Explaining Counterexamples Using Causality
... respect to one of the occurrences) and a non-bottom value (with respect to a different occurrence) for any state s. Before we formally define causes in counterexamples we need to deal with one subtlety: the value of ϕ on finite paths. While computation paths are infinite, it is often possible to det ...
... respect to one of the occurrences) and a non-bottom value (with respect to a different occurrence) for any state s. Before we formally define causes in counterexamples we need to deal with one subtlety: the value of ϕ on finite paths. While computation paths are infinite, it is often possible to det ...
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... form ϕ @ ψ is true iff ϕ is true and false iff ψ is false; ϕ / ψ is related to Blamey’s [7] transplication and can be read as ‘ψ, presupposing ϕ’. This formula has the value of ψ if ϕ is true, but is neither true nor false otherwise. The Π and Σ quantifiers are the duals of ∀ and ∃ and correspond to ...
... form ϕ @ ψ is true iff ϕ is true and false iff ψ is false; ϕ / ψ is related to Blamey’s [7] transplication and can be read as ‘ψ, presupposing ϕ’. This formula has the value of ψ if ϕ is true, but is neither true nor false otherwise. The Π and Σ quantifiers are the duals of ∀ and ∃ and correspond to ...
Dynamic logic of propositional assignments
... this, decidability of the satisfiability problem follows. Our result contrasts with both Miller and Moss’s undecidability result for the extension of PAL by the PDL program connectives and with Tiomkin and Makowsky’s undecidability result for the extension of PDL by local assignments. But the decida ...
... this, decidability of the satisfiability problem follows. Our result contrasts with both Miller and Moss’s undecidability result for the extension of PAL by the PDL program connectives and with Tiomkin and Makowsky’s undecidability result for the extension of PDL by local assignments. But the decida ...
071 Embeddings
... For technical explanation of this term see Monk [1976] chapters 13 to 16, and Tarski, Mostowski and Robinson ...
... For technical explanation of this term see Monk [1976] chapters 13 to 16, and Tarski, Mostowski and Robinson ...
Back to Basics: Revisiting the Incompleteness
... quantifiers can either be primitive, or – e.g. in the case where T is a rich set theory – they will be defined by restricting the theory’s native quantifiers. Either way, quantifier notation in the formal sentences below should always be read as indicating such numerical quantifiers. The standard in ...
... quantifiers can either be primitive, or – e.g. in the case where T is a rich set theory – they will be defined by restricting the theory’s native quantifiers. Either way, quantifier notation in the formal sentences below should always be read as indicating such numerical quantifiers. The standard in ...
Equality in the Presence of Apartness: An Application of Structural
... axiom for equality, the n-th given by the addition of a generalized stability axiom Sn (see Section 4 for the details). By analyzing the structure of formal derivations in the system, Van Dalen and Statman were able to prove that if a formula containing no apartness symbol is derivable in the theory ...
... axiom for equality, the n-th given by the addition of a generalized stability axiom Sn (see Section 4 for the details). By analyzing the structure of formal derivations in the system, Van Dalen and Statman were able to prove that if a formula containing no apartness symbol is derivable in the theory ...
CS5371 Theory of Computation
... Why do we use the term countable?? For a countable set S, there will be a oneto-one correspondence f from N to S. If f(k) = x, we call x the kth element of S To list out elements in S, we may list the 1st element, then the 2nd element, then the 3rd element, and so on. (Just like counting sheep when ...
... Why do we use the term countable?? For a countable set S, there will be a oneto-one correspondence f from N to S. If f(k) = x, we call x the kth element of S To list out elements in S, we may list the 1st element, then the 2nd element, then the 3rd element, and so on. (Just like counting sheep when ...
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... inductively as A0 = RM (1) and An+1 = An · A. The image of the interpretation RM together with the operations ∪, ·, ∗ , ∅, {1M } is the algebra of regular sets over M , denoted by Reg M . If M is the free monoid Σ ∗ , then RM is the standard language interpretation of regular expressions. It is know ...
... inductively as A0 = RM (1) and An+1 = An · A. The image of the interpretation RM together with the operations ∪, ·, ∗ , ∅, {1M } is the algebra of regular sets over M , denoted by Reg M . If M is the free monoid Σ ∗ , then RM is the standard language interpretation of regular expressions. It is know ...
Default Reasoning in a Terminological Logic
... The field of TLs has lately been an active area of research, with the attention of researchers especially focusing on the investigation of their logical and computational properties. Nevertheless, few researchers have addressed the problem of extending these logics with the ability to perform defaul ...
