On the specification of sequent systems
... an involutive negation and this makes it difficult to address directly dualities in object-logic proof systems. This lack of dualities is particularly unfortunate when specifying sequent calculus [Gen69] since they play a central role in the theory of such proof systems. Pfenning in [Pfn95,Pfn00] us ...
... an involutive negation and this makes it difficult to address directly dualities in object-logic proof systems. This lack of dualities is particularly unfortunate when specifying sequent calculus [Gen69] since they play a central role in the theory of such proof systems. Pfenning in [Pfn95,Pfn00] us ...
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... The importance of verifying the correctness of hardware and software designs dates back to the early realization of the prevalence of design errors, that is, “bugs”. While testing has once been considered a satisfying method for detecting bugs, today’s rapid development of complex and safety-critica ...
... The importance of verifying the correctness of hardware and software designs dates back to the early realization of the prevalence of design errors, that is, “bugs”. While testing has once been considered a satisfying method for detecting bugs, today’s rapid development of complex and safety-critica ...
self-reference in arithmetic i - Utrecht University Repository
... 2.1. First source of intensionality: coding. The coding is the bridge between properties of numbers and properties of syntactic objects such as formulae and terms. The choice of coding is primary in the sense that the satisfaction of the other two tasks depends on it. It depends on the depends on th ...
... 2.1. First source of intensionality: coding. The coding is the bridge between properties of numbers and properties of syntactic objects such as formulae and terms. The choice of coding is primary in the sense that the satisfaction of the other two tasks depends on it. It depends on the depends on th ...
On the Expressive Power of QLTL⋆
... part) can also be suppressed to the temporal operator X (“Next”), the counterpart of successor relation S(x, y). However, because in S1S the positions of words can be referred to directly by first order variables while in QLT L they cannot, it turns out that in QLT L the LT L part cannot be suppress ...
... part) can also be suppressed to the temporal operator X (“Next”), the counterpart of successor relation S(x, y). However, because in S1S the positions of words can be referred to directly by first order variables while in QLT L they cannot, it turns out that in QLT L the LT L part cannot be suppress ...
Nelson`s Strong Negation, Safe Beliefs and the - CEUR
... we consider one of these alternative approaches: logic programs can be understood as propositional theories and their answer sets are then defined as models in a formal logic system. In particular, we follow the line of research started by Pearce [17], who focused on establishing links between negat ...
... we consider one of these alternative approaches: logic programs can be understood as propositional theories and their answer sets are then defined as models in a formal logic system. In particular, we follow the line of research started by Pearce [17], who focused on establishing links between negat ...
Introduction to Formal Logic - Web.UVic.ca
... This inference fulfils condition (i): there is no possible case where its premises could be true and its conclusion false. Hence the inference is valid. But the inference also fulfils condition (ii), because its premises are true: all whales are in fact mammals, and all mammals have spinal chords. N ...
... This inference fulfils condition (i): there is no possible case where its premises could be true and its conclusion false. Hence the inference is valid. But the inference also fulfils condition (ii), because its premises are true: all whales are in fact mammals, and all mammals have spinal chords. N ...
A Contraction-free and Cut-free Sequent Calculus for
... is well-defined. Indeed, there are several equivalent axiomatisations of P DL (see for example [4, 7]), each of which is obtained by adding to classical propositional logic: (i) the distribution axiom schema, that now has the form: [α](A → B) → ([α]A → [α]B), for each program α; (ii) modus ponens and ...
... is well-defined. Indeed, there are several equivalent axiomatisations of P DL (see for example [4, 7]), each of which is obtained by adding to classical propositional logic: (i) the distribution axiom schema, that now has the form: [α](A → B) → ([α]A → [α]B), for each program α; (ii) modus ponens and ...
ANNALS OF PURE AND APPLIED LOGIC I W
... Via this association we can view PDL programs as being carried out along paths rather than as binary relations. For the reduction, however, it is more convenient to use the automata version of PDL, namely APDL [9]. The reason for this is that ‘fl’ can be handled more economically by automata than by ...
... Via this association we can view PDL programs as being carried out along paths rather than as binary relations. For the reduction, however, it is more convenient to use the automata version of PDL, namely APDL [9]. The reason for this is that ‘fl’ can be handled more economically by automata than by ...
Completeness - OSU Department of Mathematics
... • Whenever f is an n-ary function symbol h(f A (a1 , . . . , an )) = f B (h(a1 ), . . . , h(an )) for all a1 , . . . , an ∈ |A|. Notice that if = is in L, A and B respect equality and h is a homormorphism of A to B then h is 1-1 i.e. h is an embedding of A into B. When h is a homomorphism from A to ...
