Class Notes 2b: Scheme Reference
... (equal? x y) is true if the values of x and y are the same object, maybe not atomic. (null? x) is true if x is (), the empty list. (append x y) concatenates the lists x and y. ...
... (equal? x y) is true if the values of x and y are the same object, maybe not atomic. (null? x) is true if x is (), the empty list. (append x y) concatenates the lists x and y. ...
1 Non-deterministic Phase Semantics and the Undecidability of
... The term non-deterministic was introduced in [Galmiche and Larchey-Wendling 2006] in order to emphasize the fact that the composition a ◦ b may yield not only one but an arbitrary number of results including the possible incompatibility of a and b in which case a ◦ b = ∅. If (M, +, 0) is a (usual) c ...
... The term non-deterministic was introduced in [Galmiche and Larchey-Wendling 2006] in order to emphasize the fact that the composition a ◦ b may yield not only one but an arbitrary number of results including the possible incompatibility of a and b in which case a ◦ b = ∅. If (M, +, 0) is a (usual) c ...
AGM Postulates in Arbitrary Logics: Initial Results and - FORTH-ICS
... state that it forces a contraction operator to remove too little information from the KB. However, it is generally acceptable that the recovery postulate cannot be dropped unless replaced by some other constraint, such as filtering ([8]), that would somehow express the Principle of Minimal Change. A ...
... state that it forces a contraction operator to remove too little information from the KB. However, it is generally acceptable that the recovery postulate cannot be dropped unless replaced by some other constraint, such as filtering ([8]), that would somehow express the Principle of Minimal Change. A ...
programming in haskell
... x:xs patterns must be parenthesised, because application has priority over (:). For example, the following definition gives an error: head x:_ = x ...
... x:xs patterns must be parenthesised, because application has priority over (:). For example, the following definition gives an error: head x:_ = x ...
FUNCTIONAL PEARL Data types `a la carte
... Which injection should be inferred, Inl or Inr? There is no reason to prefer one over the other—both choices are justified by the above instance definitions. The functions we present here, however, do not inspect where something occurs in a coproduct. Indeed, we can readily check that eval (In (Inl (V ...
... Which injection should be inferred, Inl or Inr? There is no reason to prefer one over the other—both choices are justified by the above instance definitions. The functions we present here, however, do not inspect where something occurs in a coproduct. Indeed, we can readily check that eval (In (Inl (V ...
1 Introduction to Categories and Categorical Logic
... Note that our first class of examples illustrate the idea of categories as mathematical contexts; settings in which various mathematical theories can be developed. Thus for example, Top is the context for general topology, Grp is the context for group theory, etc. On the other hand, the last two exa ...
... Note that our first class of examples illustrate the idea of categories as mathematical contexts; settings in which various mathematical theories can be developed. Thus for example, Top is the context for general topology, Grp is the context for group theory, etc. On the other hand, the last two exa ...
Introduction to Linear Logic - Shane Steinert
... X ∈ C when X is in ob(C) and for every pair of objects X and Y , a set of morphisms, hom(X , Y ) (if f ∈ hom(X , Y ), we write f : X → Y ). These objects and morphisms must satisfy: For each X ∈ ob(C), ∃1X ∈ hom(X , X ). Morphisms can be composed: given f ∈ hom(X , Y ) and g ∈ hom(Y , Z ), then g ◦ ...
... X ∈ C when X is in ob(C) and for every pair of objects X and Y , a set of morphisms, hom(X , Y ) (if f ∈ hom(X , Y ), we write f : X → Y ). These objects and morphisms must satisfy: For each X ∈ ob(C), ∃1X ∈ hom(X , X ). Morphisms can be composed: given f ∈ hom(X , Y ) and g ∈ hom(Y , Z ), then g ◦ ...
Constructing Cut Free Sequent Systems With Context Restrictions
... Since in general rules are not invertible and we need to take care of Contraction we will follow Gentzen’s original strategy [8] when proving cut elimination and eliminate mulΓ ⇒ ∆, An Am , Σ ⇒ Π instead of cuts. Thus we also need to deal with multiple ticuts Γ, Σ ⇒ ∆, Π principal occurrences of the ...
... Since in general rules are not invertible and we need to take care of Contraction we will follow Gentzen’s original strategy [8] when proving cut elimination and eliminate mulΓ ⇒ ∆, An Am , Σ ⇒ Π instead of cuts. Thus we also need to deal with multiple ticuts Γ, Σ ⇒ ∆, Π principal occurrences of the ...
Unit 11 — Functional Programming with Haskell
... In Haskell this concept can be applied to lists, called list comprehension Allows you to generate lists that are too complex for ranges For example, out of the first five odd natural numbers, we want those whose square is not equal to 25 [ x | x <- [1,3..9], (x*x) /= 25 ] <- stands for ∈ (or is inte ...
... In Haskell this concept can be applied to lists, called list comprehension Allows you to generate lists that are too complex for ranges For example, out of the first five odd natural numbers, we want those whose square is not equal to 25 [ x | x <- [1,3..9], (x*x) /= 25 ] <- stands for ∈ (or is inte ...
Default Logic (Reiter) - Department of Computing
... Informally, this is ‘if α is in the database then β is in the database’. So, if the database content is Cn(D) (some base D, some notion of consequence Cn) then we are saying: if α ∈ Cn(D) then β ∈ Cn(D) ...
... Informally, this is ‘if α is in the database then β is in the database’. So, if the database content is Cn(D) (some base D, some notion of consequence Cn) then we are saying: if α ∈ Cn(D) then β ∈ Cn(D) ...
Chapter 2 Propositional Logic
... wff. That’s why we use the metalinguistic variables “φ” and “ψ”.2 The practice of using variables to express generality is familiar; we can say, for example, “for any integer n, if n is even, then n + 2 is even as well”. Just as “n” here is a variable for numbers, metalinguistic variables are variab ...
