,…) with fixed arity, the number of arguments they take. A term is constructed as follows. 1) A variable and a constant are terms. 2) If f is a function symbol of arity n and if t1 ,…, tn are terms, then f(t1,…,tn) is a ...
From Syllogism to Common Sense Normal Modal Logic
... ‣ Informally, a rule of inference A/B is derivable in a logic L if there is an L -proof of B from A. ‣ If there is an L -proof of B from A, by the rule of substitution there also is an L -proof of #(B) from #(A), for any substitution #. For admissible rules this has to be made explicit. ‣ A rule A/B ...
... ‣ Informally, a rule of inference A/B is derivable in a logic L if there is an L -proof of B from A. ‣ If there is an L -proof of B from A, by the rule of substitution there also is an L -proof of #(B) from #(A), for any substitution #. For admissible rules this has to be made explicit. ‣ A rule A/B ...
First-Order Intuitionistic Logic with Decidable Propositional
... with careless addition of some interaction axioms may make the combination logic collapse into classical logic [FH, SRC]. Precautions are necessary to avoid the collapsing problem. Let us have a fresh look at this problem of combining features of intuitionistic and classical logic and try to do it s ...
... with careless addition of some interaction axioms may make the combination logic collapse into classical logic [FH, SRC]. Precautions are necessary to avoid the collapsing problem. Let us have a fresh look at this problem of combining features of intuitionistic and classical logic and try to do it s ...
Loop Formulas for Circumscription - Joohyung Lee
... Given a propositional formula A and an atom q, a formula ϕ that does not mention q is called a weakest sufficient condition of q if • A |= ϕ ⊃ q, and • for any other formula ψ such that it does not mention q and A |= ψ ⊃ q, we have that A |= ψ ⊃ ϕ. For any A and q, weakest sufficient conditions of ...
... Given a propositional formula A and an atom q, a formula ϕ that does not mention q is called a weakest sufficient condition of q if • A |= ϕ ⊃ q, and • for any other formula ψ such that it does not mention q and A |= ψ ⊃ q, we have that A |= ψ ⊃ ϕ. For any A and q, weakest sufficient conditions of ...
Deep Sequent Systems for Modal Logic
... its corresponding modal formula should be. Hein, Stewart and Stouppa use the calculus of structures to give pure systems for modal logics. This formalism was developed by Guglielmi [8] and was further studied by the author [5, 3] and by Straßburger[9, 18]. It is based on deep inference, which is the ...
... its corresponding modal formula should be. Hein, Stewart and Stouppa use the calculus of structures to give pure systems for modal logics. This formalism was developed by Guglielmi [8] and was further studied by the author [5, 3] and by Straßburger[9, 18]. It is based on deep inference, which is the ...