... • Whenever f is an n-ary function symbol h(f A (a1 , . . . , an )) = f B (h(a1 ), . . . , h(an )) for all a1 , . . . , an ∈ |A|. Notice that if = is in L, A and B respect equality and h is a homormorphism of A to B then h is 1-1 i.e. h is an embedding of A into B. When h is a homomorphism from A to ...
The Relative Efficiency of Propositional Proof
... for propositional proof systems which will be used in the rest of this paper. The letter n will always stand for an adequate set of propositional connectives which are binary, unary, or nullary (have two, one, or zero arguments). Adequate here means that every truth function can be expressed by form ...
... for propositional proof systems which will be used in the rest of this paper. The letter n will always stand for an adequate set of propositional connectives which are binary, unary, or nullary (have two, one, or zero arguments). Adequate here means that every truth function can be expressed by form ...
Restricted notions of provability by induction
... k ∈ N there is a Π1 sentence Con(IΣk ) expressing the consistency of IΣk , see for example [10]. We then have: Theorem 2.3. For all k ∈ N: PA ` Con(IΣk ) but IΣk 0 Con(IΣk ). Note that this result embodies a very strong non-analyticity requirement: given any k > 1, in order to prove Con(IΣk ) not on ...
... k ∈ N there is a Π1 sentence Con(IΣk ) expressing the consistency of IΣk , see for example [10]. We then have: Theorem 2.3. For all k ∈ N: PA ` Con(IΣk ) but IΣk 0 Con(IΣk ). Note that this result embodies a very strong non-analyticity requirement: given any k > 1, in order to prove Con(IΣk ) not on ...
Classical Logic and the Curry–Howard Correspondence
... These notions of ordered pair, union etc. should be familiar to any computer scientist. Notice that the clause for implication means that a system of realizers for intuitionistic logic must include higher-order functions. In predicate logic, the quantifiers are interpreted as follows, where S is one ...
... These notions of ordered pair, union etc. should be familiar to any computer scientist. Notice that the clause for implication means that a system of realizers for intuitionistic logic must include higher-order functions. In predicate logic, the quantifiers are interpreted as follows, where S is one ...
THE AXIOM SCHEME OF ACYCLIC COMPREHENSION keywords
... weak extensionality. We will indicate briefly after the proof of the main claim how the assumption of weak extensionality could be dispensed with. Finite Axiomatization of Stratified Comprehension: We present a finite list of instances of stratified comprehension which is equivalent to the full sche ...
... weak extensionality. We will indicate briefly after the proof of the main claim how the assumption of weak extensionality could be dispensed with. Finite Axiomatization of Stratified Comprehension: We present a finite list of instances of stratified comprehension which is equivalent to the full sche ...
Version 1.5 - Trent University
... assumed to be formulas of LP unless stated otherwise. What do these definitions mean? The parentheses are just punctuation: their only purpose is to group other symbols together. (One could get by without them; see Problem 1.6.) ¬ and → are supposed to represent the connectives not and if . . . then ...
... assumed to be formulas of LP unless stated otherwise. What do these definitions mean? The parentheses are just punctuation: their only purpose is to group other symbols together. (One could get by without them; see Problem 1.6.) ¬ and → are supposed to represent the connectives not and if . . . then ...
Automated Deduction
... Normal forms of logical formulas play a very important role in automated deduction. This is due to the following reason: the central computational problem of automated deduction is that the search space (of an algorithm that searches for a proof) is very large. Avoiding different syntactic represent ...
... Normal forms of logical formulas play a very important role in automated deduction. This is due to the following reason: the central computational problem of automated deduction is that the search space (of an algorithm that searches for a proof) is very large. Avoiding different syntactic represent ...
Chiron: A Set Theory with Types
... The usefulness of a logic is often measured by its expressivity: the more that can be expressed in the logic, the more useful the logic is. By a logic, we mean a language (or a family of languages) that has a formal syntax and a precise semantics with a notion of logical consequence. (A logic may al ...
... The usefulness of a logic is often measured by its expressivity: the more that can be expressed in the logic, the more useful the logic is. By a logic, we mean a language (or a family of languages) that has a formal syntax and a precise semantics with a notion of logical consequence. (A logic may al ...
A Resolution-Based Proof Method for Temporal Logics of
... time’. Thus ϕ will be satisfied at some time if ϕ is satisfied at the next time. The connective means ‘until’. Thus ϕ ψ will be satisfied at some time if ψ is satisfied at that time or some time in the future, and Pϕ is satisfied at all times until the time that ψ is satisfied. P Of the derive ...