... wff. That’s why we use the metalinguistic variables “φ” and “ψ”.2 The practice of using variables to express generality is familiar; we can say, for example, “for any integer n, if n is even, then n + 2 is even as well”. Just as “n” here is a variable for numbers, metalinguistic variables are variab ...
Relative Completeness for Logics of Functional Programs
... Analysis) of [28] which is “an extension of Heyting arithmetic with variables and quantifiers for number-theoretic functions”. Our theory T differs from EL in two respects: it is based on classical logic and formulated in a sublanguage of L which refers only to the strict total elements of nat and n ...
... Analysis) of [28] which is “an extension of Heyting arithmetic with variables and quantifiers for number-theoretic functions”. Our theory T differs from EL in two respects: it is based on classical logic and formulated in a sublanguage of L which refers only to the strict total elements of nat and n ...
Constraint Logic Programming with Hereditary Harrop Formula
... different computation domains, whose logical behaviour is given by constraint systems. CLP languages keep all the good semantic properties of pure logic programming, including soundness and completeness results (Jaffar et al., 1996). Their implementation relies on the combination of SLD resolution w ...
... different computation domains, whose logical behaviour is given by constraint systems. CLP languages keep all the good semantic properties of pure logic programming, including soundness and completeness results (Jaffar et al., 1996). Their implementation relies on the combination of SLD resolution w ...
CUED PhD and MPhil Thesis Classes
... view, NL provides some advantages over L. On the other hand, NL can be treated as a basic substructural logic. Roughly speaking, NL is a logic which omits all structural rules. It is a pure logic of residuation. Therefore it makes sense to investigate NL and its extensions. It is not surprising that ...
... view, NL provides some advantages over L. On the other hand, NL can be treated as a basic substructural logic. Roughly speaking, NL is a logic which omits all structural rules. It is a pure logic of residuation. Therefore it makes sense to investigate NL and its extensions. It is not surprising that ...
Programming in Logic Without Logic Programming
... isa(book, item), do not include time parameters. Temporal constraint predicates, including inequalities of the form T1 < T2 and T1 T2 between timepoints, and functional relationships among timepoints, such as max(T1, T2, T) and min(T1, T2, T) have only time parameters. In KELPS, temporal constrain ...
... isa(book, item), do not include time parameters. Temporal constraint predicates, including inequalities of the form T1 < T2 and T1 T2 between timepoints, and functional relationships among timepoints, such as max(T1, T2, T) and min(T1, T2, T) have only time parameters. In KELPS, temporal constrain ...
The substitutional theory of logical consequence
... interested in logical consequence in natural language may take first-order languages as a test case. If the substitutional theory is successful for them, there is at least hope it can be extended to natural language. terminological remark. I apply the terms ‘logical validity’ and ‘formal validity’ t ...
... interested in logical consequence in natural language may take first-order languages as a test case. If the substitutional theory is successful for them, there is at least hope it can be extended to natural language. terminological remark. I apply the terms ‘logical validity’ and ‘formal validity’ t ...
Strong Normalisation for a Gentzen-like Cut
... 3. Cut(haiAx(y, a), (x)M ) −−l→ M [x 7→ y] if M freshly introduces x As can be seen, these cut-reductions are restricted so that they are applicable only if the immediate subterms of the cuts freshly introduce the names and conames corresponding to the cut-formulae. Without this restriction bound na ...
... 3. Cut(haiAx(y, a), (x)M ) −−l→ M [x 7→ y] if M freshly introduces x As can be seen, these cut-reductions are restricted so that they are applicable only if the immediate subterms of the cuts freshly introduce the names and conames corresponding to the cut-formulae. Without this restriction bound na ...
Declarative Programming in Escher
... being equations involving -expressions. Having done this, programs in all three languages are equational theories in a higher-order logic and the various languages can be compared by applying the appropriate computational mechanisms to these equations. Furthermore, depending on the language, other ...
... being equations involving -expressions. Having done this, programs in all three languages are equational theories in a higher-order logic and the various languages can be compared by applying the appropriate computational mechanisms to these equations. Furthermore, depending on the language, other ...
Adequate set of connectives
... How do we show that a given set S of connectives is not adequate? Show that some standard connective cannot be expressed by S. Example. The set S = {∧} is not adequate. Proof. To see this, note that a formula depending on only one variable and which uses only the connective ∧ has the property that i ...
... How do we show that a given set S of connectives is not adequate? Show that some standard connective cannot be expressed by S. Example. The set S = {∧} is not adequate. Proof. To see this, note that a formula depending on only one variable and which uses only the connective ∧ has the property that i ...
Let me begin by reminding you of a number of passages ranging
... In the corresponding passage from the main text, he fills this out as follows: When entering upon the study of a science, we need to have some idea, if only a provisional one, of its nature. We want to have in sight a goal to strive towards; we want some point to aim at that will guide our steps in ...
... In the corresponding passage from the main text, he fills this out as follows: When entering upon the study of a science, we need to have some idea, if only a provisional one, of its nature. We want to have in sight a goal to strive towards; we want some point to aim at that will guide our steps in ...
The Herbrand Manifesto
... Another difference concerns compactness. A logic is compact if and only if every unsatisfiable set of sentences has a finite subset that is unsatisfiable. Relational Logic with Tarskian semantics turns out to be compact. The upshot is that it is possible to demonstrate unsatisfiability in finite spa ...
... Another difference concerns compactness. A logic is compact if and only if every unsatisfiable set of sentences has a finite subset that is unsatisfiable. Relational Logic with Tarskian semantics turns out to be compact. The upshot is that it is possible to demonstrate unsatisfiability in finite spa ...