... The field of TLs has lately been an active area of research, with the attention of researchers especially focusing on the investigation of their logical and computational properties. Nevertheless, few researchers have addressed the problem of extending these logics with the ability to perform defaul ...
A Simple and Practical Valuation Tree Calculus for First
... that the reader pays a closer attention to the example given there. ...
... that the reader pays a closer attention to the example given there. ...
Completeness and Decidability of a Fragment of Duration Calculus
... logic to specify the requirements for real-time systems. DC has been used successfully in many case studies, see e.g. [ZZ94,YWZP94,HZ94,DW94,BHCZ94,XH95], [Dan98,ED99]. In [DW94], we have developed a method for designing a real-time hybrid system from its specification in DC. In that paper, we intro ...
... logic to specify the requirements for real-time systems. DC has been used successfully in many case studies, see e.g. [ZZ94,YWZP94,HZ94,DW94,BHCZ94,XH95], [Dan98,ED99]. In [DW94], we have developed a method for designing a real-time hybrid system from its specification in DC. In that paper, we intro ...
(pdf)
... This means that there for any sequence of moves of the Spoiler the Duplicator has a response that leads to a win. Similarly we will say that the Spoiler wins if there is no such sequence of moves. Exercise 2. Suppose the Duplicator is winning the game Ehr(G1 , G2 , k) after the ith turn. Show that ...
... This means that there for any sequence of moves of the Spoiler the Duplicator has a response that leads to a win. Similarly we will say that the Spoiler wins if there is no such sequence of moves. Exercise 2. Suppose the Duplicator is winning the game Ehr(G1 , G2 , k) after the ith turn. Show that ...
The Art of Ordinal Analysis
... In the above, many notions were left unexplained. We will now consider them one by one. The elementary computable functions are exactly the Kalmar elementary functions, i.e. the class of functions which contains the successor, projection, zero, addition, multiplication, and modified subtraction func ...
... In the above, many notions were left unexplained. We will now consider them one by one. The elementary computable functions are exactly the Kalmar elementary functions, i.e. the class of functions which contains the successor, projection, zero, addition, multiplication, and modified subtraction func ...
Coordinate-free logic - Utrecht University Repository
... (ii) if ϕ, ψ are formulas, then (ϕ ∧ ψ), ¬ϕ are formulas, (iii) if ϕ is a formula and x is a simple term, then ∀x ϕ is a formula. We will assume that ∨, →, ↔, ∃ are defined in an obvious way. For example, ∃x ϕ denotes ¬∀x ¬ϕ. As the definitions show, we have no terms with more than one argument-pla ...
... (ii) if ϕ, ψ are formulas, then (ϕ ∧ ψ), ¬ϕ are formulas, (iii) if ϕ is a formula and x is a simple term, then ∀x ϕ is a formula. We will assume that ∨, →, ↔, ∃ are defined in an obvious way. For example, ∃x ϕ denotes ¬∀x ¬ϕ. As the definitions show, we have no terms with more than one argument-pla ...
The Complete Proof Theory of Hybrid Systems
... it prove all that is true? A corollary to Gödel’s incompleteness theorem shows that hybrid systems do not have a sound and complete calculus that is fully effective, because both their discrete fragment and their continuous fragment alone are nonaxiomatizable since each can define integer arithmeti ...
... it prove all that is true? A corollary to Gödel’s incompleteness theorem shows that hybrid systems do not have a sound and complete calculus that is fully effective, because both their discrete fragment and their continuous fragment alone are nonaxiomatizable since each can define integer arithmeti ...
Chapter 5 Predicate Logic
... formula: (∀x)H(x, x). Here there is still only one quantifier and no connectives, but there is more than one quantified variable. The interpretation is that both arguments must be the same. This expression is true if H can pair all elements of D with themselves. This is true in the just preceding ca ...
... formula: (∀x)H(x, x). Here there is still only one quantifier and no connectives, but there is more than one quantified variable. The interpretation is that both arguments must be the same. This expression is true if H can pair all elements of D with themselves. This is true in the just preceding ca ...
Default reasoning using classical logic
... and 5 we discuss how the models presented in Section 3 can be treated as classical models of propositional logic. We present algorithms that associate for each nite default theory a classical propositional theory that characterizes its extensions. Then, in Section 6 we use constraint satisfaction t ...
... and 5 we discuss how the models presented in Section 3 can be treated as classical models of propositional logic. We present algorithms that associate for each nite default theory a classical propositional theory that characterizes its extensions. Then, in Section 6 we use constraint satisfaction t ...
