Structural Proof Theory

... structural proof theory belongs, with a few exceptions, to what can be described as computational proof theory. Since 1970, a branch of proof theory known as constructive type theory has been developed. A theorem typically states that a certain claim holds under given assumptions. The basic idea of ...

... structural proof theory belongs, with a few exceptions, to what can be described as computational proof theory. Since 1970, a branch of proof theory known as constructive type theory has been developed. A theorem typically states that a certain claim holds under given assumptions. The basic idea of ...

Goal-directed Proof Theory

... involves a great amount of data (that is what we have qualied as deductive database perspective). Moreover goal-directed deductive procedures can be used to design abductive reasoning procedures [Eshghi89], although we will not develop this point further in the present work. The goal-directed parad ...

... involves a great amount of data (that is what we have qualied as deductive database perspective). Moreover goal-directed deductive procedures can be used to design abductive reasoning procedures [Eshghi89], although we will not develop this point further in the present work. The goal-directed parad ...

Labeled Natural Deduction for Temporal Logics

... In the literature, labeled natural deduction systems have been proposed for linear-time logics [19, 103] and the branching logic CTL [20, 131], which, given its syntactic restrictions on the nesting of operators, is not Ockhamist and thus is not reported in Table 1.1. In this thesis, we propose a mo ...

... In the literature, labeled natural deduction systems have been proposed for linear-time logics [19, 103] and the branching logic CTL [20, 131], which, given its syntactic restrictions on the nesting of operators, is not Ockhamist and thus is not reported in Table 1.1. In this thesis, we propose a mo ...

A Unified View of Induction Reasoning for First-Order Logic

... strategy to perform lazy induction on particular explicit induction proofs. Kapur and Subramaniam [29] devised a method that extends schemata-based induction to deal with a special class of mutually defined functions. Courant [18] identified a class of implicit induction inference systems for which ...

... strategy to perform lazy induction on particular explicit induction proofs. Kapur and Subramaniam [29] devised a method that extends schemata-based induction to deal with a special class of mutually defined functions. Courant [18] identified a class of implicit induction inference systems for which ...

SEQUENT SYSTEMS FOR MODAL LOGICS

... rules). This ‘anti-realistic’ conception of the meaning of the logical operations is often traced back to a certain passage on natural deduction from Gentzen’s Investigations into Logical Deduction [Gentzen, 1934, p. 80]: [I]ntroductions represent, as it were, the ‘definitions’ of the symbols concer ...

... rules). This ‘anti-realistic’ conception of the meaning of the logical operations is often traced back to a certain passage on natural deduction from Gentzen’s Investigations into Logical Deduction [Gentzen, 1934, p. 80]: [I]ntroductions represent, as it were, the ‘definitions’ of the symbols concer ...

Divide and congruence: From decomposition of modal formulas to preservation of branching and eta-bisimilarity

... premises are variables that are all distinct, and that do not occur in the source. An ntytt rule is an ntyxt rule if its source is a variable, an ntyft rule if its source contains exactly one function symbol and no multiple occurrences of variables, and an nxytt rule if the left-hand sides of its pr ...

... premises are variables that are all distinct, and that do not occur in the source. An ntytt rule is an ntyxt rule if its source is a variable, an ntyft rule if its source contains exactly one function symbol and no multiple occurrences of variables, and an nxytt rule if the left-hand sides of its pr ...

proof terms for classical derivations

... of p. The second assumed p twice (tagging these assumptions with x and y), conjoined their result (in that order) and disharged them in turn (also in that order). Seeing this, you may realise that there are two other proofs of the same formula. One where the conjuncts are formed in the other order ( ...

... of p. The second assumed p twice (tagging these assumptions with x and y), conjoined their result (in that order) and disharged them in turn (also in that order). Seeing this, you may realise that there are two other proofs of the same formula. One where the conjuncts are formed in the other order ( ...

Termination of Higher-order Rewrite Systems

... I was given to spend a considerable amount of time at the Mathematical Institute of the Munich University. My host, Helmut Schwichtenberg, enabled a very intensive research. He had great in uence on the direction of this thesis and was a pleasant co-author. Special attention deserves Ulrich Berger f ...

... I was given to spend a considerable amount of time at the Mathematical Institute of the Munich University. My host, Helmut Schwichtenberg, enabled a very intensive research. He had great in uence on the direction of this thesis and was a pleasant co-author. Special attention deserves Ulrich Berger f ...

Deductive Databases with Universally Quantified Conditions

... clear soon. In the above definition, p(t1,...,tn) is referred to as the limiting predicate. Note that we do not allow the terms within a limiting predicate to be a term with another limiting predicate (it would be interesting to consider such cases and is left for future work). Definition 2.3. Given ...