... time’. Thus ϕ will be satisfied at some time if ϕ is satisfied at the next time. The connective means ‘until’. Thus ϕ ψ will be satisfied at some time if ψ is satisfied at that time or some time in the future, and Pϕ is satisfied at all times until the time that ψ is satisfied. P Of the derive ...
lecture notes in Mathematical Logic
... computer science — and to some hard open problems as well. Logic and computer science Computability theory, also called recursion theory, separated from mathematical logic during the thirties of the last century. In turned out that some parts of logic are of a special nature: they can be entirely ca ...
... computer science — and to some hard open problems as well. Logic and computer science Computability theory, also called recursion theory, separated from mathematical logic during the thirties of the last century. In turned out that some parts of logic are of a special nature: they can be entirely ca ...
Proof Search in Modal Logic
... 1.2.1 Formal systems and provability Peano Arithmetic (PA) is a formal system whose axioms are the axioms of classical firstorder logic (including those for falsum), axioms for zero and successor, recursion axioms for addition and multiplication, and the induction axiom scheme. PA’s inference rules ...
... 1.2.1 Formal systems and provability Peano Arithmetic (PA) is a formal system whose axioms are the axioms of classical firstorder logic (including those for falsum), axioms for zero and successor, recursion axioms for addition and multiplication, and the induction axiom scheme. PA’s inference rules ...
Sequent-Systems for Modal Logic
... A rule, axiom or axiom-schema is eliminablefrom a system S if the subsystem of S without this rule, axiom or axiom-schema has the same theorems as S. It can easily be shown that a rule R is admissible in a system S iff it is eliminable from the extension of S with R. Our aim now is to show that D is ...
... A rule, axiom or axiom-schema is eliminablefrom a system S if the subsystem of S without this rule, axiom or axiom-schema has the same theorems as S. It can easily be shown that a rule R is admissible in a system S iff it is eliminable from the extension of S with R. Our aim now is to show that D is ...
Divide and congruence applied to eta-bisimulation
... Labelled transition systems can be distinguished from each other by a wide range of semantic equivalences, based on e.g. branching structure or decorated versions of execution sequences. Van Glabbeek [8] classified equivalences for processes that take into account the internal action τ . Here we foc ...
... Labelled transition systems can be distinguished from each other by a wide range of semantic equivalences, based on e.g. branching structure or decorated versions of execution sequences. Van Glabbeek [8] classified equivalences for processes that take into account the internal action τ . Here we foc ...
Label-free Modular Systems for Classical and Intuitionistic Modal
... inference rules such that cut elimination is preserved. Recently, this goal has been achieved for substructural logics: In [4] it has been unveiled which classes of axioms can be transformed into equivalent structural rules in the sequent calculus, respectively hypersequent calculus, such that the r ...
... inference rules such that cut elimination is preserved. Recently, this goal has been achieved for substructural logics: In [4] it has been unveiled which classes of axioms can be transformed into equivalent structural rules in the sequent calculus, respectively hypersequent calculus, such that the r ...
Label-free Modular Systems for Classical and Intuitionistic Modal
... inference rules such that cut elimination is preserved. Recently, this goal has been achieved for substructural logics: In [4] it has been unveiled which classes of axioms can be transformed into equivalent structural rules in the sequent calculus, respectively hypersequent calculus, such that the r ...
... inference rules such that cut elimination is preserved. Recently, this goal has been achieved for substructural logics: In [4] it has been unveiled which classes of axioms can be transformed into equivalent structural rules in the sequent calculus, respectively hypersequent calculus, such that the r ...
3.6 First-Order Tableau
... are two terms with sort(s) = sort(t) then 1. if {s = t} ⇒∗SU E then any equation (s0 = t0 ) ∈ E is well-sorted, i.e., sort(s0 ) = sort(t0 ). 2. ⇒SU terminates on {s = t}. 3. if {s = t} ⇒∗SU E then σ is a unifier (mgu) of E iff σ is a unifier (mgu) of {s = t}. 4. if {s = t} ⇒∗SU ⊥ then s and t are no ...
... are two terms with sort(s) = sort(t) then 1. if {s = t} ⇒∗SU E then any equation (s0 = t0 ) ∈ E is well-sorted, i.e., sort(s0 ) = sort(t0 ). 2. ⇒SU terminates on {s = t}. 3. if {s = t} ⇒∗SU E then σ is a unifier (mgu) of E iff σ is a unifier (mgu) of {s = t}. 4. if {s = t} ⇒∗SU ⊥ then s and t are no ...
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... 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 ...