Glivenko sequent classes in the light of structural proof theory
... in isolating such classes lies in the fact that it may be easier to prove theorems by the use of classical rather than intuitionistic logic. Further, since a proof in intuitionistic logic can be associated to a lambda term and thus obtain a computational meaning, such results have more recently been ...
... in isolating such classes lies in the fact that it may be easier to prove theorems by the use of classical rather than intuitionistic logic. Further, since a proof in intuitionistic logic can be associated to a lambda term and thus obtain a computational meaning, such results have more recently been ...
Notes on Mathematical Logic David W. Kueker
... be (¬φ) for some φ ∈ Sn; this can only happen if φ = S17 ¬, and S17 ¬ ∈ / Sn since it has length greater than 1, but has no parentheses. The set Sn of sentences was defined as the closure of some explicitly given set (here the set of all sentence symbols) under certain operations (here the operation ...
... be (¬φ) for some φ ∈ Sn; this can only happen if φ = S17 ¬, and S17 ¬ ∈ / Sn since it has length greater than 1, but has no parentheses. The set Sn of sentences was defined as the closure of some explicitly given set (here the set of all sentence symbols) under certain operations (here the operation ...
Beginning Deductive Logic
... brave or foolhardy (or both!) to venture an answer to such a question, unless perhaps one has set aside enough time and space to craft at least a sustained essay, or perhaps even a book. Well, we fancy ourselves brave; here goes; all done in but one sentence: Logic is the science and engineering of ...
... brave or foolhardy (or both!) to venture an answer to such a question, unless perhaps one has set aside enough time and space to craft at least a sustained essay, or perhaps even a book. Well, we fancy ourselves brave; here goes; all done in but one sentence: Logic is the science and engineering of ...
The Development of Categorical Logic
... in the work of the Grothendieck school and in Lawvere’s functorial semantics, but it was Joyal and Reyes (see Reyes 1974) who, in 1972, identified a general type of firstorder theory, later called a coherent or geometric theory, which could be shown always to possess a classifying topos. This work w ...
... in the work of the Grothendieck school and in Lawvere’s functorial semantics, but it was Joyal and Reyes (see Reyes 1974) who, in 1972, identified a general type of firstorder theory, later called a coherent or geometric theory, which could be shown always to possess a classifying topos. This work w ...
Factoring Out the Impossibility of Logical Aggregation
... : , c(') = c( ); if ' = '1 _ '2 , ' = '1 ^ '2 ; or '1 ! '2 ; c(') = max fc('1 ); c('2 )g + 1: (This de…nition of complexity di¤ers from the standard one, which states that if ' = : , c(') = c( ) + 1.) The axiomatic system of propositional logic de…nes an inference relation S ` ' holding between sets ...
... : , c(') = c( ); if ' = '1 _ '2 , ' = '1 ^ '2 ; or '1 ! '2 ; c(') = max fc('1 ); c('2 )g + 1: (This de…nition of complexity di¤ers from the standard one, which states that if ' = : , c(') = c( ) + 1.) The axiomatic system of propositional logic de…nes an inference relation S ` ' holding between sets ...
P - Department of Computer Science
... returns True if run on some element that is in S and False if run on an element that is not in S. – A characteristic function can be used to determine whether or not a given element is in S. ...
... returns True if run on some element that is in S and False if run on an element that is not in S. – A characteristic function can be used to determine whether or not a given element is in S. ...
Remarks on Second-Order Consequence
... only that each informally proven theorem be provable by means of the calculus (in other words, when formalizing, we do not mean to be true to proofs, but to theorems). As soon as we state this demand we see the difficulty it involves, for if the notion of an informal theorem turned out to be open-en ...
... only that each informally proven theorem be provable by means of the calculus (in other words, when formalizing, we do not mean to be true to proofs, but to theorems). As soon as we state this demand we see the difficulty it involves, for if the notion of an informal theorem turned out to be open-en ...
Godel`s Incompleteness Theorem
... will be true iff x is the Gödel number of a FOL sentence, and we can show that if n is the Gödel number of a sentence, then “Sentence(n)” can be derived from PA1-6. In other words, ‘sentence-ness’ is definable (in LA) and representable (in PA). • You can also show that for any recursive set of axiom ...
... will be true iff x is the Gödel number of a FOL sentence, and we can show that if n is the Gödel number of a sentence, then “Sentence(n)” can be derived from PA1-6. In other words, ‘sentence-ness’ is definable (in LA) and representable (in PA). • You can also show that for any recursive set of axiom ...