... clear soon. In the above definition, p(t1,...,tn) is referred to as the limiting predicate. Note that we do not allow the terms within a limiting predicate to be a term with another limiting predicate (it would be interesting to consider such cases and is left for future work). Definition 2.3. Given ...

Elements of Finite Model Theory

... historical comments. I taught two courses based on this book, and students in both classes provided very useful feedback; in addition to those I already thanked, I would like to acknowledge Antonina Kolokolova, Shiva Nejati, Ken Pu, Joseph Rideout, Mehrdad Sabetzadeh, Ramona Truta, and Zheng Zhang. ...

... historical comments. I taught two courses based on this book, and students in both classes provided very useful feedback; in addition to those I already thanked, I would like to acknowledge Antonina Kolokolova, Shiva Nejati, Ken Pu, Joseph Rideout, Mehrdad Sabetzadeh, Ramona Truta, and Zheng Zhang. ...

Cut-elimination for provability logics and some results in display logic

... deductive reasoning that is employed in practice. In order to study the properties of this system, Gentzen then constructed yet another proof-system called the sequent calculus. Gentzen’s Hauptsatz or main theorem for the sequent calculus is the cut-elimination theorem which shows how to obtain a st ...

... deductive reasoning that is employed in practice. In order to study the properties of this system, Gentzen then constructed yet another proof-system called the sequent calculus. Gentzen’s Hauptsatz or main theorem for the sequent calculus is the cut-elimination theorem which shows how to obtain a st ...

? A Unified Semantic Framework for Fully

... In this section we precisely define the family of basic systems. For doing so, we define the general structure of derivation rules that are allowed to appear in basic systems. Rules of this structure will be called basic rules. Two key ideas are applied in this definition. First, we explicitly diffe ...

... In this section we precisely define the family of basic systems. For doing so, we define the general structure of derivation rules that are allowed to appear in basic systems. Rules of this structure will be called basic rules. Two key ideas are applied in this definition. First, we explicitly diffe ...

Combinaison des logiques temporelle et déontique pour la

... time on the one hand, and obligations, permissions, and prohibitions on the other hand. Indeed, we have to express for instance the permission to access a resource for a certain period, the obligation to release a resource before a deadline, or the prohibition to execute a task for a too long period ...

... time on the one hand, and obligations, permissions, and prohibitions on the other hand. Indeed, we have to express for instance the permission to access a resource for a certain period, the obligation to release a resource before a deadline, or the prohibition to execute a task for a too long period ...

preliminary version

... • We assume that distinct assumptions have distinct labels. So it is not possible to have assumptions Ax and B x in one and the same proof. • It is possible to have a certain formula A more than once as an assumption in a proof. Those assumptions do not necessarily have the same label. For instance, ...

... • We assume that distinct assumptions have distinct labels. So it is not possible to have assumptions Ax and B x in one and the same proof. • It is possible to have a certain formula A more than once as an assumption in a proof. Those assumptions do not necessarily have the same label. For instance, ...

Harmony, Normality and Stability

... rules and conversely, which is a process in which formal systems don’t come in, at least initially. Looking at the introduction rules alone we should be able to determine which elimination rules are harmonious with them, and conversely. ‘If we use an upwards justification procedure, harmony validate ...

... rules and conversely, which is a process in which formal systems don’t come in, at least initially. Looking at the introduction rules alone we should be able to determine which elimination rules are harmonious with them, and conversely. ‘If we use an upwards justification procedure, harmony validate ...

A KE Tableau for a Logic of Formal Inconsistency - IME-USP

... can we guarantee this is in fact a valuation? First, we know that there is no pair T X and F X in DS. Second, all mCi KE rules with one or more premises (except (F ◦ ¬ n ◦) rules) preserve valuations. Note that (F ◦ ¬ n ◦) rules are taken into account by the last clause in Definition 2. That is, if ...

... can we guarantee this is in fact a valuation? First, we know that there is no pair T X and F X in DS. Second, all mCi KE rules with one or more premises (except (F ◦ ¬ n ◦) rules) preserve valuations. Note that (F ◦ ¬ n ◦) rules are taken into account by the last clause in Definition 2. That is, if ...

CERES for Propositional Proof Schemata

... in the presence of induction is problematic, the proof was formalized as a sequence of proofs (πk )k∈N showing that the assumption that there exist exactly k primes is contradictory. The application was performed in a semi-automated way: CL(πk ) was computed for some small values of k and from this, ...

... in the presence of induction is problematic, the proof was formalized as a sequence of proofs (πk )k∈N showing that the assumption that there exist exactly k primes is contradictory. The application was performed in a semi-automated way: CL(πk ) was computed for some small values of k and from this, ...

logic for the mathematical

... Actually, in that argument, the word “should” is probably better left out. Mostly, we want to deal with statements which simply state some kind of claimed fact, statements which are clearly either true or false, though which of the two might not be easy to determine. Such statements are often called ...

... Actually, in that argument, the word “should” is probably better left out. Mostly, we want to deal with statements which simply state some kind of claimed fact, statements which are clearly either true or false, though which of the two might not be easy to determine. Such statements are often called ...

Sound and Complete Inference Rules in FOL Example

... • Termination with a clause which is a single answer literal Ans(c1 , . . . , cn ). In this case, the constants c1 , . . . , cn gives us an answer to the query. There might be more answers depending on whether there are more resolution refutations of Ans(v1 , . . . , vn ) ∨ ¬φ. We can go on looking ...

... • Termination with a clause which is a single answer literal Ans(c1 , . . . , cn ). In this case, the constants c1 , . . . , cn gives us an answer to the query. There might be more answers depending on whether there are more resolution refutations of Ans(v1 , . . . , vn ) ∨ ¬φ. We can go on looking ...

CUED PhD and MPhil Thesis Classes

... Substructural logics are logics which omit some structural rules, e.g. contraction, weakening, commutativity. Nonassociative Lambek calulus (NL) introduced by Lambek is a propositional logic omitting all structural rules, which can be treated as a basic core of substructural logics. NL can be enrich ...

... Substructural logics are logics which omit some structural rules, e.g. contraction, weakening, commutativity. Nonassociative Lambek calulus (NL) introduced by Lambek is a propositional logic omitting all structural rules, which can be treated as a basic core of substructural logics. NL can be enrich ...

Modal fixpoint logic: some model theoretic questions

... MSO is an extension of first-order logic, which allows quantification over subsets of the domain. It is also one of the most expressive logics that is known to be decidable on trees, whether they are binary or unranked (that is, there is no restriction on the number a successors of a node). Hence, i ...

... MSO is an extension of first-order logic, which allows quantification over subsets of the domain. It is also one of the most expressive logics that is known to be decidable on trees, whether they are binary or unranked (that is, there is no restriction on the number a successors of a node). Hence, i ...

arXiv:1512.05177v1 [cs.LO] 16 Dec 2015

... to 3ExpTime-completeness of the problem for VLDL. Moreover, strengthening VLDL by using traditional pushdown automata results in a logic with an undecidable satisfiability problem, even if we restrict ourselves to deterministic pushdown automata. Related Work There exist other approaches to using vi ...

... to 3ExpTime-completeness of the problem for VLDL. Moreover, strengthening VLDL by using traditional pushdown automata results in a logic with an undecidable satisfiability problem, even if we restrict ourselves to deterministic pushdown automata. Related Work There exist other approaches to using vi ...

LTL and CTL - UT Computer Science

... Temporal logics are a convenient way to formalise and verify properties of reactive systems. LTL is an infinite sequence of states, where each state has a unique succesor. Therefore it establishes a total order on the set of states. Hence one can understand LTL formula’s on a line. LTL is built from ...

... Temporal logics are a convenient way to formalise and verify properties of reactive systems. LTL is an infinite sequence of states, where each state has a unique succesor. Therefore it establishes a total order on the set of states. Hence one can understand LTL formula’s on a line. LTL is built from ...

Programming with Classical Proofs

... need to be in the language. Since negation is usually defined as ¬A := A ! ?, we do not necessarily have negation in mFOL. Firstly, we need to specify what language we work with. Definition 2.2.1 (The language of first-order logic). Given a signature S consisting of functional symbols and relational ...

... need to be in the language. Since negation is usually defined as ¬A := A ! ?, we do not necessarily have negation in mFOL. Firstly, we need to specify what language we work with. Definition 2.2.1 (The language of first-order logic). Given a signature S consisting of functional symbols and relational ...

Predicate logic definitions

... A derivation in PDE is a series of sentences of PLE, each of which is either an assumption or is obtained from previous sentences by one of the rules of PDE. A sentence P of PLE is derivable in PDE from a set Γ of sentences of PLE, written S ` P, iff there exists a derivation in PDE in which all the ...

... A derivation in PDE is a series of sentences of PLE, each of which is either an assumption or is obtained from previous sentences by one of the rules of PDE. A sentence P of PLE is derivable in PDE from a set Γ of sentences of PLE, written S ` P, iff there exists a derivation in PDE in which all the ...